Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Measure density with the help of buoyancy I am trying to derive a formula to calculate the density of a irregulary shaped object.
I can measure the (false) weight of the object in pure air (of known density), and the (false) weight of the object in water (of known density).
I cannot measure (directly) the volume of the... | The mass/volume terms can be cancelled out.
You know $G_{water}=G_w$ and $G_{air}=G_a$, $\rho_w$, $\rho_a$, and want to know $\rho$.
$$G_w=mg-\rho_w V g= Vg(\rho-\rho_w)$$
$$G_a=mg-\rho_a V g= Vg(\rho-\rho_a)$$
Dividing, $$\frac{G_w}{G_a}=\frac{\rho-\rho_w}{\rho-\rho_a}$$
You can rearrange the terms after this to get $... | {
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Should the electric potential of a positively charged sphere be negative? Because the indefinite integral of the electric field results in a negative value? (As the function is proportional to $r^{-2}$?
I've got to be missing something... Help please!! Thanks!
Also, I took the integral from infinity to the radius of th... | $V_f - V_i = \int_i^f \vec{E} \cdot d \vec{r}$. The dot product has a sign depending on the relative orientation between the electric field $\vec{E}$ and infinitesimal displacement $d \vec{r}$. Also note that as you move radially inwards from infinity to some point, the displacement $d \vec{r} = \vec{r_f} - \vec{r_i}$ ... | {
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Will we feel right up inside the airplane if the airplane accelerates toward earth at 20m/s^2? Suppose you are in the airplane and the airplane falls toward the earth with acceleration of $20m/s^2$(double of gravitational acceleration $g$). This double acceleration by airplane will cancel the gravitational acceleration... | With respect to the aeroplane, the acceleration will be g upward(against the earth's gravitational pull). However a person in the plane will feel weightless (till he strikes the ceiling) as there is no normal reaction of the aeroplane acting on him. There is only the gravitational force exerted on him by the earth. He ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Why is $dL = L d\epsilon$? Let's say there's a random elastic material. It's length is $L$ and it's tensile strain $\epsilon= (L-L_0)/L_0$
Now, when one pulls on it the following is true:
$dW_{tot}=FdL =\sigma AdL=\sigma A L d\epsilon$
Why can we make that last step or in other words: why is $dL = L d\epsilon$?
| From the above definition for tensile strain, $L$ is $L=\epsilon L_{0}+L_{0}$. Thus
$dL=L_{0}d\epsilon$
replacing in the above $L_{0}$ with $L_{0}=\frac{L}{1+\epsilon}$ you get
$dL=\frac{L}{1+\epsilon}d\epsilon$
Now, if the deformation of the material is very small you can expand $\frac{1}{1+\epsilon}$ as
$\frac{1}{1+\... | {
"language": "en",
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Sign convention for basic Dirac equation The dirac equation;$$(i\gamma^\mu\partial_{\mu} - m)\psi=0 $$ is just; $$(i\gamma^{0}\partial_{0} - i\gamma^{i}\partial_{i} - m)\psi=0 $$ in a (+,---) metric right?
| The expression $A^{\mu}B_{\mu}$ simply means that
$$A^{\mu}B_{\mu}=A^{0}B_{0}+A^{1}B_{1}+A^{2}B_{2}+A^{3}B_{3}$$
Using the Minkowski metric with signature $(+---)$ you write this as
$$A^{\mu}B_{\mu}=A^{\mu}\eta_{\mu\nu}B^{\nu}=A^{0}B^{0}-A^{1}B^{1}-A^{2}B^{2}-A^{3}B^{3}$$
The metric simply tells you have how the compon... | {
"language": "en",
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QFT in Quantum Computing and Control Theory? Is QFT being applied to quantum computing and control theory?
I took yesteryear a basic course on quantum computing and if I remember correctly we didn't touch on any QFT (though I think that if it were a course with prequisites of QFT, not a lot of people would have attende... | "Quantum field theory" is more or less the universally spoken language of modern quantum physics at the research level, regardless of the energy scale. There appears to be widespread confusion that it is somehow a fundamentally different theory than "ordinary quantum mechanics", but this is not so. It is a compact nota... | {
"language": "en",
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Does the Higgs mechanism address the spin statistics problem? Since the Higgs mechanism is so intimately tied to binding together massless chiral fermions, does it happen to have anything to say about the spin statistics issue?
I'm actually assuming the answer is no, but perhaps there is a deeper tie there that I'm mis... | The spin-statistics thing isn't a problem, it is a theorem (a demonstrably valid proposition), and it shouldn't be addressed, it should be understood and celebrated.
