Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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Could the acceleration of universe expansion be caused by gravity itself? Dark energy is suggested to be a repulsive force in the universe causing an accelerated expansion. If the amount of mass outside our observable universe is greater than inside (higher mass density), would it not cause an accelerated expansion from our viewpoint?
| I have been working on a similar theory for years, since the type 1a supernova project showed the unexpected acceleration. I tried to publish several times, unsuccessfully. My idea is; yes, it is gravity which causes the expansion because a static universe would be isotropic and homogeneous but a dynamic universe is iotropic but not homogeneous. It is the inhomogeneous distribution of matter (along the observer's line of sight) which drives the expansion.
If so, there would be no need for dark energy. Einstein's lambda would really be zero (he may have been right first time!).
As stated above, this would require another explantion for CMB as a gravitational expansion has no big bang. If the universe is accelerating, all the free charged particles will radiate. Hence: isotropic background radiation. I need to show that it has a 2.7° black body spectrum. That is as far as I have got for the moment on an alternative CMB mechanism.
| {
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How does one calculate the force applied on an object by a magnetic field? I've tried very hard to find an answer to this question, and every path leads me to an abstract discussion of fundamental forces. Therefore, I will propose two very specific scenarios and see if they yield the result that I am looking for.
Scenario One
Let's say that I have a vertical tube exactly 1 inch in diameter that is completely incapable of holding an electromagnetic charge and has a frictionless surface. Resting inside this tube is a steel ball also exactly 1 inch in diameter. If a cylindrical magnet, also exactly 1 inch in diameter is slowly lowered into the tube, how does one determine the exact point at which the force being applied to the steel ball by the magnet will cause the ball to overcome gravity and rise toward the magnet? Is there even any way to determine this? What further information would I need?
Scenario Two
I have the same tube from above with the cylindrical magnet resting on the bottom of the tube, north pole facing upward. Suspended by a weightless string in the tube is an identical magnet, north pole facing downward. If the bottom magnet is slowly raised, how does one calculate the exact point at which the suspended magnet will begin to move upward? Is this calculation possible? What further information would I need for this calculation?
Extra Question
How are weight capacities on magnets calculated? I.e. if a whitepaper says that a magnet is capable of lifting 25 pounds, how is the correct size magnet calculated?
| If you have a steel ball bearing with high permeability, then the magnetic potential energy of this configuration will be proportional to the square of the B-field at the position of the ball bearing. For a simple dipole magnetic field on-axis, this will be proportional to $r^{-6}$, where $r$ is the distance from the centre of the magnet.
The force associated with this will be the gradient of the potential and will go as $r^{-7}$.
The exact force I think is difficult to calculate because of geometric factors and the finite sizes of the components - e.g. integrals of the dipole field over the volume of the ball bearing.
| {
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Mathematical Universe Hypothesis What is the current "consensus" on Max Tegmark's Mathematical Universe Hypothesis (MUH) which claims every concievable mathematical structure exists, including infinite different Universes etc.
I realize it's more metaphysics than physics and that it is not falsifiable, yet a lot of people seem to be taking a liking to it, so is it something that is plausible ?
I've yet to hear any very good objections to it other than "it's crazy", but are there any real technical problems with it?
| Tegmark, Hut and Alford wrote an article at http://arxiv.org/abs/physics/0510188 . Penrose had the idea of a causal loop between matter, mind and math. The authors don't like causal loops and tried to cut them at various points. Tegmark is the Platonist who believes math is fundamental and the ground of being.
Is that really so? Math rests upon a substrate of matter. Just as Landauer showed information has to have a physical substrate, so does math. Immaterial disembodied mathematical structures don't exist. Platonic ideal forms not made up of matter?
| {
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Why is the energy density of the electric field produced by two unlike charges positive, even though their potential energy is negative? Consider two stationary charges, one positive the other negative. Their potential energy is clearly negative. So you would expect that the energy density of the associated electric field would also be negative. But it isn’t. It’s the square of the electric field and therefore positive. Why isn’t the energy density negative like the potential energy?
| Here, you should refer to the definition of energy density, the electric field energy density is defined as following:
$u = \frac{1}{2}\epsilon |\vec{E}|^2$
where $\epsilon$ is the permittivity of media.
therefore this energy density should always be positive.
The derivation of the electric energy density can be referred to the wiki post
btw, I think you made some mistake statement in your question on your example
"Their potential energy is clearly negative. "
We can look at you example of two charge particles, we can easily write out the potential everywhere in your case as
$\Phi(\vec{x}) = \frac{Q_1}{4\pi\epsilon |\vec{x}-\vec{r_1}|}+\frac{Q_2}{4\pi\epsilon |\vec{x}-\vec{r_2}|}$
where $\vec{r_1}$ and $\vec{r_2}$ are the position of these charges in space, if we assume $Q_1$ is positive, $Q_2$ is negative, then we can calculate the electric potential everywhere. Clearly, you'll notice that there is zero-potential surface perpendicular to the line connects these two charges, on the side of $Q_1$, potential is always positive, and on the side of $Q_2$, potential is always negative, then the electric potential energy at these two charges are both positive ($U_1 = \Phi_1Q_1 >0$, since $\Phi_1>0$, similarly, $U_2 = \Phi_2Q_2 >0$, cause $\Phi_2 <0$ and $Q_2<0$). So the electric potential energy is positive. Also you should note that this energy is only a part of total electric energy.
| {
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What is a black hole? Is there a definition of a black hole in a generic spacetime? In some books, for example Wald's, black holes are defined for asymptotically flat spacetime with strong asymptotic predictability, although the definition makes sense without the second condition. Is there a notion of a black hole in general spacetime, not necessarily asymptotically flat? Or is it the case that there is not a "natural" or agreed upon definition?
| What the definition needs to capture is that a black hole is not (1) a naked singularity, or (2) a big bang (or big crunch) singularity. We also want the definition to be convenient to work with so that, for example, it's possible to prove no-hair theorems.
Since we want to exclude naked singularities, it's natural that we require an event horizon. Event horizons are by their nature observer-dependent things. For example, if we have a naked singularity, we can always hide its nakedness by picking an observer who is far away from it and accelerating continuously away from it. Such an accelerated observer always has an event horizon, even in Minkowski space. This example shows that it makes a difference what observer we pick.
Actually, we can't have a material observer at null infinity, since timelike infinity, not null infinity, is the elephants' graveyard for material observers. However, the choice of null infinity is the appropriate one because a black hole is supposed to be something that light can't escape from.
Of course the actual universe isn't asymptotically flat, but that doesn't matter. In practice, all we care about is that the black hole is surrounded by enough empty space so that the notion of light escaping from it is well defined for all practical purposes.
There are other possible ways of defining a black hole, e.g., http://arxiv.org/abs/gr-qc/0508107 .
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Could a ship equipped with Alcubierre drive theoretically escape from a black hole? Could a ship equipped with Alcubierre drive theoretically escape from a black hole?
Also, could it reach parts of the universe that are receding faster than the speed of light from us?
| The drive works by warping normal space, creating a bubble that kind of surfs through space time. I don't know what would determine the speed such a ship could achieve so not sure if a natural law would limit the ability to travel beyond visible space. Black holes exist because their extreme mass has warped space beyond to point where light can escape. It seems like it might take an infinite amount of energy in this case to keep the warp bubble from collapsing.
| {
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Problem with an electricity / thermodynamics assignment I've been trying to figure this one out for a while on my own, so I'd like to ask for your help if you could offer some.
The task states:
A heater made out of a wire with a diameter $R = 0.2\text{ mm}$, length $4\pi\text{ m}$ and electrical resistivity of $0.5\times 10^{-6}\ \Omega\;\mathrm{m}$ is connected to a voltage source of $220\text{ V}$, sinked in the water.
Which mass of water will it heat up from $20^{\circ}\mathrm{C}$ to $50^{\circ}\mathrm{C}$ in the time of 10 minutes? (C of water = $4200\ \mathrm{J\;kg}/\mathrm{K}$)
I know I have the electrical properties of the wire and the thermodynamic properties of the water, but I don't know how to proceed from there. We've been studying electricity and I am not really aware how I can connect it with thermodynamics?
| Find the resistance of the particular wire. Then calculate the power it uses. Assume this power is dissipated as heat. Find how much energy is converted to thermal energy by heat in 10 minutes. Use the equation Q = mc ΔT to find the mass of the water.
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Momentum and indirect bandgap When an electron is excited from valence band to conduction band, it has to have a finite momentum in the case of indirect bandgap. Does that mean that the electron cannot be created at rest ? does it have a non zero "speed" ?
| It means that to fulfill momentum conservation law, the momentum should be taken from somewhere: from phonons, impurities, etc.
The speed is a derivative of the energy (remember Hamilton's equations), thus the speed is zero at the minimum of the energy, wherever this point is located in k space
| {
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Is the cooling rate of a (very) cold object, sitting next to an AC higher or lower? In more detail:
If i have two soda cans, both are cooled to exactly 4 degrees celsius,
And i put one in a 25 degrees room, and the other next to an AC vent set to 16 degrees.
After three minutes, which one should be colder than the other and why?
Edit: To clarify - if I have a cold soda can, should I place it near the AC vent or not (if I like my drink cold)? Which location will cause faster heating?
| The answer for the question as posed is easy:
It could be either
This applies for the initial rate of heating. Of course, over time, the story is quite different. The temperature of the one in front of the AC could initially become higher than the one in the static room air. However, given sufficient time, of course the can in front of the AC will be the cooler can and will remain that way. Simple Newton's law of cooling:
$$\frac{dT_{can}}{dt} = C_{air} \left( T_{air} - T_{can} \right)$$
$$\frac{dT_{can}}{dt} = C_{AC} \left( T_{AC} - T_{can} \right)$$
The solution to both of these is simple. Written for both cases, they are:
$$T_{can}(t) = T_{air} - ( T_{air}-T_{can}(0)) exp(-C_{air} t)$$
$$T_{can}(t) = T_{AC} - ( T_{AC}-T_{can}(0)) exp(-C_{AC} t)$$
I can not make any statements about the exact values, and I don't think that's likely to be valuable for this exercise, but allow me to make simple relative statements.
$$ T_{air} > T_{AC}$$
$$C_{air} < C_{AC} $$
It's taken as a given that the initial can temperature is the same for both cases. Should the above inequalities have the same direction, then the problem would have an absolute answer. One can would be hotter at all times, $t>0$. In the problem presented, the signs are different leading to two possibilities, which is that the AC case initially leads to a higher temperature, or that the AC case is always of lower temperature. This can be formalized by the following inequality.
$$C_{AC} \left( T_{AC} - T_{can}(0) \right) \stackrel{?}{\le} C_{air} \left( T_{air} - T_{can}(0) \right)$$
If the above inequality is true, then the can in front of the AC is always a lower temperature. If it is false, then the can in front of the AC initially becomes hotter, then the other can becomes hotter.
| {
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Which subjects in physics should I choose if I want to help tackling today's energy and environment related problems? I was wondering what subjects a freshman in mathematics ought to choose in the future if s/he wanted to help working on energy and environment-related issues we are currently facing, and will very likely be even more profound in the future.
I am currently day-dreaming about using the skills I will acquire as a mathematician to create more efficient solar cells or wind-mills, but I guess a little understanding of the physics involved wouldn't hurt.
Which subjects should I choose to enhance my knowledge on this subject? Could you perhaps suggest a rough pathway by means of which I could deepen my understanding of the subject matter?
| Can I recommend hot fusion, instead of cold fusion?
It works in the sun and other stars and may yet work here on earth.
If it works well, it will have many advantages.
A standard general physics sequence will be a good start for this.
Pay particular attention to electromagnetism.
Eventually you would specialize in nuclear physics.
| {
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Does conservation of momentum really imply Newton's third law? I often heard that conservation of momentum is nothing else than Newton's third law.
Ok, If you have only two interacting particles in the universe, this seems to be quite obvious.
However if you have an isolated system of $n$ ($n > 2$) interacting particles (no external forces). Then clearly Newton's third law implies conservation of total momentum of the system. However presuppose conservation of total momentum you only get:
$$
\sum_{i\neq j}^n \mathbf F_{ij} = \frac{d}{d t} \mathbf P = 0
$$
Where $\mathbf F_{ij}$ is the forced acted by the $i$th particle upon the $j$th particle and $\mathbf P$ is the total linear momentum.
But this doesn't imply that $\mathbf F_{ij} = -\mathbf F_{ji}$ for $j \neq i$.
So does conservation of momentum implies Newton's third law in general or doesn't it? Why?
| No, but Newton's third law of motion implies the conservation of momentum. In other words, Newton's third law is a special case of the more general law, which is the conservation of momentum.
| {
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Is Thirring model a particular case of Gross model? In the Wikipedia entry for the Gross–Neveu model, it is said that
If one takes $N=1$ (which permits only one quartic interaction) and makes no attempt to analytically continue the dimension, the model reduces to the massive Thirring model (which is completely integrable).
But the aditional term in the Thirring model is
$$\frac{g}{2}\left(\overline{\psi}\gamma^\mu\psi\right) \left(\overline{\psi}\gamma_\mu \psi\right).$$
I think this is different from $\frac{1}{2}g \left(\overline{\psi} \psi\right)^2$, the additional term of the Gross–Neveu model. So I think Wikipedia is wrong. Am I right?
If you do not mind, I would like that you respond to this question also: Could this model have soliton solutions?
Thanks in advance.
| Yes, because the grassman expansion of a quartic fermi interaction can only be $\psi_1\psi_2\bar{\psi}_1\bar{\psi}_2$ in 2d, because there are only four grassman fields, so all other quartics are zero.
| {
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Are specific heat and thermal conductivity related? Are there any logical relationship between specific heat capacity and thermal conductivity ?
I was wondering about this when I was reading an article on whether to choose cast iron or aluminium vessels for kitchen.
Aluminium has more thermal conductivity and specific heat than iron ( source ).
This must mean more energy is required to raise an unit of aluminium than iron yet aluminium conducts heat better than cast iron.
Does it mean that aluminium also retains heat better ?
How does mass of the vessel affect the heat retention?
| TLDR - heat capacity comes from how phonons behave in the harmonic approximation, while thermal conductivity comes from how anharmonic phonons are.
In more detail, thermal conductivity comes primarily from two channels:
(1) Conduction electrons carrying heat
(2) Anharmonicity of the crystal lattice (i.e. phonons scattering from one another because the crystal lattice isn't a perfect quadratic potential).
On the other hand, near ambient temperature, the main contributer to the heat capacity is the harmonic behavior of phonons (i.e. their frequency, which is their energy cost to excite).
Generally, there is no rule relating the harmonic to anharmonic terms of the lattice potential/phonons, so the heat capacity and thermal conductivity typically have very different behavior and knowing one does not usually tell you much about the other.
| {
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Chance of "macro tunneling"? We know that subatomic particles can and do tunnel through barriers, so it is theoretically "possible" somewhat that a grain of sand could tunnel through a paper, but Id like to get some perspective on it.