The Higgs field gives us interactions between chiral fermions and the Higgs, $yh\cdot \chi_\alpha\eta^\alpha$ which produces mass terms $m \chi_\alpha\eta... | {
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A simple model that exhibits emergent symmetry? In a previous question Emergent symmetries I asked, Prof.Luboš Motl said that emergent symmetries are never exact. But I wonder whether the following example is an counterexample that has exact emergent spin rotational symmetry.
Just consider the simplest Ising model for... | I think the simplest example is very closely related to your suggestion of the two-site Ising model. Instead, consider the two-site XX-chain:
$$ H = \sigma_1^x \sigma_2^x + \sigma_1^y \sigma_2^y. $$
Clearly the Hamiltonian has $U(1)$ symmetry (generated by $\sigma^z_1 + \sigma^z_2$) but it does NOT have full $SU(2)$ sy... | {
"language": "en",
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Newtonian Mechanics and Quantum mechanics Why isn't Newtonian mechanics valid in Quantum world? Suppose you isolate an alpha particle and accelerate it in absolute vacuum. Why it doesn't follow the equation $F=ma$? If Newtonian mechanics is invalid in quantum world, what is the guarantee that Quantum mechanics is valid... | It seems to me that the answers given to the question that has been posed are not correct at all and do not address the question: Is quantum mechanics valid in in the classical limit? The answer is NO. The example of sending an electron in a vacuumtube is actually a very good example. It turns out that the differential... | {
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Is the movement of electrons truly random? The result of rolling dice is considered pseudo-random because it depends on an almost endless list of factors (how you roll it, the terrain it lands on, etc.), but it is not TRULY random. Is the movement of electrons TRULY random or just pseudo-random? If it is just pseudo-ra... | The result of rolling dice is not pseudo-random because the dice is interacting with surrounds, and this is a LPS with Poincaré resonances. As a consequence of the existence of Poincaré resonances, the classical system is unpredictable even if the initial state is known with complete precision.
The same about electrons... | {
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Work as an integral of mass over velocity? As I've understood it, the area under $F$-$s$-graph is the work done, so then :$$W(s)=\int{F(s)ds}$$
I am also given this equation ($W_k$ is kinetic energy, which is equal the work done to set the object in motion):
$$W_k=\frac{1}{2}mv^2$$
Does this mean that work is also the ... | What is the net force? Newton:
$F_{net}=ma=m\frac{dv}{dt}$
So the net work done to accelerate a particle from $v_0=0$ to final velocity $v_f$ is
$W_{net}=\int F_{net}ds=\int m\frac{dv}{dt}ds=m \int_{v_0}^{v_f} v dv=\frac{1}{2}mv_f^2$
where in the last step I used $ds/dt=v$ and $m$ constant. If you had some crazy system... | {
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"as measured in a local Lorentz frame"? I've seen the phrase "as measured in a local Lorentz frame" tagged on the end of so many sentences.
What does it mean precisely?
To give an explanation with an example, consider the context of measuring the speed of an object, if this applies.
| You consider processes where you have no geodesic deviation over the whole area, in the appropriate time frame.
Notice that in every point $p$ of your pseudo-Riemannian manifold, you can find a coordinate system such that $g(x)=\eta+\mathcal{O}(x^2)$. Mathematically, in any neightborhood $U_p$ around $p$, you can't tur... | {
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Deriving torque from Euler-Lagrange equation How could you derive an equation for the torque on a rotating (but not translating) rigid body from the Euler-Lagrange equation? As far as I know from my first class in Classical Mechanics, there is no potential defined for a rotating body, so the only term I see in the Lag... | The question isn't clear to me, the torque is zero if there's not an applied torque, and it's nonzero if it's not zero, but that's just a tautology. Do you mean that you just want to derive something/anything torque-esque given an applied force? If so, here's one way.
This isn't a job for a potential, it's the job for ... | {
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Magnetostatics of Current-Carrying wire A question has been nagging at me about Faraday's Law as related to a wire with a constant current:
If you have a circular loop of wire with some small resistivity, connected to a battery so that it has a constant DC current, the electric field inside the wire is going to be long... | If you apply Faraday's law here, the net emf around the loop is zero (as expected). You forget that there is a battery in the loop, which has a large, opposite electric field across it. So $\varepsilon=\varepsilon_{loop}+\varepsilon_{battery}=0$. Which means that $-\frac{d\Phi_B}{dt}=0$, so $B=constant$, which means th... | {
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Photon on null geodesic If given an FRW metric
$ds^2 = -dt^2 + a^2(t)[dx^2+dy^2+dz^2]$
and for the trajectory followed by a photon (null geodesic; $ds^2=0$) with affine parameter $\lambda$, know that
$g_{\mu\nu}\,\frac{dx^{\mu}}{d\lambda}\,\frac{dx^{\nu}}{d\lambda} = 0.$
How does one find
$\frac{dt}{d\lambda} = \frac... | I'll use dots for derivative with respect to affine parameter. The FRW metric has Killing vectors $\partial_x, \partial_y, \partial_z$ each of which leads to a conservation equation:
\begin{align}
c_x &= \dot x\cdot\partial_x = a^2\dot x \\
c_y &= \dot y\cdot\partial_y = a^2\dot y \\
c_z &= \dot z\cdot\partial_z... | {
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A charged sphere with pulsing radius
Radius increases and decreases periodically (as a pulse).And so does the charges on the surface of sphere.
I can't get what is gonna happen.the EM waves are produced perpendicularly to motion of the charges,but here where is the perpendicular ? Will the EM waves get produced into ... | RADIATION FROM A PULSATING ELECTRICALLY CHARGED SPHERE
Here is a modest approach to find an answer to this problem which I hope will generate some discussion in the right direction, and someone will enrich and extend.
There is no point in rediscovering the wheel, so we can use some well known results from classical ele... | {
"language": "en",
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What is the speed of acceleration of the inflation of the universe? Is the inflation speed of the universe accelerating or is it a constant speed of expansion proportional to distance between objects.
| See At what speed does our universe expand? and Speed of Universe Expansion for related questions, but I take your question to be specifically asking if the rate of expansion is increasing. The answer is that the expansion rate is increasing, and this was measured experimentally in 1998 by Perlmutter and Riese's group... | {
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Reducing deposition on power supply wires From my house there are two wires, i.e. a live and a ground wire which are placed on the power supply lines by a hook. They are not tightly jointed because of some reasons. These two wires are also in contact with air. So, after a few days some material gets deposited on the wi... | If the joint is not tight, there can be arcing which will damage the metal surfaces.
If the joint is not airtight, most metals will corrode, for some metals, their oxides are less conductive.
If two dissimilar metals are in contact in a wet environment there can be an electrochemical corrosive reaction.
You need a tig... | {
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Units and Dimensions - Use of proportionality constant
In units and dimensions we learn about Establishing a Formula :
(example) : to establish a relationship between T (Time Period) , m (Mass) , l (length of the string) and g(acc. due to gravity) - Case of Pendulum :
$$T \propto m ,\quad T \propto l ,\quad T \pr... | The only way to get the constant of proportionality is to solve the equations of motion for your system, or do an experiment to get an approximate value.
The period of the pendulum is frequently used to teach dimensional analysis because it is simple to understand and works well. However in real life physicists don't o... | {
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How Do Electrons "Resonate Against the Restoring Force Of Positive Nuclei?" While reading "Surface Plasmon Resonance," I came across the following:
"The resonance condition is established when the frequency of light photons matches the natural frequency of surface electrons oscillating against the restoring force of p... | The effective potential that a valence electron of a large atom sees vaguely resembles a harmonic oscillator. The "restoring force" is just the valence electron's attraction to the positively charged parent ion.
| {
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How old is SUN ☉? How do we know/calculate the exact age of sun ☉ ? ie. 4.57 billion years. What is the way to calculate it?
| I'm not familiar with nucleocosmochronology (mentioned by Gugg), but traditionally the age of the Solar System, and therefore of the Sun, has been estimated by dating meteorites using uranium decay.
The material that makes up the Solar System was formed in supernovae, and this forms the two isotopes of uranium, U-238 a... | {
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How do Doppler Effect and Time Dilation differ?
Above, I have drawn a diagram showing Doppler Effect (here we are using space-time but in a non-relativistic sense. Time and distance are the same for A and B).
Edit: I am adding a relativistic space-time diagram below this with
lines of simultaneity drawn. I am also ... | I have given this question much thought. I am providing a link to what I think is the answer, as putting the answer here would be too long.
Please note that I am new to physics so my thinking may be wrong. I would appreciate your comments on whether you think this idea is correct or not.
Thanks!