Can anybody give any sort of estimate of how long one would have to wait to expect to see a grain of sand tunnel through a sheet of paper? (for instance2 10 times the life of the universe")
| That is hard even to estimate correctly. For a grain of sand to tunnel through a sheet of paper the probability is so small not because of the tunnel barrier but because it would require the whole grain to move in one direction spontaneously.
Starting with the Transmission probability (from Wikipedia)
$$
T = \frac{e^{-2\int_{x_1}^{x_2} dx \sqrt{\frac{2m}{\hbar^2} \left( V(x) - E \right)}}}{ \left( 1 + \frac{1}{4} e^{-2\int_{x_1}^{x_2} dx \sqrt{\frac{2m}{\hbar^2} \left( V(x) - E \right)}} \right)^2}$$
With the barrier height V and the energy of the incoming particle E you can plug in different numbers for the width of the barrier (x1-x2) and the height V which for realistic estimates will be tiny.
Then you have to estimate the likelihood of all $$\approx 10^{23}$$ atoms doing that at the same time, so in the order of $$T^{10^{23}}$$ with $$T \ll1 .$$
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How do we perceive hotness or coldness of an object? Some objects, especially metallic ones, feel cold on touching and others like wood, etc. feel warm on touching. Both are exposed to same environment and are in their stable state, so some kind of equilibrium must be being reached. What is this equilibrium?
And how do we perceive hotness or coldness of an object? Does skin have some kind of heat sensors, etc., which transmit signals to brain? Like, how do the eye transmit/convey an image formed on the retina to brain?
| I don't know the actual biological process, but I know that our body detects heat transfer, not temperature. That's why an object with a higher heat transfer coefficient (such as metallic objects) appear colder than wood at the same temperature, for example.
| {
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Why are infinitesimal rotations commutative, whereas finite rotations are not? Infinitesimal rotations commute and every finite rotation is the composition of infinitesimal rotations which should logically mean they also commute; but they don't. Why?
| Infinitesimal rotations don't commute exactly if you're accurate enough. An infinitesimal rotation may be written as
$$ \exp( i a A ) $$
where $a$ is an infinitesimal "angle" and $A$ is a combination of generators. Such an object doesn't commute with the analogous object $\exp(ibB)$ in general. Instead,
$$ \exp(iaA) \exp(ibB) = \exp(ibB) \exp(iaA) \exp(-ab [A,B] + O(a_i^3)) $$
where $[A,B]=AB-BA$ is the ordinary "commutator" of operators i.e. the generators (of the bases "vectors" $A,B$ of the Lie algebra associated with the Lie group). The equation above may be verified by carefully expanding the exponentials on both sides to the second order in $a$ or $b$, ignoring cubic and higher-order terms, but being careful about the ordering of $A$ and $B$ etc.
The failure of the infinitesimal rotations to commute is only expressed by a smaller angle $ab$ which is second order but the accumulation of these $O(a_i^2)$ terms is what makes finite rotations "obviously noncommuting". Why? Because if you want to interchange $N$ copies of $\exp(iaA)$ in $\exp(iNaA)$ with $M$ copies of $\exp(ibB)$ in $\exp(iNbB)$, you need to make $MN$ similar permutations, so assuming that $Ma$ and $Nb$ are finite, the factors of $MN$ (large) and $ab$ (small) cancel and you get a finite difference between the products written in the opposite orders.
| {
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Numeric method to calculate the charge distribution on a conducting surface? If I have an arbitrary (closed?) conducting surface and a nearby charge density, is there a simple numeric way of computing the induced charge distribution on the surface?
| There is no simple way. The "standard" way is to solve Poisson equation with proper boundary conditions (constant $\varphi$ at the surface). Out of potential distribution it is easy to extract charge distribution.
For simple shapes (infinite plane, sphere, etc) it is possible to solve the problem analytically. For arbitrary shape there is no simple solution.
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Is there any anti-gravity material? I want to know if there is any anti-gravity material. I am thinking of making flying vehicles which are made up of anti-gravity material so that they will not experience any gravity on them and can easily take off and be more fuel efficient. Is there any such thing? Or any workaround?
| When you say "anti-gravity material", the closest thing I can think of is the hypothetical concept of negative mass:
In theoretical physics, negative mass is a hypothetical concept of
matter whose mass is of opposite sign to the mass of the normal
matter. Such matter would violate one or more energy conditions and
show some strange properties such as being repelled rather than
attracted by gravity. It is used in certain speculative theories, such
as on the construction of wormholes. The closest known real
representative of such exotic matter is a region of pseudo-negative
pressure density produced by the Casimir effect.
But it gets more complicated because there are actually three different kinds of mass: gravitational mass, passive grativational mass, and inertial mass:
Thus objects with negative passive gravitational mass, but with
positive inertial mass, would be expected to be repelled by positive
active masses, and attracted to negative active masses. However, any
difference between inertial and gravitational mass would violate the
equivalence principle of general relativity. For an object where both
the inertial and gravitational masses were negative and equal, we
could cancel out mi and mp from the equation, and conclude that its
acceleration a in the gravitational field from a body with positive
active gravitational mass (say, the planet Earth) would be no
different from the acceleration of an object with positive passive
gravitational and inertial mass (so a small negative mass object would
fall towards the Earth at the same rate as any other object).
In any case, there does not exist any such thing, to the extent of human knowledge.
| {
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Should annealed disorder be characterized by the average of the partition function? Most of the literature says that for a quenched average over disorder, an average over the log of the partition function must be taken:
\begin{equation}
\langle \log Z \rangle,
\end{equation}
while for the annealed average, it's
\begin{equation}
\langle Z \rangle.
\end{equation}
But a while ago, I came across a book that said that the annealed average is not $\langle Z \rangle$, though I don't remember what it said should be calculated instead.
Does anyone know which book this is, or what they might want to calculate instead of $\langle Z \rangle$ for the annealed average?
| If your disorder is not annealed but quenched, the distribution of impurities does not obey the thermal (Gibbs) distribution. Non-annealed, "quenched" disorder is an externally given background, and you need to average extensive quantities, such a free energy ($\propto \log Z$). It is hard for me to guess the exact context of your annealing, but I assume it to be thermalization of some kind. If it brings the disorder to thermal equilibrium with the rest of the system, then disorder averaging merges with thermodynamic averaging and you can use $\langle Z \rangle$.
| {
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About the Ether Theory acceptance Why was the Ether Theory refused by Modern Physics? If you please explain me, I just wanted to understand it more.
| There never was really any experimental or observation evidence that the luminous aether ever existed in the first place. It was merely an invention to pave over a gap in the Newtonian model of light.
Up until the mid-19th century everyone thought that Newton's experiments with prisms and slits had conclusively demonstrated that light was a wave phenomena i.e. it behaved just like waves in water. That, however, raised the question of "waves in what?" The answer was the "aether", something that couldn't be seen, touched or measured but whose motion transmitted light just like the motion of water transmitted the force of a wave.
Basically, the luminous aether was just a concept that the era's scientist pulled out of their collective… er, hats.
By contrast, Leibniz argued, purely on philosophical grounds, that light must be made of particles. Leibniz lost the argument because his hypothesis didn't have the experimental predictive power that Newton's did. However, if Leibniz's idea had been the dominate idea, the concept of luminous aether would have never been invented in the first place.
Prior to the late-19th century the idea of "subtle fluids" was evoked to explain many phenomena such as heat and electricity. That is why even today we talk of electricity "flowing" in "currents."
The history of science is filled with such inventions that paper over gaps in existing knowledge.
| {
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"url": "https://physics.stackexchange.com/questions/16596",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Why is beta negative decay more common than beta positive? In simple terms, why is beta negative decay more common than beta positive?
I know it's something to do with occuring inside/outside the nucleus - but I can't find a simple, easy to understand explanation!
| The beta-decay may be "locally" reduced to a decay of a proton or a neutron inside a nucleus. The beta-minus decay contains the microscopic process
$$ n\to p + e^- + \bar \nu_e + O(1{\rm \,MeV})$$
where the last term indicates the rough increase of the kinetic energy of the decay products. On the other hand, the beta-plus decay contains the process
$$ p + O(1{\rm \,MeV})\to n + e^+ + \nu_e $$
which means that the proton has to acquire some extra energy if it wants to decay to a neutron and a positron. In realistic beta-plus decays, it takes it from the surrounding nucleons in the nucleus.
Obviously, decays to lighter products where the energy conservation allows to give the final products some extra kinetic energy are more frequent than decays which only occur if an extra energy is found at the beginning. For example, among the bare processes above (for free nucleons), only the decay of the neutron may occur. The proton is stable (ignoring very infrequent processes linked to grand unification).
At the end, the inequality between the two types of the decay boils down primarily to the fact that the neutron is heavier than the proton which subsequently boils down to the fact that the down-quark is heavier than the up-quark (the rest masses).
| {
"language": "en",
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Can there be black light? I mean is it possible to devise a machine that outputs darkness? I understand there are various colours that light can have. But i was wondering why there is no 'black' light. What is the logical explanation for this? I mean I am expecting an answer that goes beyond mentioning the spectrum details. All I could think of was a machine as powerful as a blackhole; it could bend the light so hard that all we would see is darkness. But is there any other way?
P.S. I am a programmer and didn't study much Physics beyond high school. This question is not a goof. I am not asking this question for fun. I seriously have this curiosity.
| There are a couple of things here to clarify.
What is darkness, just (a shade of) black color (as per @anna's answer), yes a machine can do that.
Is it a about a machine that outputs passive darkness (as per @DavidZ's answer), yes it can be done (just dont output anything, it outputs darkness).
Is it about active darkness in the sense that it can annihilate light? Then this would relate directly to a perpetuum mobile of the 2nd kind (the entropy related one) and it would be impossible (as per @Ron's answer)
Note that black holes are supposed to actually radiate (precisely due to entropic/2nd Law considerations).
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Reality constraint What is the "definition" of a reality constraint and why is it called that way?
(I mean how it is used for example in quantum field theory and string theory.)
| A reality constraint typically cuts a quantity with complex degrees of freedom down to the same number of real degrees of freedom.
An example of a reality constraint is to impose that a complex $n\times n$ matrix $M\in{\rm Mat_{n\times n}(\mathbb{C})}$ should be Hermitian $M^{\dagger}=M$.
A bit more abstractly, one could also call the conditions $M^{\dagger}=-M$ (anti-Hermiticity) and $M^{\dagger}M={\bf 1}_{n\times n}$ (unitarity) for reality conditions, because they also cut the degrees of freedom in half.
In quantum theory, one for instance imposes that observables are Hermitian operators and evolution is unitary.
| {
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How long does it take for expanding space to double in size? I have been reading about Hubble's constant and trying to make 'sense' of the theory of the expanding Universe. Is is possible that space in the universe expands uniformly? If so, absent of other forces (ie gravity), how long does it take for the distance between any to dimensionless points in the universe to double in length?
I've tried to work the math as follows:
$\frac{74.2 \text{ km}}{\text {s Mpc}}\times\frac{\text{Mpc}}{3,261,564\text{ ly}}\times\frac{\text{ly}}{9,460,730,472,581 \text{ km}}\times\frac{31,557,600\text{ s}}{\text{yr}} = \frac{1}{13,177,793,645 \text{ yr}}$
Using the continuously compounding interest formula
$FV= Pe^{rt}$
$2 = 1\times e^{(1/13,177,793,646)t}$
$\ln(2) = \frac{1}{13,177,793,646}\times t$
$t = 9,134,150,511 \text{ yr}$
So it would take 9 billion years for the distance between any two points in space to double in length?
If this is so, when two points in 3D space double in distance apart, the space itself increased by $2^3 = 8$ so the time it would take for space itself to double in size would be
$t = 1,141,768,813 \text{ yr}$
Since the Universe is only about 15 billion years old and started from a singularity of volume $0$,
I would have to assume that the rate of expansion of space isn't constant over time?
Does the time for the distance between two points to double in length vary based on the original distance between those to points?
| You are correct that the expansion of the universe hasn't been constant over time. A period of the very early universe is considered to have undergone exponential inflation doubling in size continuously to result in a $10^{78}$ increase in size between $10^{-36}$ to $10^{-32}$ seconds after the big bang.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Desperately Need Help with Grade 9 Static Electricity I am preparing to teach Grade 9 Static Electricity next week and am going crazy trying to figure out what is happening in one of my experiments. I have a short piece of PVC pipe, 4 inches diameter, and I rub it with wool to charge it negatively. I can observe excellent repulsion when I touch it with my foil bit (dangling from a thread). Here is the problem: I am holding down the PVC pipe down on a wooden base with two brass-plated wood screws, and these screws somehow collect an INSANE amount of positive charge, even when I am very careful not to touch them with the wool. The foil bit is strongly attracted to the screws, and when it touches them it bounces off more violently than anything I've seen in any of my other static electricity experiments.
Can anybody explain what I am seeing?
| Great question. If you can rule out magnetism (based on the material of the foil and the material of the screw under the brass plating, or by demonstrating that there is no force on the foil when the PVC pipe is not charged), you are seeing the effect of induced electric charge on the screws. They are electrically neutral overall, and when they are surrounded by the negative charge of the PVC pipe, electrons in the screw are repelled slightly inward from the surface, leaving positive charge at the surface. This is called polarization. The resulting electric forces are strongly concentrated near the screw, and since the foil is negatively charged, it is attracted to the screw.
Why does the foil bounce when it touches the screw? The electrons of the foil are attracted to the screw, and some of those electrons flow onto the screw at the contact point. You then have a slight positive foil, which is repelled by the screw.
You can test this explanation: The foil should be positively charged after it bounces from the screw, and then it should be attracted to the negatively charged pipe.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/16922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Time, what is it? If you ask any person about time, she/he will give you some answer.
I suspect that it is extremely difficult, (if not impossible) to define time.
Is there a definition of what it is in physics? Is it an "axiom" that has to be taken as it is, without explanations? I also noticed that the tag "time" has no pop-ups with comments/definition/explanations.
| Neuroscience tells us our perceptions are stored in short term memory and after much processing, are backdated in time to give the perception of "now" at a later time. This can take many many milliseconds later, even close to a second. Or imagine watching a family movie filmed years or decades ago and becoming so engrossed in it so that it seems to be happening all over again in the present. When is now? The neurosurgeon Penfield stimulated parts of the brains of some patients and they relived events from years or decades before as if they were happening in the present. Now is not when you think it is.
In near death experiences, a person reviews memories of their life flashing by. How do you know your now isn't actually happening much later and backdated? Let's say decades in the future, you upload your mind to a computer and a replay of your life memories is made on some playback machine to be observed by some computer mind. It would seem backdated to 2011, would it not, even if it's much later.