Link to analysis:
https... | {
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The rule breaker, emissivity + reflectivity = 1 If emissivity and reflectivity are inversely proportionate, why does glass have a high emissivity of around 0.95-0.97 as well as being very reflective for IR Radiation?
normally it works but not with glass!
Can anyone explain this?
| In an isothermal steady state condition, meaning when the temperature is uniform and not changing with time,
%Reflected + %Transmitted + % Absorbed = 100%
For opaque system,
%Reflected + % Absorbed = 100% .........(1)
Now, if the object absorbs infrared radiation, its energy (and thus temperature) will increase, but ... | {
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What prevents bosons from occupying the same location? The Pauli exclusion principle states that no two fermions can share identical quantum states. Bosons, one the other hand, face no such prohibition. This allows multiple bosons to essentially occupy the same space, a phenomenon that has been theorized responsible ... | Er ... nothing prevents this. That's what a Bose-Einstein condensate is: lots of bosons in the same place and quantum state.
You are observing that the state is not perfectly localized, but that is a consequence of the state not being exactly zero momentum. Ultimately the Heisenberg principle puts a lower limit on how ... | {
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Is there a symbol for "unitless"? I'm making a table where columns are labelled with the property and the units it's measured in:
Length (m) |||| Force (N) |||| Safety Factor (unitless) ||| etc...
I'd like not to write "unitless" on several columns...and I'm quite surprised I can't seem to find a symbol for it. Any ... | I've seen "(1)" used. Radians (and steradians) are also "unitless" but they're clearly not appropriate here.
| {
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How do you actually define an orbit? How do you actually define an orbit?
I believe, Newtonian Mechanics describes an orbit as one object in free fall around another where projectile paths become elliptical. I think, Einstein describes an orbit as an object taking the shortest distance through curved space. And in Qua... | You could generalize to something like
The relative motion of two bodies under a mutual interaction with fixed angular momentum and a lower bound on the radial separation.
If you don't feel the need to examine asymptotically free interactions you could add an upper bound on the separation as well, but historically ma... | {
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Why pulse waves results in wave packets? I was doing experiments of measuring sonic velocity and I generate pulse waves from sensor 1, but when they are received by sensor 2, I saw wave packets on the oscilloscope, can you explain why?
I was told that it might be related to Fourier transform but I don't understand.
| Yes this is related to Fourier transforms. If you have a signal $s(t)$ that you measure over a time window $\Delta t$, then you will have a resulting dispersion in the frequencies i.e. a $\tilde{s}(\omega)$ that differs from a delta function for a purely harmonic signal. The "physical" reason for this is that for your ... | {
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How does the freezing temperature of water vary with respect to pressure? I know when the pressure is reduced, the boiling temperature of water is reduced as well. But how does the pressure affect the freezing point of water?
In a low-pressure environment, is water's freezing temperature higher or lower than $0\sideset... | Here is an interesting article that shows how water was frozen at high temperature under pressure
http://www.azom.com/news.aspx?newsID=8016
Here is an extract
Sandia Convert Water to Ice in Nanoseconds
Published on March 19, 2007 at 1:15 AM
Sandia’s huge Z machine, which generates temperatures hotter than the sun, has... | {
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Electron Spin Resonance and Free electrons When performing an experiment to observe electron spin resonance, we use DPPH molecules as they contain an unpaired electron on one of the N atoms.
My question is, why cant free electrons be used in this experiment? What is wrong with a beam of electrons or a metal?
Any help ... | The physical issue you are interested in is the Zeeman effect lifting the degeneration of energy states. For that you need an unpaired electron that is tightly bound in a system.
A free electron is... free. You cannot see energy state splitting if there are no quantized energies. (See free electron Laser for something ... | {
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Difference between slanted indices on a tensor In my class, there is no distinction made between,
$$
C_{ab}{}^{b}
$$
and
$$
C^{b}{}_{ab}.
$$
All I know, and read about so far, is the distinction of covariant and contravariant, form/vector, etc. etc. But what is this slanted business all about?
| An easy way to see that they are distinct is to consider what happens upon raising (or lowering) all indices.
For example, upon lowering,
$$
T_{ab}{}^{cde}
$$
becomes $T_{abcde}$, whereas
$$
T_{a}{}^{cd}{}_{b}{}^{e}
$$
becomes $T_{acdbe}$, and similarly
$$
T_{a}{}^{cde}{}_{b}
$$
becomes
$$
T_{acdeb}.
$$
You need to "sl... | {
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How to express a Hamiltonian operator as a matrix Suppose we have Hamiltonian on $\mathbb{C}^2$
$$H=\hbar(W+\sqrt2(A^{\dagger}+A))$$
We also know $AA^{\dagger}=A^{\dagger}A-1$ and $A^2=0$, letting $W=A^{\dagger}A$
How can we express $H$ as $H=\hbar \Big(\begin{matrix}
0 & \sqrt2 \\
\sqrt2 & 1
\end{matrix} \Big)$
S... | As @MichaelBrown has pointed out in the answer, to get the matrix element you just have to sandwich the operator between two states. So in the case of your Hamiltonian $H$, the matrix elements are given as
$$H_{ij} = \langle i|H|j \rangle $$
I should point out that the $i$'s that you use should be the basis set that y... | {
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Why isn't this capacitor charging? Let's say you have a parallel plate capacitor and you connect one plate to the positive terminal of a battery and the other plate to the negative end. So this is like a static situation, you have a large positive charge (the cathode in the battery, which keeps recharging)) connected t... | Well, suppose the charges equalize so half the charge of the cathode goes to the plate connected to it. What happens next? The redox reaction inside the battery charges the cathode back up to full value (the value needed for electric field inside the battery to balance the chemical gradient). But it's connected to that... | {
"language": "en",
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Mass-energy equivalence and Newton's Second Law of motion According to Einstein's Mass-energy equivalence,
$ E = mc^2$ OR
$ m = \frac E{c^2}$..... (1)
and According to Newton's Second Law of motion,
$ F = ma$ OR $m = \frac Fa$ ..... (2)
If we compare eq. (1) and eq. (2), we obtain;
$\frac E{c^2} = \frac Fa$..... (3)
I... | Your symbolic manipulations are correct, but the relations you write down do not properly describe Newton's second law in the context of special relativity.
In the context of special relativity, the relativistic momentum of a particle is defined as
$$
\mathbf p = \gamma m \mathbf v, \qquad \gamma = (1-\mathbf v^2/c^2... | {
"language": "en",
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Is it possible to "add cold" or to "add heat" to systems?
Amanda just poured herself a cup of hot coffee to get her day started.
She took her first sip and nearly burned her tongue. Since she didn't
have much time to sit and wait for it to cool down, she put an ice
cube in her coffee and stirred it with a metal... | Yes. The fluid is in contact with the spoon, and because the former is hotter than the latter, heat is transferred, resulting in the spoon getting warmer. About the ice cube: technically "adding cold" does not make much sense. The ice cube will melt due to heat transfer from the hotter to the colder medium, a process w... | {
"language": "en",
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Why does a temperature increase on a fixed volume increase entropy? I heard that this statement is correct. However, it seems odd to me. The number of possible microstates is still the same, so isn't the entropy constant?
| The number of microstates that may represent a macroscopic configuration behaves like
$$ N\sim\exp(S/k)$$
so it is not constant if the entropy increases. What you're probably missing is that the microstates are defined not only by positions but by velocities (or momenta), too.
Take $N$ molecules of a gas. They may be s... | {
"language": "en",
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Classical results proved using quantum mechanics Are there any results in classical mechanics that are easier to show by deriving a corresponding result in quantum mechanics and then taking the limit as $\hbar\rightarrow0$?
(Are there classical results that were first discovered through taking the classical limit of qu... | It frequently happens in classical relativity that it's much easier to carry out certain reasoning by talking about photons.
An example of some historical interest is in Einstein's 1905 paper on SR, section 8, "Transformation of the Energy of Light Rays. Theory of the Pressure of Radiation Exerted on Perfect Reflectors... | {
"language": "en",
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Why cosmic background radiation is not ether? why cosmic background radiation is not ether? I mean it's everywhere and it' a radiation then we can measure Doppler effect by moving with a velocity.
| Why cosmic background is no ether is not a question it is a statement.
A question would be is the cosmic background an ether?
For one thing it is an inertial system.
This inertial system can be found by anyone in the universe.
One can find(calculate) his or her velocity to it.
Since a inertial system is sufficiant to d... | {
"language": "en",
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If photons can be absorbed by electrons, wouldn't that mean light has a charge? I am a biochemistry and molecular biology major. If photons can be absorbed by electrons, wouldn't that mean light has a charge? Electrons only attract positive charges. Isn't it?
| The simple answer is "no". Absorption and attraction are two different things.
The simplest "absorption" of part of the photon's energy comes in Compton Scattering: A photon scatters on an electron and gives up part of its energy, changing wavelength in the meantime.