This process and go on over and over until the end of time. Maybe "now" is a playback at the end of time at the end of the universe and we have been fooled about the actual date. Whatever it is observing the playback at the end of the world sustains us over and over again by the act of observation. Being, with no becoming.
| {
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"url": "https://physics.stackexchange.com/questions/17056",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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AdS space - Poincare Patch How can I work out in detail the explicit coordinate transformation formulas needed to go from the "canonical" coordinates to the "Poincare patch"? I'm reading about AdS but the text takes the validity of the Poincare patch for granted, which troubles me. It only has a drawing showing the Poincare patch covering half the whole space, which doesn't help.
| I can recommend the article Introduction to the Maldacena Conjecture on AdS/CFT, arXiv:hep-th/9902131, by Petersen. I suspect that the first 8 pages would be helpful.
| {
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Inertial Mass of a scalar field Does it make sense to talk of the inertial mass of a scalar field? By the equivalence principle, it must be equal to its gravitational mass. We know that the scalar field contributes towards the stress-energy tensor, so, shouldn't it have an inertial mass too?
| Yes, indeed
This is exactly why it is assumed in the Standard Model that mass is the result of an interaction with a scalar "world potential" (The Higgs field) while for instance the changes in energy/momentum of a charge are due to the electromagnetic vector potential $A^\mu$
Regards, Hans
| {
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How do we know Dark Matter isn't simply Neutrinos? What evidence is there that dark matter isn't one of the known types of neutrinos?
If it were, how would this be measurable?
| Cold neutrinos which clumped together would form a Fermi-Dirac condensate. Unlike electrons in an atom there would be no mutual repulsion and the quantum numbers could increase truly "astronomically". For a large concentrate all but the early neutrino contributors would be far from cold. Such a concentrate would behave like a huge heavy ball of unobservable, very rareified liquid which is exacty what you see in a barred spiral galaxy, the bar is in the liquid where g varies as r and the spiral arms are outside, subject to the inverse square law. Cold neutrinos may have been around since the early universe but another source could be black holes where they may pour out like Hawkinge radiation or as a result of accretion disc annihilation at the event horizon. Either way they would be very cold by the time they had crawled away from the hole.
| {
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Proof of the Hydrostatic weighing equation? I've been trying to derive this equation for about two hours:
$$\frac{\text{density of body}}{\text{density of water}} = \frac{\text{weight of body}}{\text{weight of body} - \text{weight of immersed body}}.$$
But I can't seem to get it. In my class, we are using it to accurately measure the volume of irregulary-shaped objects, but I completely missed the derivation.
I'm not looking to leech off of somebody's proof, but any help with this would be greatly appreciated. Thank you in advance!
Just in case anyone would like to see my implementation of Mark's answer:
$$
\begin{align}
w_{\text{dry}} &= m_{\text{dry}}g, \\
m_{\text{dry}}g &= \rho_{o}Vg, \\
\rho_{o} &= \frac{m_{\text{dry}}}{V}.
\end{align}
$$
The forces acting on the same object, but submerged fully in a fluid are
$$
\begin{align}
w_{ \text{wet}} &= m_{\text{dry}}g - B, \\
&= m_{\text{dry}}g - \rho_{f}gV. \\
m_{\text{dry}} - m_{\text{wet}} &= \rho_{f}V, \\
\rho_{f} &= \frac{m_{\text{dry}} - m_{\text{wet}}}{V}.
\end{align}
$$
The specific gravity of the object in relation to the fluid it is submerged in is therefore
$$
\begin{align}
S_{g} &= \frac{\rho_{o}}{\rho_{f}}, \\
&= \frac{\frac{m_{\text{dry}}}{V}}{\frac{m_{\text{dry}} - m_{\text{wet}}}{V}}, \\
&= \frac{m_{\text{dry}}}{m_{\text{dry}} - m_{\text{wet}}}.
\end{align}
$$
| If you place an object with volume $V$ in the water, the buoyant force on it is $\rho_w g V$, with $\rho_w$ the density of water and $g$ gravitational acceleration. This is because the force on the body from the water is the same as it would be on a thin bag of the same shape but filled with water; the surrounding water will push on anything the same way. The bag wouldn't go anywhere since water doesn't move on average, so the buoyant force would have to equal the weight of the bag. This buoyant force is the difference between the weight of the body and the "weight" of the immersed body.
The force on the body when out of the water is $mg$, which is the same as $\rho_b V g$, with $\rho_b$ the density of the body.
Therefore, the right hand side of your equation is
$$\frac{\rho_b V g}{\rho_w V g}$$
canceling $Vg$ we have
$$\frac{\rho_b}{\rho_w}$$
However, the equation you wrote up is a little imprecise. The "weight" of something is the gravitational force on it. That's unchanged as you put it underwater, so "weight of immersed body" is not really the right term.
Additionally, the most important correction to the formula comes from our failure to account for the buoyancy of the atmosphere, which is about .1% the density of water, so we'll be off by that much (depending on how we measure the density of water).
| {
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RG of the Gaussian Model: Finding the scaling factor I'm studying how the Renormalization Group treatment of the simple Gaussian model,
$$\beta H = \int d^d r \left[ \frac{t}{2} m^2(r) + \frac{K}{2}|\nabla m|^2 - hm(r)\right]$$
In momentum space, the Hamiltonian reads
$$\beta H = \frac{1}{(2\pi)^d} \int d^d q \left[\frac{t + q^2 K}{2} |m(q)|^2\right] - hm(0)$$
and the $q$-integral runs from $0$ to some long-wavelength cut-off $\Lambda$.
The coarsening is done by splitting this integral into one from $0$ to $\Lambda/b$ and one from $\Lambda/b$ to $\Lambda$, and because the Gaussian model is so simple, the two integrals don't mix and decouple nicely. The high-momentum integral contributes just a constant additional term to the free energy, so we ignore it, and then we are left with
$$\beta H = \frac{1}{(2\pi)^d} \int_0^{\Lambda/b} d^d q \left[\frac{t + q^2 K}{2} |m(q)|^2 \right] - hm(0).$$
The rescaling is done by introducing a new momentum $q' = bq$. For the order parameter $m(q)$, one makes the scaling assumption $m'(q') = m(q)/z$. Then I can rewrite $\beta H$ in terms of the new momentum variable $q'$, and then I demand that the rescaled Hamiltonian has the same functional form as the old Hamiltonian, which allows me to read off
$$t' = b^{-d} z^2 t$$
$$K' = b^{-d-2} z^2 K$$
$$h' = zh$$
Now we don't know $z$, and in the literature I found that one somehow demands that $K' = K$, so that $z = b^{d/2 + 1}$, and I don't really understand why we can make that demand, and if there are other possibilities. Could we also demand that $t' = t$ and read off a different $z$?
| You have to remember what $t$ means in the Hamiltonian. Usually it stands for $t\equiv(T - T_c)/T_c$, i.e., $t$ is the reduced temperature. Now, since we are interested in the critical behavior, the physical relevant fixed points have to have $t = 0$.
| {
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units and nature I am wondering whether the five$^1$ units of the natural unit system really is dictated by nature, or invented to satisfy the limited mind of man?
Is the number of linearly independent units a property of the nature, or can we use any number we like? If it truly is a property of nature, what is the number? -five? can we prove that there not is more?
In the answers please do not consider cgs units, as extensions really is needed to cover all phenomenon. - or atomic units where units only disappear out of convenience. - or SI units where e.g. the mole for substance amount is just as crazy as say some invented unit to measure amount of money.
$^1$length, time, mass, electric charge, and temperature (or/and other linearly independent units spanning the same space).
| Even seasoned professionals disagree on this one. Trialogue on the number of fundamental constants by M. J. Duff, L. B. Okun, G. Veneziano, 2002:
This paper consists of three separate articles on the number of
fundamental dimensionful constants in physics. We started our debate
in summer 1992 on the terrace of the famous CERN cafeteria. In the
summer of 2001 we returned to the subject to find that our views still
diverged and decided to explain our current positions. LBO develops
the traditional approach with three constants, GV argues in favor of
at most two (within superstring theory), while MJD advocates zero.
Okun's thesis is that 3 units (e.g. $c$, $\hbar$, $G$) are necessary for measurements to be meaningful. This is in part a semantic argument.
Veneziano says that 2 units are necessary: action $\hbar$ and some mass $m_{fund}$ in QFT+GR; or a length $\lambda_s$ and time $c$ in string theory; and no more than 2 in M-theory although he's not sure.
Finally, Duff says there is no need for units at all, all quantities are fundamentally subject to some symmetry, and units are merely conventions for measurement.
This is a very fun paper and answers your question thoroughly.
| {
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What is the physical sense of the transition dipole moment? So if the states are the same we achieve the expectation value of the dipole moment for a given state. I mean
$ \langle \mathbf{\mu} \rangle = \langle \psi \vert \hat{\mathbf{\mu}} \vert \psi \rangle$
But I don't feel the physical sense in the case of transition dipole moment when psi-functions on both sides are different
$\langle \psi_{1} \vert \hat{\mathbf{\mu}} \vert \psi_{2} \rangle$
Help me to understand, please.
| The dipole transition matrix element has a classical interpretation as the time Fourier series of the classical dipole moment of the Bohr orbit corresponding to one of the energy levels. The interpretation is only exact at high levels, at the correspondence limit, and the m,n matrix element is the m-n-th Fourier series coefficient for either orbit m or orbit n (the difference is higher order in h). This is covered in Wikipedia's page on Matrix Mechanics.
When an operator x(t) is varying in time in a stationary state, that means it has off diagonal matrix elements. On diagonal operators are constant in a stationary state. The electron is orbiting the nucleus, so the position is a function of time x(t), the dipole moment in a certain direction has Fourier components, and these Fourier components are the off diagonal matrix elements. This correspondence was the main tool used by Heisenberg to construct his matrices.
To give a simple example, the x(t) operator in the harmonic oscillator is $a+a^{\dagger}$, so it has off diagonal matrix elements purely between the state of frequency $\omega$ higher and $\omega$ lower, of magnitude about $\sqrt{n}$ (the exact relations are $a|n\rangle = \sqrt{n} |n-1\rangle$, $a^{\dagger}|n\rangle = \sqrt{n+1}|n+1\rangle$. This means that the classical x motion corresponding to the n-th state has exactly two fourier components, one of which is at frequency $\omega$, the other at frequency $-\omega$, both of size $\sqrt{n}$. This means X is sinusoidal with period $2\pi\over \omega$ of size $\sqrt{E}$, and this is indeed the classical harmonic oscillation motion.
The same holds for Rydberg orbits of the H atom, and for all off diagonal matrix elements--- they correspond to the time Fourier series of the classical quantity in the Bohr orbit version of the stationary state. They don't actually make an orbit, because a real orbit has a Fourier series which is multiples of the fundamental frequency, while the quantum system doesn't have exactly equally spaced energy levels, they are only approximately equally spaced at large N in the correspondence limit.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is it possible to recover Classical Mechanics from Schrödinger's equation? Let me explain in details. Let $\Psi=\Psi(x,t)$ be the wave function of a particle moving in a unidimensional space. Is there a way of writing $\Psi(x,t)$ so that $|\Psi(x,t)|^2$ represents the probability density of finding a particle in classical mechanics (using a Dirac delta function, perhaps)?
| @Arnoques
Sorry, but I think there is an error in your answer:
The spatial extent of the particle wave-function, must be much smaller (and not longer) than the variation length-scale of the potential, to transform $\langle \nabla V(x)\rangle$ turns into $\nabla V\left(\langle x\rangle \right).$
Only in this case, it is possible to make a Taylor series of $V(X)),$ because $V(X)$ is slowly varying in the domain where the wave function is not null, and you can take the mean expectation :
$$\nabla{\mathrm i}\, V(X) = \nabla{\mathrm i}\, V(\langle X\rangle ) + (X_j - \langle X_j\rangle ) \nabla j \,\nabla{\mathrm i} \,V(\langle x\rangle )\, +\,\textrm{negligible higher order terms in}\,\, (X_j - \langle X_j\rangle)$$
So, $\langle \nabla{\mathrm i}\, V(X)\rangle = \nabla{\mathrm i} \,V(\langle X\rangle),$ because $\langle X_j - \langle X_j\rangle \,\rangle = 0\;.$
| {
"language": "en",
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That 10km/day error predicted if GPS satellite clocks not corrected for relativity Some authorities have stated publicly and without explanation that if the theories of Special and General Relativity were not taken into account in the design of the GPS (by building the satellite clocks to run 38us/day slower than GPS time before launch aka 'the factory offset), the position indicated by an earthbound GPS user device would drift by about 11km/day. I've considered this for various GPS models but can only predict much smaller effects. That multiplying the 38us/day uncorrected difference from GPS time by the speed of light yields 11.6km/day, does not for me seem to relate to GPS receiver function. I'd be very glad for any pointers.
| I looked at the 10km/day-if-38us/day-uncorrected claim several years ago and found it was based on a model of one or all-but-one of the GPS satellites used to fix the observer's position having an uncorrected clock. This model bears no relation to any sensible GPS system. Many lecturers appear not to realise this so 10km/day became received wisdom. Not offsetting satellite clocks by 38us/day would yield only mm location errors but would use up the current +/-1ms time correction range of the current correction system described below.
I also learnt that all satellites also transmit data on their own clock error, derived from earthbound time standards and transmitted up to each satellite from ground stations, allowing the observer to determine precise time.
Hope this fills in the picture on timing a bit. I would have loved to know more but could not find anything on how those satellite time error signals are derived.
| {
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Is there a limit to the resolving power of a mirror telescope? Like, if you grabbed the asteroid 16 Psyche and hammered it out into a disc of 1 mm thick iron foil and curved it into a telescope mirror with 2.4x the radius of the Sun, could you resolve details on the surface of an exoplanet? At what resolution?
Could you make the mirror arbitrarily bigger and continue to get better resolution?
What are the formulas for calculating this and what are the limits, if any?
If you make a big enough mirror could you see individual houses on Proxima b? Could aliens with big telescopes have videos of the formation of the moon in their libraries?
| Some theoretical answers were provided, but here's a practical answer from an astronomer's point of view.
(First off, the resolving power is given by diameter, not surface area. So we will talk about diameter here.)
For visible light, the practical resolving power of a 100mm diameter mirror is 1 arcsec. A 200mm mirror: 0.5 arcsec. And so on. This is the rule used by astronomers.
A mirror "with 20x the surface area of the Sun" (apparent area, I assume) would have approx 4.5x the diameter of the Sun, which is 6.3 x 10^6 km. Such a mirror would have a resolving power of 1.6 x 10^-11 arcsec.
At 10 light years distance, such a mirror would resolve details as small as 7 meters.
Please note this discussion is entirely theoretical, as there are no known technologies to manufacture such a big mirror with the required precision for visible light astronomy optics - which means the surface error cannot be bigger than 100 nanometers - in fact, for a good mirror, the acceptable error is 4x ... 5x smaller. There's no way to maintain such precision across millions of kilometers of reflective surface.
Currently, the biggest monolithic mirror is 6m in diameter and it never performed very well. The biggest well-performing monolith mirror is 5m. The biggest segmented mirrors are 10m in diameter, with a 40m project having had its initial funding approved very recently.
| {
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Sinusoidal vs exponential wave functions with Schrodinger's equation When solving Schrodinger's equation, we end up with the following differential equation:
$$\frac{{d}^{2}\psi}{dx^2} = -\frac{2m(E - V)}{\hbar}\psi$$
As I understand it, the next step is to guess the wave function, so let $\psi = {e}^{i\kappa x}$ or let $\psi = \sin(\kappa x)$, both of which I understand as they satisfy the differential equation, but when would you use one over the other? My textbook seems to use both in different scenarios, but I can't seem to figure out what the conditions are for using exponential over sinusoidal and vice versa.
| The most general solution is $Ae^{-ikx} + Be^{ikx}$
Depending on the coefficients $A$ and $B$ this can be equal to either of the example wave functions you gave, and these coefficients will be determined by boundary conditions.
| {
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What is escape velocity? In reality, how can something no longer be under the gravitational influence of something else? Isn't G a continuous function and although you leave the immediate vicinity of the earth with an escape velocity won't it always exert a force, however small it may be. Won't that force eventually pull the object back to the earth (assuming the absence of other objects)
| No, because the fall-off of gravity is $1/r^2$, the force becomes weaker and weaker as the object moves away, at such a rate that it will never pull the object back in, even though the force is never actually zero. If the gravitational force only fell off as $1/r$ then you would be right, it would always pull the object back in eventually.
My guess is your discomfort is equivalent to some form of Zeno's Paradox.
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When has the speed of light been measured, recently? Yes, it is weird, absurd, but I can't stop thinking that the would-be superluminal
neutrino speed has been computed by an arithmetic operation (space/time) and not by direct comparison with a simultaneous light ray running in "parallel".
So the "unspeakable", outrageous question: is the speed of light increasing in the last months? When it has been measured the last time?
Also: is the speed of light measured in a one-way or two-ways (forward and back) method?
It is a "politically incorrect" question, but its logic is rock-solid, I think.
| A $2 \times 10^{-5}$ shift in the speed of light would have shown up in a variety of ways. For instance, the laser ranging that is used to monitor the distance to the moon.
The reason no one has attempted to run side-by-side speed test is related to the difficulties of doing neutrino speed measurements. Between needing a high-intensity, moderately high energy accelerator at one end; a physically large detector at the other end; the need to shield the detector from cosmic rays; and wanting a long enough distance to between them to tolerate nano-second scale timing uncertainties and still get the necessary precision the line of flight for the neutrinos has to follow a chord though the Earth. Accordingly the only way to run a side by side trial would be to dig a multi-hundred kilometer tunnel that goes very deep and is straight enough for a uninterrupted line of sight. Good luck getting funding for that.
| {
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Do multiple permanent magnets aggregated together approach the same strength as a single magnet of the same size? Here's an applied physics question. ;) If I buy some cube or sphere magnets like these, can I aggregate them together to create a stronger magnet (almost as strong as a single magnet)?
| Although the combined strength will be larger, it will not be exactly the same as the strength of single large magnet. This is because the magnets lower down in the stack are being shielded slightly by the magnets above and below them.
| {
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Define Pressure at A point. Why is it a Scalar? I have a final exam tomorrow for fluid mechanics and I was just looking over the practice exam questions. They do not provide solutions. But pretty much I have to define pressure at a point and also say why pressure is scalar instead of a vector.
I am thinking pressure at a point is $P=\lim_{\delta A \to 0} \frac{\delta F}{\delta A}$. Please let me know if I am wrong.
But I do not know at all why pressure is a scalar instead of a vector. I know it has something to do with $d \mathbf{F}=-Pd \mathbf{A}$
| A gedanken experiment to illustrate the scalarness of pressure:
*
*take a rubber glove over a glass container and make it hermetic so that water will not penetrate.
*take the glass container under water. Measure the inward curvature of the rubber surface and take it as a rough measurement of the force that water is exerting over the surface opening of the container
Now question yourself: how does this curvature varies in function of different spatial orientations of the mouth of the glass container that is hermetically covered with a rubber surface?
it is a property of all liquids and gases that they don't care about the orientation of your container, it will exert the same pressure (eventually you might end up learning that this is because liquids and gases are phases with rotational symmetry).
pressure in liquids and gases is a scalar because of this rotational invariance. If you have taken a class of linear algebra you might be familiar with matrices and how they are understood as operators over vectors. You can think of pressure in general as such linear operators that can be represented as $3 \times 3$ matrices that are a property of the material. In the case of liquids and gases, this operator multiplies vectors in the same way as a scalar would, so they can be think as the identity matrix times a scalar, called scalar pressure.
Since you are in a fluid dynamics class, which is about liquids, the above applies
| {
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Difference between Clausius-Clapeyron and Van't Hoff equation I was wondering what is the difference between the Clausius-Clapeyron equation and the Van't Hoff equation. They appear to have the exact same physical meaning and are often used interchangeably.
| They're two forms of the same equation, but Clausius-Clapeyron uses vapor pressure ($p^*$) where Van 't Hoff uses the reaction equilibrium constant ($K$).
Why does this work out? Well, think of vaporization as a chemical reaction:
$$ \text{X} (l) \longrightarrow \text{X} (\text{g}) $$
The equilibrium constant is defined in terms of activity (a):
$$ K=\left.\frac{a_\text{X(g)}}{a_{\text{X}(l)}}\right|_\text{equil} $$
For an ideal liquid solution at modest pressure, a is just the mole fraction x. And for an ideal gas, a is just its partial pressure in bar:
$$ K=\left.\frac px\,\right|_\text{equil}$$
One equilibrium condition is $x=1\ $ and $p=p^*,\ $ so...
$$ K=p^* $$
Or $p^*$ is the equilibrium coefficient for vaporization. Now since K is characteristic of a reaction at a given temperature, this would imply that if we change x, then the new partial pressure at equilibrium would change according to
$$ K=p^*=\frac{p}{x} \qquad\Rightarrow\qquad p=xp^*$$
And that's exactly what happens. Neat, huh?
| {
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Are Uncertainties in Measurements Important? In the first lecture of MIT's Classical Mechanics Prof. Lewin highlights the importance of uncertainties in measurements by quoting "Any measurements, without the knowledge of uncertainty is meaningless."
He measures the length of a student and an aluminium bar both in their vertical and horizontal positions respectively and notes their lengths in centimeters and calculates the difference between the two positions.
Question:Is it so important to make such serious considerations while making measurements?
Do professional physicists make such considerations when making real world measurements?
Aren't we well equipped to just ignore these uncertainties?
| For the purpose of solving problems in physics class, uncertainties are not that important, as the solution will usually be stated to 2,3, or 4 significant figures. However, it is important to understand the concept of uncertainty to be able to do lab work, and to understand if your data are reasonable or not. Uncertainty is usually mentioned in the beginning of textbooks, to be quickly forgotten, but professional physicists actually use uncertainties. Engineers often report their parts with tolerances of +- 0.01 cm, for instance, and in this context, uncertainty is important, because the part will actually be different from the nominal size. For instance, G, the universal gravitational constant and g, the gravitational acceleration of Earth, both come with uncertainties. g = 9.80665..., and the last two digits are usually written in parentheses as uncertain.
| {
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What does a unitary transformation mean in the context of an evolution equation? Let be the unitary evolution operator of a quantum system be $U(t)=\exp(itH)$ for $t >0$.
Then what is the meaning of the equation
$$\det\bigl(I-U(t)e^{itE}\bigr)=0$$
where $E$ is a real variable?
| I would like to elaborate Adam Zalcman's answer, from a physical angle. What Adam actually showed is that all the eigenvalues of the time-evolution operator are of the form $e^{i\phi}$ where $\phi$ is real. The mathematical implication is that $U$ does not change the norm of states.
Let's look at the systems eigen-states, $\{|n\rangle\}$, which are defined by
$$H|n\rangle=\epsilon_n|n\rangle$$
These states span the whole Hilbert space, so knowing how $U$ acts on them tells you everything you need to know about time evolution of an arbitrary state. Note that these states are also eigenvectors of $U$, because
$$\begin{align}
U(t)|n\rangle&=e^{\frac{i}{\hbar} H t}|n\rangle = \sum_k \frac{\left(\frac{i}{\hbar}Ht\right)^k}{k!}|n\rangle\\
& = \sum_k \frac{\left(\frac{i}{\hbar}\epsilon_n t\right) ^k }{k!}|n\rangle = e^{\frac{i}{\hbar} \epsilon_n t}|n\rangle
\end{align}$$
and, indeed, each eigenvalue is of the form $e^{i\phi}$. This means physically that each eigenstate evolve in a very simple way - simply by changing its phase. An arbitrary state is of the form
$$|\psi\rangle=\sum_n c_n |n\rangle$$
and its norm is
$$\sqrt{\langle\psi|\psi\rangle}=\sqrt{\sum_{n,m}c_m^*c_n \langle m|n\rangle}=\sqrt{\sum_{nm}c_m^*c_n\delta_{mn}}=\sqrt{\sum_n |c_n|^2}$$
Since application of $U$ changes each $c_n$ only by its phase, it does not change the norm of $|\psi\rangle$.
BTW, since the absolute phase is not measurable, this implies that if the system is in a pure an eigenstate, it does not evolve in time. However, if the system is in a superposition of eigenstates, each eigenstate evolves with a different phase, according to their different energies, their relative phase changes and this is what causes stuff to change. Here's a nice applet that allows you to play with that.
| {
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Deterministic quantum mechanics I came across a very recent paper by Gerard 't Hooft
The abstract says:
It is often claimed that the collapse of the wave function and Born's rule to interpret the square of the norm as a probability, have to be introduced as separate axioms in quantum mechanics besides the Schroedinger equation. Here we show that this is not true in certain models where quantum behavior can be attributed to underlying deterministic equations. It is argued that indeed the apparent spontaneous collapse of wave functions and Born's rule are features that strongly point towards determinism underlying quantum mechanics.
http://de.arxiv.org/abs/1112.1811
I am wondering why this view seems to unpopular?
| Even great physicists sometimes write weak paper, and this is the case. Any attempt to find some classical deterministic theory behind quantum mechanics failed, so far. And that is because there is not any.
| {
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Projective Transformations in GR A Thought Experiment:
We are in flat spaceime provided with a reference frame—a rectangular Cartesian frame. The coordinate labels[the spatial labels] are visible to us. Each spatial point is provided with a clock—and the different clocks are synchronized wrt to each other. Gravity is now turned on and made to vary upto some final state. During this process of experimentation the physical separations change while the coordinate labels remain fixed to their own positions. [Coordinate separations remain unchanged]. The length and the orientation of a vector changes in this process both in the 3D and in the 4D sense. We are passing through different/distinct manifolds in our thought experiment and if the 4D arc length does not change we are simply having a transition between manifolds for which ds^2 is not changing but the metric coefficients are changing.We consider the option of $ds^2$ changing in this posting.
Query:
Our experiment indicates at projective transformations operating in the physical sense[considering changes in the metric and in the value $ds^2$].A time dependent field is being observed where the metric coefficients are not being preserved. Is it important to include projective transformations[concerned with the non-preservation of the metric] in the mathematical framework of GR?
| Interesting discussion.
This is my 2 cents contribution:
1- as @RonMaimon stated above, the term "projective transformations" is probably inappropriate and misleading in this context. "Projective" has specific meanings in mathematics, none of which seems to me to be applicable in this case. The OP should perhaps justify the use of the term
2- In GR literature there are examples of the kind of transformations described by the OP, where you have a parametrized family of manifolds M($\epsilon$) with metrics g($\epsilon$) that are related by a $\epsilon$-family of isomorphisms. These transformations are used in (higher order) perturbative GR to covariantly describe perturbations of exact solutions.
For example see: http://arxiv.org/PS_cache/gr-qc/pdf/0607/0607025v1.pdf
or other papers by David Brizuela and Jose M. Martın-Garcıa
The conclusion is that the language of differential geometry is powerful enough to deal with these kinds of transformations which describe perturbations of the underlying manifold.
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Complex numbers in optics I have recently studied optics. But I feel having missed something important: how can amplitudes of light waves be complex numbers?
| Isaac, let me say you are not the only one who feels this way. I had recently been tutoring an undergrad course (on nonlinear optics) and was almost shocked to find most of the students getting muddled in the usage of complex no.s. In fact, in the process, they had a hard time in also understanding/appreciating the beautiful physics.
In my opinion, it's disappointing that the lecturers/professors do not emphasize enough that the complex representation of electric field amplitude contains an additional term
c.c. (or sometimes H.c.)
c.c. stands for complex conjugate; H.c. means Hermetian conjugate and you can see that the addition of this term would make the overall quantity (on LHS) as real.
So, an electric field of the form $ E(z,t) = E_0 e^{i(kz - \omega t + \phi_0 )} + c.c.$ = $ 2\cdot E_0 \cos(kz - \omega t + \phi_0)$ indeed describes a real/physical wave.
Of course, while doing the maths it may become cumbersome to carry around the c.c. term through a series of equations, and so it is dropped (but implicitly, it is still there).
Perhaps the simplest reason for justifying the usage of this representation is that multiplication of two or more light waves - which can be encountered in several phenomena such as interference - can be simply understood by the addition or subtraction of the terms in the exponent.
As in, two waves $e^{i\omega_1 t}$ and $e^{i\omega_2 t}$ will produce terms $\propto$
$e^{i(\omega_1+\omega_2) t}$, $e^{i(\omega_1-\omega_2) t}$ etc.
Compare this with having to use trigonometric identities and you'll understand the beauty of using complex no.s in optics.
| {
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Causality and anti-particles How can I quantitatively and qualitatively understand the fact that there is a relevance between the existence of anti-particles and causality?
| This is mainly an issue of the complex Klein Gordon field (There's no such requirement for the Dirac field for instance) It's most easily shown with the self propagator of the complex Klein Gordon field using plane waves in the x-direction:
The Klein Gordon equation is.
$\frac{\partial^2 \psi}{\partial t^2} ~~=~~ \left(\frac{\partial^2}{\partial x^2} - m^2 \right)~\psi$
A straightforward generator of time evolution would be.
$\frac{\partial \psi}{\partial t} ~~=~~\pm i \sqrt{-\frac{\partial^2}{\partial x^2} + m^2}~\psi$
With a (-) sign for particles and a (+) sign for anti-particles.
However, this generator is non local since it corresponds to an infinite series of derivatives. In fact it's equal to a convolution with a Bessel K function.
$\frac{\partial \psi}{\partial t} ~~=~~\pm i \sqrt{-\frac{\partial^2}{\partial x^2} + m^2}~\psi ~~=~~ \frac{m}{x}K_1(mx)~*~\psi$
source: wikipedia
This means instantaneous propagation since $\partial\psi/\partial t$ depends on non local values of $\psi$. This problem then enters the general time evolution operator for arbitrary $t$.