Scattering means two trajectories meeting and an ex... | {
"language": "en",
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Analytical solution of two-level system driving by a sinusoidal potential beyond rotating wave approximation A quantum mechanical two-level system driven by a constant sinusoidal external potential is very useful in varies areas of physics. Although the widely used rotating-wave approximation (RWA) is very successful i... | You could use the Floquet theory to go beyond the RWA.
See the paper by Shirley[J. H. Shirley, Phys. Rev. B 138, 974 (1965)].
If the amplitude of the driving field isn't too strong, you could use
a simpler perturbative method such as the
averaging method to second-order. Look here at this manuscript: Rabi oscillations ... | {
"language": "en",
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Physical Interpretation of the Integrand of the Feynman Path Integral In quantum mechanics, we think of the Feynman Path Integral
$\int{D[x] e^{\frac{i}{\hbar}S}}$ (where $S$ is the classical action)
as a probability amplitude (propagator) for getting from $x_1$ to $x_2$ in some time $T$. We interpret the expression $... | "It's certainly not a probability amplitude of any sort because it's modulus squared is one." This does not follow... Anyway, an (infinite) normalisation factor is hidden away in the measure. The exponential has the interpretation of an unnormalised probability amplitude. Typically you don't have to worry about the nor... | {
"language": "en",
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Why does in string theory the amount of supersymmetry have to be $\cal{N} \leq 2 $? Why is this, that in string theory the maximum amount of supersymmetry is $\cal{N} = 2$, whereas in supergravity one can have up to $\cal{N} = 8$ ?
| For higher numbers of supersymmetries on the sigma model's world sheet, the target space dimension becomes negative. (Proof is a boring computation.) It's not particularly clear how to interpret this -- e.g., should the theory have a supergravity limit? -- so these theories are typically discarded from the classifica... | {
"language": "en",
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What would happen to the Moon if Earth is turned into a black hole? Assume that all of sudden the Earth is turned into a black hole. And the moon revolves around the Earth (before turning into a black hole). What would happen to the Moon after earth changes to black hole will it be sucked to the black hole or continue ... | Simply, the moon remains in its orbit. It is a common misconception that black holes always "suck" which is quite wrong. They're just like any other celestial body - round, held by gravity, spinning, etc. Only because they're compressed to a much small size, they (in rough words) don't let go of the light (nor any othe... | {
"language": "en",
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Why must the angular part of the Schrodinger Equation be an eigenfunction of L^2? I was reading about the solution to the Schrodinger Equation in spherical coordinates with a radially symmetric potential, $V(r)$, and the book split the wavefunction into two parts: an angular part and a radial part. When dealing with t... | It has to do with the Laplacian $(\nabla^2)$.
When trying a separable solution
$$\psi(r,\theta,\phi) = R(r)\Theta(\theta)\Phi(\phi)$$
You will get two ODE's, a radial and an angular equation. They must be equal to separation constants. Turns out the angular part is the $L^2$ operator. You can rewrite
$$H \psi = E \psi... | {
"language": "en",
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Introductory examples of AdS/CFT duality I would like to know, what are the simplest/starting/basic examples that are typically used to introduce students to how AdS/CFT really works? (not the MAGOO paper, as I am not sure it has concrete examples that a beginning student can work through)
For example, are the PhD. pa... | One review I found helpful relates Type IIB superstrings on a maximally supersymmetric plane wave background to N=4 SYM:
Lectures on the Plane–Wave String/Gauge Theory Duality by Jan Christoph Plefka
It might be more conversational than what you're looking for, but this review by Polchinski has some great discussion an... | {
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Two masses attached to a spring I'm trying to understand the solution of the following problem.
Two masses $m_{1}$ and $m_{2}$ slide freely in a horizontal frictionless track and are connected by a spring whose force constant is $K$. Find the the frequency of oscilatory motion for this system.
In the solutions manual... | You really need to start with a bit of intuition about what is going on not just regurgitate equations. Try to imagine in your head those 2 masses oscillating. First imagine if the masses are equal. Then if unequal. YES of course the center of mass is totally important. Assume first the center of mass is not moving, ju... | {
"language": "en",
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Quantum Mechanics - Hidden Variables In Steven Weinberg's Lecture on Quantum Mechanics (p. 342), he writes:
The correlation between the spins of the two particles can be
expressed as the average value of the product of the $\hat{a}$
component of the spin of particle 1 and the $\hat{b}$ component of the
spin of p... | You should read Bell's original paper, where he constructs such a model explicitly. The key point is that for any single pair of directions one can construct such a model. Bell's theorem states that there is no single local hidden variable model that can reproduce the quantum mechanical correlation for every pair of d... | {
"language": "en",
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What is metric of spherical coordinates $(t,r,\theta,\phi)$? In spherical coordinates the flat space-time metric takes:
$$ds^2=-c^2dt^2+dr^2+r^2d\Omega^2$$
where $r^2d\Omega^2$ come from when the signature of metric $g_{\mu\nu}$ is (-,+,+,+)?