$\psi(t) ~~=~~ \exp\left\{ \pm i \sqrt{-\frac{\partial^2}{\partial x^2} + m^2}\right\}~\psi$
However, now comes the trick:
The sum of the particle and anti particle propagators is local (within the light cone)
$\psi(t) ~~=~~ \frac12\Big(\exp\left\{ + i ...\right\} +\exp\left\{ - i ...\right\}\Big)\psi ~~=~~ \cos\left\{ \sqrt{-\frac{\partial^2}{\partial x^2} + m^2}\right\}~\psi$
because the Taylor series expansion of the cosine only contains even powers of the argument there is no more square root operator.
It has to be said that the part outside the light cone is small to begin with, in the order of the Compton radius of the particle. But it also shrinks further as propagation progresses. For an electron it's about $10^{-13}m$ at the start of the propagation but only $10^{-20}m$ after a lightmicron (the time in which light propagates 1 $\mu m$). It shrinks further away linear with time.
This issue does not occur for the time evolution operator of the correct equation for the electron : The Dirac equation. This equation is linear and $\partial\psi/\partial t$ does not contain the above square root.
Hans.
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How do we visualise antenna reception of individual radiowave photons building up to a resonant AC current on the antenna? I am a chemical/biological scientist by trade and wish to understand how quantum EM phenomena translates to our more recognizable classical world.
In particular, I want to get a mechanistic picture of what is going on when a tuned antenna is interacting with a photon of the desired frequency? I believe an individual electron on the antenna (many electrons) accepts a photon; but how does the eventual process of a measurable AC current build up on the dipole (or 1/4 wavelength, for example) to be fed with no reactance onto the transmission line?
"When photon meets antenna" is a great meeting ground for a quantum/classical bridge.
Unfortunately, I do not have a serious maths background, but will try anything suggested. I have read and listened to many of the Feynman's popular quantum discussions which only increases my thirst for a better understanding of how quantum EM translates to our more visible world.
| Here is an experimentalist's view of the question:
1) one photon hits the antenna and raises a molecular electron band to a higher energy level, and it will fall back to its lower one, with the characteristic electromagnetic transition time of the order of 10^-16sec, giving the energy to the antenna grid of molecules.
One photon will just disappear.
2)a stream of photons that carry a signal means: a) that there is enough amplitude, b) there is coherence between photons: photons carry spin and thus polarization and in order to carry a signal the phases between all photons must be fixed and be coherent in time and space. Coherent means that there are fixed phases in the whole bunch.
When such a bunch of photons hits an antenna the coherence will be transferred to the individual photon absorptions and de-excitations by conservation of spin, building up a corresponding electromagnetic wave on the molecular Fermi conduction level which can be detected further as a signal.
3) It is simpler for such problems to use the classical EM picture.
| {
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Software to simulate and visualize atoms? Not sure if this is a physics or chemistry question. But if the motion of atoms and it's particles can be described by quantum mechanics, then is there a software that simulate full atoms and it's boundings, in a way you can visualize them, and that can be used, for instance, to throw 2 molecules together and watch them reacting?
| There are many, many algorithms and pieces of software to do this. In addition to Molecular Dynamics, there are also methods based on statistical simulations in Quantum Monte Carlo, and density functional theory as implemented in programs like Quantum Espresso. It is a simple and worthwhile exercise to program these things yourself - if you wish to study the oscillatory behavior of a molecule subject to some arbitrary external potential, you can do this quite readily using basic programming and visualization tools provided you establish the proper functions and equations to describe your system.
I will note that these algorithms all have explicit ranges of validity and underlying assumptions, and one must very carefully understand the limitations before interpreting the results. In many cases, the accuracy and precision of the algorithms will be questionable, because assumptions at some level have to be made to reduce the system size since not even the most powerful supercomputer can handle a calculation with anything approaching a macroscopic number of particles. Nevertheless, they can provide some sense of the starting point and can give insight into trends.
Edit to add: See giant list of software applications for Quantum Chemistry
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Why do physicists believe protons and electrons are present in equal numbers? I tended to consider that negative and positive charges are present in equal numbers in the universe to be a known, obvious fact. But is it so? How can we rule out the possibility that there is some kind of asymmetry in the numbers of protons and electrons? Of course, matter is neutral and even tiny deviations would give rise to enormous forces... But not all protons and electrons are in atoms in the universe. More importantly, is there some conceptual reason why we believe that this equality is exactly perfect or do modern theories allow tiny deviations?
| I think one of the reasons could be that charge cannot be created. So if initially universe was neutral (at the time of big-bang) it must remain neutral.
| {
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Why, intuitively, must a solution in physics be unique? When solving Laplace's equation or Poisson's equation say, we require that the solution must be unique, which can be shown.
*
*In general, what is the physics behind seeking a unique solution?
*Are there differential equations which, for some mathematical reasons, cannot have unique solutions and hence cannot be used to model any physical phenomena?
| (1) There is a correlation between symmetries, conservation laws and boundary conditions (in the language of differential equations) such that if you have enough symmetries (i.e. enough conservation laws) you will obtain a unique solution.
If you have a system with some free parameters left over and there is genuinely no physics left to be put into the model, then you would say that the physics is actually happening in the full space that your equations are working in modulo the free parameters. And there would be a corresponding set of equations that operate only on that quotient space.
The point of integration is killing off dimensions -- if you don't have a boundary condition to fix the integration constant, then the dimension has lived and the integration was pointless.
Maybe here would be a good place to start: Integrable system (Wikipedia).
An example for (2) would be the differential equations that govern what happens when you drop an object without specifying how you drop it.
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Why is it concluded that the cosmos is expanding when in fact the observations are for events further back in time? Why is it concluded that the cosmos is expanding at an ever increasing rate when looking further out and therefore further back in time?
Surely, if the time were reversed to play forwards, the conclusion would be that the expansion of the universe is actually deaccelerating, and the need to postulate 'Dark Energy' would be avoided.
| You should try to think of the Universe expanding in terms of the space itself expanding. Not objects moving away from each other in a fixed space. The accelerated expansion refers to the way space in between objects is growing. This is where quantities like proper and comoving distances become very important. If you want more details on that I would refer you to Friedman equations and the scale factor therein. The bottom line is, the space itself between objects is growing therefore the Universe is expanding. The rate at which it is growing is also a positive quantity therefore it is accelerating as well.
| {
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If humans were able to catch all sun energy reaching the earth for their use, will the climate change? I guess that energy will be used up and, at the end, will contribute
to heat the earth, so I see no big differences...
please explain your point of view.
| Taking the question at face value, the answer is yes. In order to collect all the solar energy arriving at the Earth you would have to completely cover its surface in solar panels. These solar panels would have to encase the entire atmosphere, because otherwise some energy would be reflected by clouds. This would leave no energy left to drive photosynthesis, which would very rapidly change the atmosphere's composition - we'd run out of oxygen unless we used some of the energy to make more. There would also be no energy to drive the water cycle, the ocean circulation or the wind. In short there wouldn't be a climate any more.
However, let's instead assume a more sensible scenario and say that humans only decide to extract all the solar energy that reaches the surface of the Earth in places that are currently deserts, so that most of the plants on Earth are still able to survive and we could still feed ourselves using agriculture. This would produce far more energy than we currently use. Whether this would have a substantial effect on the climate is not obvious. It would heat those regions up a bit (since deserts currently reflect quite a bit of sunlight back into space) and maybe this would cause changes in weather patterns. However, the main consideration would be what effect this would have on greenhouse gases. That depends on what we use all that extra energy for, and whether we keep on using fossil fuels at the same time, so it's pretty hard to answer.
| {
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Can the quartic oscillator Hamiltonian be made quadratic? I'm interested in turning a quadratic + quartic oscillator Hamiltonian,
$$H = \frac{p^{2}}{2m} +\frac{kx^{2}}{2} + \lambda x^{4},$$
into a purely quadratic problem. One of the simplest things I can think of is replace the quartic term by
$$x^{4} \rightarrow \langle x^{2}\rangle x^{2}.$$
Has anyone come across such an approximation? If yes could you give me some reference to the work where this approximation has been used. Is there any other simple approximation which can turn the above Hamiltonian into a quadratic problem?
| This is an extraordinarily famous approximation, perhaps the most famous one, called the "Hartree Fock" or "Self-consistent field" approximation. It is the earliest and most famous approach to solving complicated self-interacting systems.
But in the case you are using it, it is very poor. The problem you state can be solved much better by Bohr-Sommerfeld quantization, or BKS, to a far better accuracy than a self-consistent field. What you do with the self-consisten field is replace the quartic by a best-fit quadratic potential, which will give completely wrong wavefunctions, and not good energy levels (BKS will certainly be better for everything).
The point of this approximation is that it works for many-body systems, where you expect that many-body interactions do take the form of a self-consistent field to a good approximation. This idea is verified in both atomic and nuclear physics.
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Is there a known optical design for a beam compressor? With my little knowledge of optics I have come across some 'known' designs such as the Double Gauss for example, is there a 'beam compressor'.
My requirements are to reduce an incoming parallel wavefront (source at infinity) to either an outgoing parallel beam (ie afocal), by a factor of around 100 within 10cm or coming to focus at an f-number around 4 or 5.
I have access to Zemax but am currently just searching in the dark worried I might be re-inventing the wheel...
| You can use two parabolic mirrors (either one concave + one convex or two concave) in an afocal Gregorian-telescope arrangement but with multiple reflections. Every second reflection would give you an extra magnification.
For example, if you take two parabolic mirrors with 100mm and 20mm focal length (5:1) then you will get after the second reflection 5x, fourth reflection 5x5x=25x, and sixth reflection 5x5x5x=125x magnification.
No aberrations if you use parabolic mirrors but at every reflection you will loose some power. With a 95% reflection mirror (at some wavelength), after the sixth reflection you will end up with a total loss of 11.5%.
Alignment is not easy either.
| {
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Does a moving escalator make it easier to walk up the steps? I was discussing with my colleagues why it feels easier to walk up an escalator when it is moving. My natural assumption was that the movement of the escalator imparts some extra acceleration on the rider that helps to move up the stairs. But my colleagues insisted that this was nonsense, and that the affect is purely psychological (i.e. it just seems easier).
We actually came up with three contradictory hypotheses, and I'm not sure which is right:
1. The escalator is constantly accelerating the rider since without constant acceleration the body wouldn't be able to counteract the force of gravity (i.e. my theory).
2. The rider is not accelerating since no acceleration is needed to maintain a constant velocity.
3. The acceleration of the escalator actually makes it harder to get to the next step since it pushes the rider against the current step.
Which of these is correct?
| Once you get yourself moving, the escalator does not accelerate you and does not assist your running up hill. The only advantage an escalator gives you is that you keep moving up even if you don't put any effort into it.
Next time you feel like running up an escalator (which is not entirely safe), you might consider repeating the experiment with your eyes closed. That will eliminate the visual effect but you will still feel the air flow on your face.
This is basic Newton's law on inertial frames. You cannot detect steady motion. This is why you can throw dice in the gambling compartment of a luxury airliner that is moving at 400 miles per hour. Or why you can exist on the earth's surface without realizing that it is moving (due to the earth's turning) at an even faster rate (depending on latitude).
| {
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What is the connection between Poisson brackets and commutators? The Poisson bracket is defined as:
$$\{f,g\} ~:=~ \sum_{i=1}^{N} \left[
\frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}} -
\frac{\partial f}{\partial p_{i}} \frac{\partial g}{\partial q_{i}}
\right]. $$
The anticommutator is defined as:
$$ \{a,b\} ~:=~ ab + ba. $$
The commutator is defined as:
$$ [a,b] ~:=~ ab - ba. $$
What are the connections between all of them?
Edit: Does the Poisson bracket define some uncertainty principle as well?
| Both the commutator (of matrices) and the Poisson bracket satisfy the Jacobi identity,
$[A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0$.
This is why Dirac was inspired by Heisenberg's use of commutators to develop a Hamilton-Jacobi dynamics style of Quantum Mechanics which provided the first real unification of Heisenberg's matrix mechanics with Schroedinger's wave mechanics. The Jacobi identity is also the basic law of Lie algebras, which are useful for symmetry groups in quantum theory.
In classical mechanics, the dynamical variables are the functions $f$ on phase space, and they get a non-trivial algebraic structure from the Poisson bracket. They are the classical « observables ». In quantum mechanics, the observables are matrices, these are the dynamical variables, but they receive a similar algebraic structure from the commutator.
As already pointed out, the anti-commutator is not analogous to the Poisson bracket, it is a distinctly new quantum phenomenon with no classical analogue.
| {
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Infinite quantum well width $L$ to $2L$ adiabatic process If we change width of the infinite quantum well $L$ to $2L$ slowly enough, how it does change energy levels.
| After the process ends, the new energy levels will be just the normal energy levels of infinite quantum well with width 2L.
moreover,
by the Adiabatic theorem:
A physical system remains in its instantaneous eigenstate if a given
perturbation is acting on it slowly enough and if there is a gap
between the eigenvalue and the rest of the Hamiltonian's spectrum
So if for example, you have a particle in the nth level it will stay in this level during the process and his energy will be the updated energy for nth level in the new well.
| {
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Does the scientific community consider the Loschmidt paradox resolved? If so what is the resolution? Does the scientific community consider the Loschmidt paradox resolved? If so what is the resolution?
I have never seen dissipation explained, although what I have seen a lot is descriptions of dissipation (i.e. more detailed pathways/mechanisms for specific systems). Typically one introduces axioms of dissipation for example:
entropy $S(t_1) \geq S(t_0) \Leftrightarrow t_1 \geq t_0$ (most often in words)
These axioms (based on overwhelming evidence/observations) are sadly often considered proofs. I have no problem with useful axioms (and I most certainly believe they are true), but I wonder if it can be proven in terms of other (deeper and already present) axioms. I.e. is the axiom really independent? or is it a corollary from deeper axioms from say logic (but not necessarily that deep).
(my opinion is that a proof would need as axioms some suitable definition of time (based on connection between microscopic and macroscopic degrees of freedom))
| Although summarized as an objection of macroscopic irreversibility when microscopic laws are reversible, Loschmidt's objection originally points that there has to be something breaking the time reversal symmetry in Boltzmann's derivation of the $H$-theorem.
I think that Boltzmann's answer was to say that high $H$ states (in absence of external driving) are more the exception than the rule. This is betrayed by the fact that inverting time in the $H$-theorem still leads to a decrease in $H$.
I think it is important to stress that Boltzmann's equation (from which derives the $H$-theorem) only looks at a very coarse grained quantity, namely the one-particle density and most rationals for the asymmetry are put at this coarse grained level.
Yet, mathematicians are still working on the problem (see here and there ).
But as a physicist, and for a picture beyond physics of gases, I think that this article on relevant entropies gives a lot of insights about these things in general.
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How does physics scattering experiments relate to real life? And what does the scientist gain from such experiments? How does physics scattering experiments relate to real life? And what does the scientist gain from such experiments? I am having a hard time figuring the answer out. Please help.
| Depending on how you define scattering, its not only scientists who do these experiments, it is you, especially your eye.
Most likely you are referring to things like Rutherford backscattering or what is done at CERN, DESY, LHC & Co.?
In the end, the entire world can be seen as particles that are getting scattered by each other. E.g. electrons on atomic nuclei, cosmic radiation (i.e. "weird" particles) on atoms, and sunlight (i.e. photons) on you.
Physicists like those individual scattering experiments because they allow to isolate and study one of those myriads of possible scattering events that happen all the time. In such controlled environments, they can compare theory with reality/experiment.
E.g. suppose you have a theory on what an electron and a proton are, and how they behave. Then you just shoot one on the other (under various angles, with various energies, etc.) and look whether the results are according to theory (which is good) or not (which is likewise good as this will lead to an improved theory or even overthrow fundamental concepts, both are seen as scientific progress).
Without the understanding from such experiments things like a transistor would not have been possible. Without a transistor, you would not have an iPod/Pad, no Mac/PC, no internet, ...
Actually, the physicist Werner Heisenberg, in his later years, even tried to model a theory of everything based on nothing else but scattering (S-Matrix, where "S" stands for ...).
| {
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Physics book for 15 year old boy
Possible Duplicate:
List of good classical physics books
my name is Bruno Alano. As stated in the title, I'm 15 years old (I'll do 16 on 7 of Feb) and much love Computer Science (C, C++), Mathematics and Physics.
Some information may have been unnecessary, but my question is: What is the suggestion of a good physics book for a teenager of my age? I know basic things (speed, shoveller these issues and basic primary and secondary).
A good reason for this is my Awe in mathematics and physics. Besides that maybe one day be useful in what I really want a career (science or computer engineering).
And another question: It is interesting physics in the area I want to go? I'm at an age that would be good to learn beyond what is taught in common schools?
| Even if it doesn't answer your question directly i think these videolectures are a great place to start and really motivating (and for free):
*
*Walter Lewin ot the MIT:
http://web.mit.edu/physics/people/faculty/lewin_walter.html go to --> take a class Physics I, II, III
*Foundations of Modern Physics: Leonard Susskind
http://academicearth.org/courses/foundations-of-modern-physics
| {
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How are the Pauli matrices for the electron spin derived? Could you explain how to derive the Pauli matrices?
$$\sigma_1 = \sigma_x =
\begin{pmatrix}
0&1 \\ 1&0
\end{pmatrix}\,, \qquad
\sigma_2 = \sigma_y =
\begin{pmatrix}
0&-i\\ i&0
\end{pmatrix}\,, \qquad
\sigma_3 = \sigma_z =
\begin{pmatrix}
1&0\\0&-1
\end{pmatrix}
$$
Maybe you can also link to an easy to follow tutorial ?
| The Hilbert space for spin 1/2 is two-dimensional - there are two possible values spin can take: $\hbar/2$ or $-\hbar/2$ (this is taken from experiment).
Now, in two-dimensional Hilbert space spin operator has to be self-adjoined (this comes from foundations of QM). Furthermore, sum of its eigenvalues has to be 0 - because sum of eigenvalues is just a sum of possible results of measurements - in this case $\hbar/2-\hbar/2 = 0$. Therefore the operator corresponding to the measurement of spin in a given direction has to be 2x2 complex hermitan traceless matrix (no commutation relations used so far!). This family of matrices is a three-parameter one (one constant on diagonal, two off-diagonal for real and imaginary part, rest is determined by hermicity and tracelessness).
Furthermore, every such an operator has to be diagonal in its eigenbasis - which corresponds to the measurement in a given direction. Let us name this direction $z$ direction. We see, that the only possible traceless hermitean diagonal matrix is a multiplicity of Pauli matrix $\sigma_z$. Now, we write the remaining two operators as $L_x$ and $L_y$.
Next, we have to use commutation relations of angular momentum: $[L_y,\sigma_z]=L_x$, $[L_x,\sigma_z]=-L_y$ and $[L_x, L_y] = \sigma_z$. Remembering that $L_y$ as well as $L_x$ are determined by 3 real constants each, we need 6 linear equations to solve for them all at once. The above commutation relations give exactly 6 equations (3 matrix equations, two-dimensional matrices) so we can solve for $L_x$ and $L_y$ which are exactly the remaining Pauli matrices.
| {
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What is the physical definition of causality? Maxwell's equations give a physical relationship between the electric and magnetic fields $\vec E$, $\vec B$ at the same time, which some interpret as changes in one causes changes in the other etc. I find this confusing because to me, the cause of both is charge and cause should precede effect.
Therefore, how do physicists determine if there is a causal relationship between two physical quantities?
| If we define some event which we will call $p$ to be the cause and $q$ to be its effect, then $p$ and $q$ should satisfy the following rules,
*
*$p$ implies $q$ but $q$ doesn't imply $p$.
*in the absence of $p$, $q$ shouldn't exist either.
*$p$ and $q$ shouldn't be simultaneous events as viewed from any inertial frame of reference.
| {
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Synchronising the Earth's rotation via mass redistribution How much material would have to be moved per year from mountain-tops to valleys in order to keep the Earth's rotation synchronised with UTC, thus removing the need for leap seconds to be periodically added? Would it be a feasible project to undertake in order to resolve the current disagreement about the future of the leap second?
| That seems like a fun question!
According to Wikipedia the day is currently 2ms too long, so that's a factor of 2.31e-8. So we need to reduce the angular momentum of the earth by this factor.
To make life easy consider a mountain on the equator, with a mass $m$, treat it as a point mass and assume we manage to move it $d$ meters nearer the centre of the earth. The change in angular momentum is:
$$\Delta L = m(r_e - d)^2 - m(r_e + d)^2 = -4mr_e d$$
where $r_e$ is the radius of the Earth. Assuming the Earth is a uniform sphere it's angular momentum is:
$$L_e = \frac {2}{5} M r_e^2$$
so I get the fractional change of the angular momentum to be:
$$\frac {\Delta L}{L} = 10 \frac{d}{r_e} \frac {m}{M}$$
Bearing mind that we're modelling the mountain as a point mass, I'd say about 10km was a reasonable distance to move it, i.e. from 5km above sea level to 5km below sea level, so taking $d$ as 10km and $r_e$ as 6380km and setting the change equal to 2.31e-8 gives:
$$\frac {m}{M} = 1.5 \times 10^{-6}$$
so if the mass of the Earh is about $6 \times 10^{24}$kg, you'd need to move about $10^{19}$ kg of mountain.
For comparison, a quick Google suggests the mass of Mount Everest is of the order of $10^{15}$ to $10^{16}$ kg so that's somewhere between 1,000 and 10,000 Mount Everests.
| {
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Was the universe a black hole at the beginning? Big bang cosmology, as far as I understand it, says that the universe was super hot and super dense and super small. It looks like that all the current matter, seen and unseen, were compressed to infinitesimal distance, which means it was a black hole.
*
*Is the big bang, and our universe expansion in this case, hence is nothing but a black evaporation via Hawking radiation?
*Are we living inside that primordial black hole explosion?
| The observable universe exists inside a black hole created by the total mass present in the observable universe.
If we find the size at which the mass distribution with the average mass density of the present universe forms a black hole.
The above expression means that if the present universe has a critical mass density ρ_c(the order of 5~6 hydrogen atoms per 1/m^3) value and the size is approximately R_UB=14.3Gly or more, this region becomes a black hole.
Currently, we estimate that the size of the observable universe is larger than 14.3 Gly, and the entire universe is estimated to be larger than the observable universe 46.5Gly, so our observable universe inevitably exists inside a huge black hole called the universe.
The size of the black hole created by this 46.5 Gly mass distribution is 491.6 Gly. The horizon of event created by the observable universe is approximately ten times larger than the observable universe.
When considering the expansion in the early high-density state of the universe, there is a problem that people mistakenly think that this event is the escape of matter from the inside of the black hole created by the total mass of the universe to the outside to form galaxies or stars.
The event horizon of black hole created by the total mass of the universe is very large compared to the area where the total mass of the universe is gathered. In this case, matter does not escape the universe black hole, but has not yet reached the event horizon of the universe black hole.
If the Schwarzschild equation is correct..., But...
https://www.researchgate.net/publication/359192496
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Are matrices and second rank tensors the same thing? Tensors are mathematical objects that are needed in physics to define certain quantities. I have a couple of questions regarding them that need to be clarified:
*
*Are matrices and second rank tensors the same thing?
*If the answer to 1 is yes, then can we think of a 3rd rank tensor as an ordered set of numbers in 3D lattice (just in the same way as we can think of a matrix as an ordered set of numbers in 2D lattice)?
| *
*All scalars are not tensors, although all tensors of rank 0 are scalars (see below).
*All vectors are not tensors, although all tensors of rank 1 are vectors (see below).
*All matrices are not tensors, although all tensors of rank 2 are matrices.
Example for 3: Matrix M (m11=x , m12=-y , m21=x^2 , m22=-y^2) .This matrix is not tensor rank 2. Test matrix M to rotation matrix.
| {
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What is the velocity area method for estimating the flow of water? Can anyone explain to me what the Velocity Area method for measuring river or water flow is?
My guess is that the product of the cross sectional area and the velocity of water flowing in a pipe is always constant. If the Cross sectional area of the pipe increases at a particular point, then the velocity decreases so that the product $AV$ is a constant. Am I right?
If so, how can we extend this to pipes where the water is accelerating & does not have a constant velocity? For example, the system may be under the action of gravity & hence the acceleration of the water is $g$, the acceleration due to gravity?
| What you refer to is conservation of mass under some assumptions:
*
*Constant density
*A steady state flow
I'll bring us back to your equation by starting with the very fundamental mass accounting for a given fluid flow. To be comprehensive, we need to recognize that velocity isn't constant over the entire area, but we will assume that it is. Take the flow rate to be $\dot{m}$.
$$\dot{m} = \rho V A$$
Now, if we have a steady state flow along a single flow path, then this quantity will be constant over the entire path, $\dot{m}=const$. Water in the cases you are concerned about is sufficiently incompressible so $\rho = const$. This results in your conclusion that $VA$ is constant.
Gravity may or may not shift the balance from $V$ to $A$ or vice versa. It depends on if there are rigid boundaries to the flow. If you have a flow fall freely in air or flow downward in a trench (like a river) then the boundary of the fluid may change freely. If you have a pipe with a given flow area, then the velocity is fully determined from that. Anyway, there are laws that conserve other things - like energy. So in a rigid pipe flowing downward (absent friction) the pressure will increase as you go down in elevation, which results directly from gravity.
| {
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How can a human eye focus on a screen directly in front of it? I am asking this question here because I think the answer has something to do with the way light is bent as it's captured through the eye.
I saw a show a while ago about tiny screens on contact lenses to pull up data on objects you see in the real world, I also just saw this article about Google testing the same idea with screens in the lenses of sunglasses.
The part I do not understand is how your eye would focus on a screen that is so close. My eyes (and I believe most others) cannot focus on anything closer than a couple inches away. Yet the lenses on glasses are much closer to the eye than a couple inches.
So I would guess they would have to use some special technology to separate the light rays in a way that your brain could make an image from it. If that's correct, how would it work? if not, how would they get your eye to focus on a screen so close?
| The answer is that your eyes would be focusing not at the concrete distance where the mechanism is placed, but at a virtual image which appears farther away.
This is just basic optics.
It is (one aspect of) what happens when you use a magnifying glass, for instance.
The virtual image of some point viewed through an optical instrument is that location in space to which the rays of light which are coming from that point appear to converge to.
So in the case of the magnifying glass, it assists our vision in two ways. Not only does the image appear larger than the original object, it is also farther away, so that our eyes can focus on it. Glasses for correction of far-sightedness are magnifying lenses, yet their job isn't to make the world bigger, but rather to enable closer focus.
A screen that is placed very close to the eye simply has to contain the right optics so that a focused virtual image ends up on the retina. This will be done by somehow ensuring that the light rays emanating from the screen diverge at a sufficiently acute angle that they appear to come from a point that is much farther away than the screen really is: far enough away so that the eye can comfortably focus on that distance.
| {
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Why can't we think of free fall as upside down rocket? /\
/ \
| |
| m |
| |
------ <--- floor (Rocket A)
This rocket is accelerated (g) upwards then mass(m) falls on the floor.
------ <--- floor
| |
| m |
| |
\ /
\ /
\/ (Rocket B)
This rocket is accelerated (g) downwards then mass(m) falls on the floor.
----- <--- ceiling
| |
| m |
| |
----- <--- floor (Elevator E)
This elevator is falling freely on the earth. Acceleration due to gravity is g. The mass stay in the midair.
Why? Why can't we think of the elevator as upside down rocket? Why doesn't mass go to the ceiling of the falling elevator?
NOTE: Principle of equivalence of this document is what I am trying to understand.
| In fact, in case B, if the rocket really accelerate at g, downward, in the vicinity of earth, it is "free-fall" : m stay midair.
Except this detail, the main difference between the rockets and the elevator is the thrust.
No thrust pushes the elevator during free fall.
No thrust means no force applied on its inhabitants, so no feeling of gravity.
But, if you attach downwards rockets on your free falling elevator, you may bonk your head on the ceil :)
| {
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What formulations of QM are there? It is usually said that there are different formulations of QM, for example historically there was Schrodinger's (wave mechanics), and Heisenberg's (matrix mechanics), then Dirac's (which showed they are equivalent)
Since they are all physically equivalent I have a few questions:
1-Is Dirac's formulations considered more fundamental in the sense that it can be reduced to one of the 1st two?
2-I also hear about the path integral and density matrices, are they another formulations?
3-Are there more formulations less known to undergraduate students but known by researchers because they are technically advanced?
| http://www-physique.u-strasbg.fr/cours/l3/divers/meca_q_hervieux/Articles/Nine_form.pdf
See this article by Styer et al entitled Nine formulations of quantum mechanics
| {
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Discreteness of Spacetime and Violation of Lorentz symmetry It is usually said that existence of discrete spacetime violates Lorentz symmetry. What quantity is used to quantify such violation? I mean could someone points a reference for a derivation that shows such analysis.
My other question is: if Lorentz symmetry is violated, does that imply space-time is discrete? or not necessarily?)
| There are nontrivial discrete subgroups of the Lorentz group. It is easy to construct an SO(3,1) matrix that has only integer entries and yet is not just a simple rotation. A rectangular lattice in Minkowski space is invariant under the group of these transformations. Different space-time lattices have different discrete subgroups of Lorentz under which they are invariant.
This might be what you are looking for, even if it does not answer the question ...