what is signature of spherical metric?
this is signature of spherical coord... | The trickiness is what you mean by a spherical metric. What you've written down is the metric of flat space in spherical coordinates, which can be thought of as a warped product of the flat minkowskian two space $(t,r)$ with the unit sphere. This space is equivalent to the normal $(t,x,y,z)$ coordinates of standard s... | {
"language": "en",
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Perfect fluids in cosmology? In cosmology, it is often assumed that the equation of state of a cosmological fluid is of the form $p=w\rho$. Why is this? Is it the equation of a perfect fluid?
Why does $w=0$ for matter $1/3$ for radiation and $-1$ for a cosmological constant?
| The Wikipedia article on the equation of state discusses this.
Matter
Consider non-relativistic matter first, because this is easy. We write the pressure as:
$$ p = \rho_m \space RT = \rho_m \space C^2 $$
where $\rho_m$ is the mass density, $RT$ is the thermal energy of the matter (or whatever) in your fluid, and $C$ i... | {
"language": "en",
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When does Thevenin's theorem not apply (modelling a power source with a ohmic internal resistance) Most physics text books say that a power source can be modelled as an EMF with a internal resistance. This is also know as Thevenin's theorem or Norton's theorem. However I have read in some sources that this is not alway... | There are actually two slightly different versions of Thevenin's theorem. I think what you are describing is the weaker of the two: you can replace any circuit with a single voltage/current source and a single resistor. That version holds for any two-terminal network made up only of voltage/current sources and ohmic re... | {
"language": "en",
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Inserting metal into parallel plate capacitor
The plates of an isolated parallel plate capacitor with a
capacitance C carry a charge Q. The plate separation is d.
Initially, the space between the plates contains only air.
Then, an isolated metal sheet of thickness 0.5d is inserted
between, but not touching, th... | If there is a charge $Q$ and $-Q$ on each plate of the capacitor , when you insert a perfect conductor between the plates (parallel), you simply will have a charge $+Q$ on one side(facing negative plate of capacitor) and $-Q$ on other side of the inserted plate. (Given that the area of plates is large enough to assume... | {
"language": "en",
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Alpha particle and helium nucleus The symbol for the alpha particle is α or $α^{2+}$, it can be written as $He^{2+}$.
What I want to know is that, are they same? I mean alpha particle and helium nucleus are same or any subtle difference exists?
| They are exactly the same, with the different notations arising in different contexts. You could start with a bunch of helium gas and heat it up or shine UV light on it to turn it into a plasma, and then you'd probably say you have $\mathrm{He}^{2+}$ (or $\mathrm{He}\ \mathrm{III}$ if you are an astronomer). The symbol... | {
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How can Ohm's law be correct if superconductors have 0 resistivity? Ohm's law states that the relationship between current ( I ) voltage ( V ) and resistance ( R ) is
$$I = \frac{V}{R}$$
However superconductors cause the resistance of a material to go to zero, and as I understand it, as $R \to 0$, $I \to \infty$. Does ... | No, because the voltage drop across the device also goes to zero.
| {
"language": "en",
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Rolling sound of thunder
When sound of thunder travels to the listener from its location, different parts of the same wave have to travel through air of different moisture levels, densities, and molar masses.
This is what I found when I wanted to know about the reason for the rolling sound of thunder (the sound which... | Thunder
Thunder is the sound caused by lightning. Depending on the nature of the lightning and distance of the listener, thunder can range from a sharp, loud crack to a long, low rumble (brontide). The sudden increase in pressure and temperature from lightning produces rapid expansion of the air surrounding and withi... | {
"language": "en",
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Can false origin be anywhere in a graph? Can I start labeling my $x$ and $y$ axes from non-zero values when drawing a graph? Or is there any convention to only label $x$-axis from non-zero value when using a false origin and not $y$-axis? Please clarify this.
| According to the blog Jacobs Physics: Resources for teachers and students of introductory physics:
It's perfectly acceptable, and sometimes desirable, not to begin an axis at zero.