However, there is a snag: it is extremely difficult to construct any non-trivial dynamical model of nature (quantum, classical, anything) that transforms into itself under these discrete transformations, even if their lattice does. Versions of string theory, adapted to this lattice, may give you the best promises.
So my advice is: don't believe the no-go theorems.
| {
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How is capacitance defined for three concentric spheres? If we have a configuration of metal concentric spheres (each of negligible thickness) of radii $r_1,r_2,r_3$ respectively and $r_1<r_2<r_3$, and we are given the potentials of the spheres to be $0, constant,0$ respectively. How might we find the capacitance of the configuration?
EDIT: My question: How is capacitance defined for such a system?
Added: I think I have managed to find the charge on each shell :) What is missing now is just a definition of capacitance for such a system.
| Although there is a theoretical ambiguity because there are 3 conductors, the particular specification of the potentials here implies that the desired capacitance $C$ is the ratio of the charge $Q$ on the middle sphere to its voltage $V$: $C=Q/V$.
Note that the inner and outer spheres are grounded so $V$ is the (only) potential difference in the problem.
Note also that the total charge on all the spheres must add to zero: otherwise there would be electric fields external to the sphere-assembly, contrary to the assumption that the outer sphere is grounded.
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Degree of freedom paradox for a rigid body Suppose we consider a rigid body, which has $N$ particles. Then the number of degrees of freedom is $3N - (\mbox{# of constraints})$.
As the distance between any two points in a rigid body is fixed, we have $N\choose{2}$ constraints giving $$\mbox{d.o.f} = 3N - \frac{N(N-1)}{2}.$$ But as $N$ becomes large the second term being quadratic would dominate giving a negative number. How do we explain this negative degrees of freedom paradox?
| These constraints are not independent.
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Reynolds number with hyper-viscosity Is it possible to evaluate a Reynolds number when viscosity operator is substituted by hyper-viscosity operator at the power H (Laplacien to the power H) in the incompressible Navier-Stokes equations ?
| For the equation:
\begin{equation}
\partial_t u_i + u_j \partial_j u_i=-\partial_i p+ \nu_{hyper} \Delta^H u_i
\end{equation}
with $u$ the velocity in $m.s^{-1}$ and is characteristic order $U$, $p$ the pressure in $m^{2}.s^{-2}$, $\nu$ the hyper-viscosity in $m^{2H}.s^{-1}$. The characteristic length scale is note $L$ in $m$.
Following the non-dimensionalizing, the number Reynolds like is:
\begin{equation}
Re_{hyper}=\frac{U L^{2H-1}}{\nu_{hyper}}
\end{equation}
| {
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Speed of a dynamic hydraulic system We all have noticed that changing the temperature of the water in the shower is not instantaneous, rather the result is felt when the water that was in the tap works its way up to the showerhead. However, changing the pressure does feel instantaneous. I wonder at what speed the change in pressure propagates in the direction of flow in a flowing fluid. My guess would be at the speed of sound in the medium, based on the fact that it is faster than the speed of the flow of the water (like the temperature is), yet obviously at or below C. I am obviously missing the correct keywords to google for as I cannot find any references to this phenomenon. What is the answer to the query, or better yet, how could I have found this information (short of empirically)?
Thanks.
| The pressure wave does indeed travel with the speed of sound of the media. This is much higher for water then for air with a speed of $\approx$ 1500 m/s.
This effect is well known and feared under the name water hammer. Basically by rapidly closing a valve you create a longitudinal pressure wave that can be powerful enough to damage to destroy metal pipe joints. The kinetic energy of the flowing liquid causes this phenomenon.
| {
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Mechanism by which electric and magnetic fields interrelate I read that force due to electric field on some particle in one reference frame can exhibit itself as force due to magnetic field in some other reference frame and that electric and magnetic fields are two aspects of same underlying electromagnetic field.
My question is what is the mechanism which can explain how an electric field becomes/creates magnetic field in some other reference frame. Is there any such explanation available in relativity theory? I am not looking for mathematics but a physical explanation.
Wikipedia article http://en.wikipedia.org/wiki/Relativistic_electromagnetism
explains something about origin of magnetic forces in a wire as a consequence of lorentz contraction and motion of electrons in the wire
Calculation of the magnitude of the force exerted by a current-carrying wire on a moving charge is equivalent to calculating the magnetic field produced by the wire. Consider again the situation shown in figures. The latter figure, showing the situation in the reference frame of the test charge, is reproduced in the figure. The positive charges in the wire, each with charge q, are at rest in this frame, while the negative charges, each with charge −q, are moving to the left with speed v. The average distance between the negative charges in this frame is length-contracted to:
where is the distance between them in the lab frame. Similarly, the distance between the positive charges is not length-contracted:
Both of these effects give the wire a net negative charge in the test charge frame, so that it exerts an attractive force on the test charge.
But this still does not explain origin of magnetic field in case when there are no positive charges.
| Fundamentally they are both the interaction between charges. Magnetic fields are the results of moving charges. When charges move, its time slows down (relativity), therefore, the effect (field) of the charge is weakened. Eg. electrons are moving in a wire relative to a stationary proton, the field of the electron is weakened to the proton, therefore, the wire will be positive to the proton. A repulsion force therefore occurs.
| {
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Long-Life High Altitude Balloon Normally high-altitude balloon experiments end with the balloon popping and the payload falling back down to be reclaimed.
But if a second balloon was attached to the payload, one which was only partially inflated at launch, then could you keep the balloon aloft for a very long period of time? A sort of extremely-cheap very-low-orbit satellite.
And if so, then does anyone do this?
| There are two limitations to your proposal: the volume of the balloon will be very large at higher altitudes and the gas such as hydrogen and helium will leak even through solid walls with time.
| {
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What determines color -- wavelength or frequency? What determines the color of light -- is it the wavelength of the light or the frequency?
(i.e. If you put light through a medium other than air, in order to keep its color the same, which one would you need to keep constant: the wavelength or the frequency?)
| Colour is defined by the eye, and only indirectly from physical properties like wavelength and frequency. Since this interaction happens in a medium of fixed index of refraction (the vitreous humour of your eye), the frequency/wavelength relation inside your eye is fixed.
Outside your eye, the frequency stays constant, and the wavelength changes according to the medium, so I would say the frequency is what counts more. This explains why objects' colour don't change when we look at them under (transparent) water ($n=1.33$) or in air ($n=1$).
| {
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The speed of tachyons The other day I was wondering: When a tachyon is coming towards you faster than the speed of light, will you see it before it hits you? Then I thought of course not, since the light waves aren't traveling faster than the tachyon then how could you see it before it hits you? Now I thought today, if an tachyon is traveling away from you faster than the speed of light, would you see it?
If you fire a ball at an initial velocity of 20mph south out of a car that is going 50mph north, the final velocity of the ball would be 30mph north, is this also how light acts when the initial velocity of the object it is reflecting off is not equal to 0?
So in my case, if the speed of light were 100mph (dummy math) and a tachyon was traveling at 110mph north that means the light reflecting off the tachyon would be traveling at 10mph north, so then really would you be able to see it?
More generally, how does relativistic addition of velocities work for tachyons?
update:
This question is a hypothetical question: IF tachyons exist, then what would happen? After a few hours of research I see why a usual massive object CAN'T travel faster than (or even reach) the speed of light, but this question is about tachyons.
| If you fire a beam of photons at an object receding away from you at a speed greater than the speed of light, your photons will never reach it to reflect off it
OTOH, if such an object emits photons, you should eventually be able to see the object as it was at the time the photon was emitted.
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Example in the book: A simple accelerometer A simple accelerometer
You tape one end of a piece of string to the ceiling light of your car and hang a key with mass m to the other end (Figure 5.7). A protractor taped to the light allows you to measure the angle the string makes with the vertical. Your friend drives the car while you make measurements. When the car has a constant acceleration with magnitude a toward the right, the string hangs at rest (relative to the car), making an angle $B$ with the vertical.
*
*(a) Derive an expression for the acceleration $a$ in terms of the mass m and the measured angle $B$.
*(b) In particular, what is a when $B$ = 45? When $B$ = 0?
I don't care about the answers, the important thing is the following:-
The book says The string and the key are at rest with respect to the car, but car, string, and key are all accelerating in the +x direction. Thus, there must be a horizontal component of force acting on the key.
That's the reason the book decided to consider a force in the $+x$ direction, but I'm looking for a better explanation: how would I find detect the force in the $+x$ direction in another way? To me, when I draw the free body diagram of the string, there looks to be no force acting on the $+x$ direction! I understand it starts with noticing that the string is attached to the ceiling of the car, and that the car has force causing acceleration in one direction, but I don't know how to go further than that.
| Did you forget to include the D'Alembert forces in the Free Body Diagram?
http://en.wikipedia.org/wiki/D'Alembert's_principle
| {
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Have red shifted photons lost energy and where did it go? I think the title says it. Did expansion of the universe steal the energy somehow?
| Energy isn't a nice concept in GR, so all I'm giving is an intuitive way of looking at it.
For gravitationally redshifted stuff:
A photon has energy, thus it gravitates (as energy can gravitate analogous to mass from $E=mc^2$), thus it has some (negative) gravitational potential energy when on the surface of a planet. If it's emitted, its GPE eventually becomes 0. So, this increase in GPE had to come from somewhere: the photon's redshift gave the energy. It's pretty much the same thing that happens when you throw a ball up. It loses kinetic energy (slows down).
The GPE in relativity is basically related to the energy stored in spacetime curvature; in a complicated way that I don't know.
For a normally redshifted photon from a moving body: Energy need not be conserved if you swith frames. Energy is different from each reference frame.
See the answers to the question provided by Qmechanic above as well. Over there, they're talking about the entire universe, though, which leads to additional issues.
| {
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Can you safely draw sparks from the nose of an electrified boy? From Purcell's Electricity and Magnetism
A spectacular conclusion of one of the popular exhibitions of the time
was likely to be the electrification of a boy suspended by many silk
threads from the rafters; his hair stood on end and sparks could be
drawn from the tip of his nose.
(pp 88 of second edition)
It sounds as if this wasn't something that hurt the boy in the act, but if there are sparks, there must be significant current running through his body, right? Were the sparks simply too small to cause serious damage, or is there a reason that drawing sparks from the boy doesn't affect his physiology?
| What is required for sparks across a gap is a high voltage, while what determines whether something will do you harm is the amount of current.
You can theoretically survive almost any voltage so long as the current is sufficiently low. It's just that most sources of electricity that have high voltages tend also to have high current.
| {
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Is sea water more conductive than pure water because "electrical current is transported by the ions in solution"? Apparently, electrical charge is transported by the ions dissolved in water, is this true?
| Yep. Pure water is an extremely bad conductor of electricity, it has very few ions. Water with an electrolyte (like NaCl) is a much better conductor of electricity; as the ions can migrate. Migration of ions is just like migration of electrons. If you place an imaginary surface inside the cell, there will be net negative charge crossing over to the positive terminal and vice versa. This is just like a current. Since there is net current inside, its conducting.
The equivalent conductance (a loco* chemistry concept) of a solution is simply the sum of the conductances of its constituent parts (Kohlrausch law). Here, $\Lambda$ denotes equivalent conductance of a portion of the solution, and $\lambda$ is the same for ions. Just a notation.
For pure water,
$$\Lambda_{H_2O}=\lambda_{H^+}+\lambda_{OH^-}$$
Now, since the concentrations of $H^+$/$OH^-$ are small ($10^{-7} M$ at STP), the $\lambda$s and thus $\Lambda_{H_2O}$ are pretty tiny.
For water with salt in it, we get
$$\Lambda_{soln}=\Lambda_{H_2O}+\Lambda_{NaCl}=\lambda_{H^+}+\lambda_{OH^-}+\lambda_{Na^+}+\lambda_{Cl^-}$$
Since $NaCl$ nearly dissociates completely, we get large $\lambda$s, and thus $\Lambda_{soln}$, which can be related to conductivity (in the aforementioned loco way).
So, pure/distilled water is an extremely bad conductor, while impure water with ions in it is a good conductor
* Loco because they assume a 1 m cell throughout, and don't keep the necessary $\text{m}^{-1}$ or whatever in their units. Due to this fixing of parameters, yes, we get that $\text{Area of plates}=\text{volume}$, which lets us relate it to concentration; but this gives us predictions for a specific case; when length of the cell is 1 m only. For some reasons these predictions are blindly applied to the general case. The whole thing gets confusing if you try to visualise it.
| {
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Prove that negative absolute temperatures are actually hotter than positive absolute temperatures Could someone provide me with a mathematical proof of why, a system with an absolute negative Kelvin temperature (such that of a spin system) is hotter than any system with a positive temperature (in the sense that if a negative-temperature system and a positive-temperature system come in contact, heat will flow from the negative- to the positive-temperature system).
| For the visually inclined, this article explains it simply. The maximum hotness definition is the middle image instead of the expected right image:
Due to the unintuitive definition of heat, a sample that only includes hot particles is negative kelvin / beyond infinite hot, and as clear from the image would give energy to colder particles.
| {
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What do the dimensions of circulation mean, and how is circulation related to action? The dimensions of circulation $\int_C \vec{v}\cdot d\vec{r}$ seem strange, but if you include
(even a constant) density $\rho$, then $\int_C \rho\vec{v}\cdot d\vec{r}$ has dimensions
the same as action/volume. Is there any significance to that? Is there any heuristic way to
think about circulation which helps understand the dimensions?
| The best way to gain intuition about circulation is to think of it as a measure of the number of vortex lines through a surface. By Stokes' theorem, $\int_C \vec{v} \cdot \vec{dr}$ = $\int_A (\vec{\nabla} \times \vec{v}) \cdot \hat{n} \, dA$. The quantity $\nabla \times \vec{v}$ called the vorticity, and you are threading the surface $A$ with it. From this perspective, the units of circulation make sense as [vorticity]$\times$[area].
Note the analogy you can make to Ampere's law in E&M. Vorticity is analogous to the electric current density, and velocity is analogous to the magnetic field.
Circulation is a useful concept in fluid dynamics primarily because it obeys a conservation law in the absence of viscosity (Kelvin's circulation thoerem). This conservation law is stated without reference to the fluid's density, and for this reason I don't think there is much physical significance to the quantity $\int_C \rho \, \vec{v} \cdot \vec{dr}$ or its units of action/volume.
EDIT: I thought about this some more, and now I realize that $\int_C \rho \, \vec{v} \cdot \vec{dr}$ is physically meaningful after all - it is the angular momentum density per volume. So instead of understanding the units via action as you had suggested, think about angular momentum instead. And the conservation law for circulation that I mentioned above (Kelvin's) can be interpreted as conservation of angular momentum for the fluid.
| {
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Are there more bosons or fermions in the universe? The question is in the title: are there more bosons or fermions in the universe? Or is there the same number of bosons and fermions?
I think there is the same number but I don't know why exactly.
| The question may be unanswerable because I'm pretty sure the number of photons is variable depending on your relativistic frame. At least that was an interesting observation that Feynman made, almost apologetically, in some of his early work; I'd have to look up the specific reference. (Hmm. Feynman's idea there would require variable-count photons to be generated in spin-cancelling pairs...)
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Is ground energy of interacting fermions always higher that that of bosons? Consider two systems, each made of $N$ particles. In both systems particles interact pairwise and the interaction is given by the same Hamiltonian for both systems. Any other constraints and/or requirements you'd like to add should be the same. Except for one -- the only difference between these systems is that the first particles are bosons and the second particles are fermions.
I'm interested in ground states of these systems. My intuition tells me that the ground state energy of bosons should always be lower than the ground state of fermions -- no matter what kind of interactions or other external properties we've chosen. But I cannot think of any reasonably general proof for that statement.
Maybe, someone knows how to prove that?
Or, maybe I'm wrong and there is a counterexample?
Edit: If you worry about spins of those particles, then you are free to make this difference too. I'll even give you all the following degeneracies, but the interaction shoud be the same -- spin-independent.
| The question makes sense in a nonrelativistic setting, where either symmetry or antisymmetry can be imposed on the wave function.
The symmetric ground state always has lower energy, as it is also a ground state of the non-symmetrized system. (Proof: The symmetrization of an arbitrary ground state is again a ground state.)
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Differentiating inside an integral sign I'm reading John Taylor's Classical Mechanics book and I'm at the part where he's deriving the Euler-Lagrange equation.
Here is the part of the derivation that I didn't follow:
I don't get how he goes from 6.9 to 6.10 by partial-differentiating the term inside the integral. If this is allowed, I was probably missed my calculus class the day it was covered. Can someone tell me more about this? Which part of calculus is this from?
| It's known as the Leibniz integral rule. As long as $\alpha$ is not the variable being integrated over, then $$\frac{\mathrm{d}}{\mathrm{d}\alpha}\int f(x,\alpha) \mathrm{d}x=\int\frac{\partial f(x,\alpha)}{\partial \alpha}\mathrm{d}x$$
$x$ will not be present outside the integral anyways (due to limits of the integral). As it is, while differentiating wrt $\alpha$, $x$ is constant. So it becomes a partial derivative inside.
You may want to check out the proof and more complicated forms (involving limits as functions of $\alpha$) on the linked wiki page.
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Ropes and Pulleys - Really unintuitive answer I usually don't want to do this, but please go to this link, the solution is too big to post it here
http://engineering.union.edu/~curreyj/MER-201_files/Exam2_2_26_09_Solution.pdf
And go to page 5 of the pdf.
Briefly, the problem say
Determine the velocity of the 60-lb block A if the
two blocks are released from rest and the 40-lb block B
moves 2 ft up the incline. The coefficient of kinetic friction
between both blocks and the inclined planes is $\mu_k$ = 0.10.
Things I am confused with the solution
1.First of all, I seriously thought lb was mass not force. after some googling, it turns out they are used interchangeably...
2.Where did they even get $2s_a + s_b = 0$ from? Why did they determine the change in distance this way? My first assumption was that if block B moved up 2ft, then block A should move down 2ft (the rope must "move" 2ft too right?). Then I wasn't sure, so I did a few triangles and found that the angle made a difference
3.Where did $2v_A = -v_B$ come from?
4 The FBD for block A is confusing, why is the friction force $F_A$ in the direction of the ropes? I thought it was block B that is going down? Am I the only one who had trouble deducting that the pulley and block A are the same object?
5.Look at the final answer, how could $v_b$ be negative? The problem says block B goes UP.
If you are wondering, this is not homework. I am just interested in this problem and it is out of curiosity and very confused with the concepts.
I've had at most introductory physics experience, but I think I should have been able to solve this still.
Thank you for reading
| 2 - the block 'A' is on a pulley, when it moves 1 ft the rope moves 2ft. It has a 2:1 mechanical advantage, normally you would use this to move block 'A' half the distance you pull the rope and with twice the force - the point of a pulley is to use a smaller force for a longer distance.
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Factors affecting torque and RPM of a motor I am not a physics guy, so not even the basic concept of a DC motor is easy for me. My question is as follows:
How do these parts of a motor affect its RPM and Torque?
I had my research a while ago so I filled out some of it; please correct if there is something wrong.
More turns - less RPM, more torque, less battery consumption
Less turns - more RPM, less torque, more battery consumption
More winds (number of wires) - less RPM, more torque, ?
Less winds (number of wires) - more RPM, less torque, ?
Thicker wire - ?, ?, ?
Thinner wire - ?, ?, ?
Stronger magnets - less RPM, more torque, ?
Weaker magnets - more RPM, less torque, ?
Bigger commutator - ?, ?, ?
Smaller commutator - ?, ?, ?
Also, if you know any tutorials for a guy who has the brain of a six-year old boy regarding a DC motor, please let me know. Thanks and apologies for my English!
Update:
I'm working with constant voltage 2.4v (2 Ni-MH AA Batteries)
| The RPM is restricted by frictional losses in the engine and the tendancy of the engine to explode if you rotate it too fast. In that sense the RPM limit is down to how well the engine is made and what it's made of rather than any fundamental EM properties.
The torque is dependant on how large a magnet field the engine generates, and this is mostly dependant on the current it draws and the number of turns in the coil. So the thickness of the wire has no effect except if it affects the resistance and therefore the current drawn. Likewise the commutator won't have any effect.
Have you had a look at http://en.wikipedia.org/wiki/Electric_motor#Performance. This gives a pretty thorough description and isn't too technical.
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Why the shape of rainbow is semicircular after rain why not the whole atmosphere is colorful? I have a very simple question.
Everyone must have seen the rainbow after rain. According to the theory the rainbow is created due to the passing of sunlight from small drops of water in the atmosphere(means by dispersion of light).
Now my what I want to know is that after rain the rain drops are present in the entire atmosphere. So the whole atmosphere should look colorful.
Why only a semicircular shape is formed (or is colorful).
| This is just a guess, as I have no formal training.
My guess would be that it is due to the spherical nature of the atmosphere the light is passing through. If you have a glass globe and shine a light through it the light is bent as it passes through and the pattern that is made on the object on the other side of the globe is circular (or semi-circular).
It's also a question of relative position. If you took a snapshot at every point that can view the phenomena and super-impose them over each other, you would find that in a 3-D model that the whole area IS colored. It's just that you can only see it from your singular point of view. It's like a light-bulb, when you look at it straight on it is one color, we will go with white. That same white light is being passed through the entire room, but you only see the photons that are traveling straight at you. If you could 'see' the photons as they were in an instant it would light (or in the case of your question, color) the entire view-able area...
Is this close?
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How can people do music with Tesla coils? I saw a lot of videos of Tesla coils doing music on YouTube. And I wonder how can they do that sort of things.
How they can calculate what tone it is going to do? And what are the factors to consider?
| In short, the Tesla coil is converting air into plasma, which changes its volume, which causes pressure waves to spread out in all directions (sound).
The sparks are created by a self-oscillating coil, which happens at a high, inaudible frequency (20 to 100 kHz).
The self-oscillating coil, though, is driven by pulses from a spark gap or lower-frequency oscillator, which is in the audible range. By adjusting the frequency of this pulse generator, the rate at which plasma bursts are generated can be varied, which varies the frequency of the sound produced.
I've seen this done in person and it's pretty awesome, but I've always wondered why they play only constant tones. I think they could use something like pulse density modulation to produce any arbitrary waveform and turn the coils into giant loudspeakers.
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Can black holes be created on a miniature scale? A black hole is so powerful to suck everything into itself. So is it possible that mini black holes can be created? If not then we could have actively disproved the rumors spread during LHC experiment.
| The search is on at the running LHC experiments for signatures of black holes from large extra dimensions. Despite what @Ronmaimon claims in his answer, experimentalists are not convinced that the probability of some of the models that expect large extra dimensions to be right is zero.
A search for microscopic black hole production and decay in pp collisions at a centerof-
mass energy of 7 TeV has been conducted by the CMS Collaboration at the LHC,
using a data sample corresponding to an integrated luminosity of 35 pb1. Events
with large total transverse energy are analyzed for the presence of multiple highenergy
jets, leptons, and photons, typical of a signal expected from a microscopic
black hole. Good agreement with the expected standard model backgrounds, dominated
by QCD multijet production, is observed for various final-state multiplicities.
Limits on the minimum black hole mass are set, in the range 3.5 – 4.5 TeV, for a variety
of parameters in a model with large extra dimensions, along with model-independent
limits on new physics in these final states. These are the first direct limits on black hole
production at a particle accelerator.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/22578",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Difference in timbre between 'quiet' and 'far away' I'd like to know what are the differences in timbre - or the acoustic properties of a sound - that allow us to differentiate between a sound which is quiet (but close-by) and one which is far away.
For example, you can tell when someone near to you is playing an instrument quietly even without looking to see where they are - they don't sound 'far away'.
Hearing a loud gig or a car stereo playing from the next street doesn't sound like it's quiet - it sounds loud, but far away.
But other times we can't differentiate - I sometimes hear a siren on TV and think it's on the street!
I thought only the amplitude (i.e. volume) of a sound wave diminished with distance - does the shape/frequency alter too?
Is this ability just to do with having two ears to locate the source - surely someone who is deaf in one ear can still tell an orchestra is playing a diminuendo and not gradually getting further away?!
| Mostly the effect is due to things other than "timbre", such as reverberation.
The one thing that does affect timbre is the fact that air absorbs higher frequency sound more than lower frequency sound.
http://www.sfu.ca/sonic-studio/handbook/Sound_Propagation.html
So as the waves travel to you, higher frequencies will be attenuated more than lower frequencies. This affects the timbre of the sound, and is part of the reason why loud sounds from very far away are almost entirely bass.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/22665",
"timestamp": "2023-03-29T00:00:00",
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Do neutron stars reflect light? The setup is very simple: you have a regular ($1.35$ to $2$ solar masses) evolved neutron star, and you shine plane electromagnetic waves on it with given $\lambda$. Very roughly, what shall be the total flux of absorbed/scattered EM radiation?
Shall the result change if the neutron star is young and not evolved?
| Martin, the infalling light is blue-shifted, and red-shifted on reflection. No overall change, I think.
However, a suitably mechanically strong light-source on the surface of the neutron star (!) will be seen to emit light that is redder than usual.
If the neutron star had its normal matter scraped off (left as an exercise for the student) then I don't see how light would interact at all with it. Hard gamma rays would be absorbed, but anything else? Nah.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/22722",
"timestamp": "2023-03-29T00:00:00",
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Why is the conductor an equipotential surface in electrostatics? Since the electric field inside a conductor is zero that means the potential is constant inside a conductor, which means the "inside" of a conductor is an equal potential region.
Why do books also conclude, that the surface is at the same potential as well?
| The change in potential between two points is
$$ \Delta V = \int_a^b \mathrm{d}\vec{\ell} \cdot \vec{E} $$
but inside the conductor $\vec{E} = 0$ so that integral between any two interior points is also zero, accordingly the interior is all at the same potential.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/22776",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Decoherence when no one is looking? I understand that in the single-electron-at-a-time double slit experiment, if a detector is placed before the slit, the interference pattern vanishes.
Suppose I left the detector on, but put a bag over its screen (I can't tell what state the electron is before it passes the slits), does the interference pattern come back?
If so, does that mean the electron "knows" I'm not "looking" and proceeds to interfere with itself?
Edit: I had photons passing through the slit initially. I've updated it to electrons to reflect a more common setup.
| In the following, I have replaced "photon" going through slits with "electron", and the measuring device with a "photon". This is the traditional Heisenberg setup.
The interaction of the photon and the electron entangles the photon and the electron, so that the electron cannot make the interference pattern. It has nothing to do with looking at the screen. The looking at the screen only serves to let you know what the result of the observation was.
The entanglement of measuring devices and particles is what causes collapse relative to a detector state, independent of interpretation. The only question is when the collapse turns into a definite outcome, so that the detector state becomes a definite thing, as opposed to an indefinite superposition.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/22824",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is there an explanation for the 3:2:1 ratio between the electron, up and down quark electric charges? I understand that the NNG formula relates $Q$, $I_3$, and $Y$ and can be derived in QCD; does this unambiguously predict the electric charge ratios without making assumptions about the definitions of isospin and hypercharge?
If so, this is unintuitive to me! Why does a particle carrying $SU(3)$ color charge care what charge it has under the totally separate electroweak
$U(1)\times SU(2)~$ symmetries?
If not, is there a name for the "problem" of explaining the charge ratios?
| There is a nontrivial relation between the electric charge and the strong business, namely that there are instantons which will cause proton decay. So it is not completely true that there are no relations--- the requirement of anomaly cancellation requires that the proton decay process conserve charge, and so relates the total charge on the proton to the total charge on the electron.
The U(1) numbers are completely crazy. The only sensible explanation is that they come from an SU(5) GUT (or SO(10) or E6 or some higher version of the SU(5) idea). The reduction of charges from SU(5) is explained in this answer:
Is there a concise-but-thorough statement of the Standard Model?
This gives the 1,2,3,6 ratios of the hypercharge assignments in nature, and completely explains the crazy quark charges. It is also an automatic way of ensuring anomaly cancellation. This, and approximate coupling constant unification, are the two strongest bits of evidence for a GUT at a scale of $10^16$ GeV or thereabouts.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/22921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Can one introduce magnetic monopoles without Dirac strings? To introduce magnetic monopoles in Maxwell equations, Dirac uses special strings, that are singularities in space, allowing potentials to be gauge potentials. A consequence of this is the quantization of charge.
Okay, it looks great. But is this the only way to introduce magnetic monopoles?
| Below it is explained how a string free monopole theory can be constructed.
Monopoles have not been found in experiments. Therefore the main issue to be defined
are the principles used as cornerstones for the required monopole theory.
The first step is to formulate a theoretical definition of monopoles. This is done by the well known duality transformation. Applying this transformation to the Maxwellian system of electric charges and electromagnetic fields, one obtains a Maxellian-like system of monopoles and electromagnetic fields (without charges). The next step is to construct a unified charge-monopole theory.
Let us examine two postulates that pertain to the required charge-monopole theory:
(A) For chargeless systems the unified theory must take a form which is completely dual to the theory of charges and fields and for systems without monopoles it must take the form of Maxwellian electrodynamics.
(B) Electromagnetic fields of a system of monopoles and those of a system of charges have identical dynamical properties.
One may be tempted to use both postulates (A) and (B) as fundamental elements of the theory. However, it turns out that this course is unattainable because different sets of equations of motion are obtained from postulate (A) (without (B)) and from postulate (B) (without(A)).
It can be proved that postulate (A) yields a string-free charge-monopole theory which is consistent with the variational principle and postulate (B) yields Dirac's strings.
Information on scientific articles that discuss these issues can be found in the page pointed out by the following link.
http://www.tau.ac.il/~elicomay/mono.html
E. Comay
| {
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