And:
Students will attempt to demand a hard-and-fast rule about scaling graphs from the origin, but such a rule does not exist. The sca... | {
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What defines the adiabatic flame temperature? What defines the adiabatic flame temperature?
In a case I have to solve, I need to describe the combustion of natural gas (Groningen natural gas, to be specific). However, I am having some problems understanding the adiabatic flame temperature, since on the internet I find ... | Your understanding about adiabatic flame temperature is correct conceptually, though the mathematical representation of adiabatic flame temperature you have given is just an approximation which again is sufficiently accurate for most of the practical purposes. You can make it more accurate by expressing the specific he... | {
"language": "en",
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Are gravitational time dilation and the time dilation in special relativity independent? There are two kinds of time dilation:
*
*One because the other clock moves fast relative to me (special relativity).
*Another one because the other clock is in a stronger gravitational field (general relativity), or acceleratin... | From an engineer: I interpreted the answer from the practical point of view of how to combine effects from velocity-related time dilation and gravity-related time dilation. They can be treated independently, and then either multiple the factors together or, since the factors are usually very close to 1, you can add (... | {
"language": "en",
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How to determine the direction of medium's displacement vectors of a standing wave? Consider the following problem taken from a problem booklet. My questions are:
*
*What is displacement vector?
*And how to determine the direction of displacement vector at a certain point?
*Where is the position with zero displace... | Any material between two nodes is displaced by the same direction. So the direction of B and C has to be the same as well as the direction of A and D due to symmetry. In addition, the direction of A must be the opposite of B since they are across from a node. Similarly the direction of C and D must be opposite.
So the ... | {
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How is the distance to a $\gamma \mathrm{-ray}$ burst (GRB) measured in just a few days? Recently the Fermi Gamma-ray Space Telescope recorded the most energetic Gamma Ray burst (GRB 130427A) yet observed with a peak $\gamma \mathrm{-ray}$ energy of $94\, \mathrm{GeV}$. Various sources have reported that the burst was... | Gamma Ray Bursts are a special case of Supernovas. Supernovas mostly happen in areas of heavy star formation, which means that they happen inside a galaxy and are enveloped in or obscured by clouds of hydrogen (both neutral and ionized) and various amounts of other elements inside the host galaxy. These clouds leave ab... | {
"language": "en",
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How are qubits better than classical bit if they collapse to a classical state after measurement? Classical computers store information in bits, which can either be $0$ or $1$, but, in a quantum computer, the qubit can store $0$, $1$ or a state that is the superposition of these two states. Now, when we make a "measure... | Some calculations are too large for conventional (classical) computational capacity. These problems are classified as BQP, or bounded-error quantum polynomial time. The best way to formulate the question is "what is the use of quantum computing?", and that's where a qubit becomes useful.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is the difference between lattice vectors and basis vectors? Google has not been very useful in this regard. It seems no one has clearly defined terms and Kittel has too little on this.
| Basis vectors and lattice vectors are alternative ways to represent vectors in a vector space.
In mathematics (linear algebra,) basis vectors are mutually orthogonal and form a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space. A set of basis vecto... | {
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How do objects change their axis of rotation? If I hold a pencil at its end and spin it, throwing it upwards, it will spin about its end, but will soon start spinning around its center. How is this?
I would draw the following torque diagram for while it's in the air:
*
*Object: uniform thin rod with length $\ell$ an... | This looks like an example of the Tennis Racket Theorem. Some axes of rotation for a rigid body are more stable than others. If the initial rotation axis does not correspond to one of the principal axes, a wobble can grow and cause the rotation axis to move to a principal axis. This is a result of Euler's Equations of... | {
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Is it possible to have a singularity with zero mass? A singularity, by the definition I know, is a point in space with infinite of a property such as density.
Density is Mass/Volume.
Since the volume of a singularity is 0, then the density will thus become infinite because Mass/0 = undefined
However, is it possible to ... | I recall a presentation (many years ago at DAMPT) where the presenter claimed that focussing gravity waves could produce a curvature singularity that bore some similarities to a black hole. I've done a quick Google and found this paper, that references two papers by Alekseev:
*
*Alekseev, G. A. and Griffiths, J. B. ... | {
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North and south of magnetic field
The current I is flowing upward in the wire in this figure. The direction of the magnetic filed due to the current can be determined by the right hand rule.
Can we determine the north and the south of the magnetic field produced by the current I by using a hand rule?
| There is no North or South pole in the field around wire. You must understand that the field itself has no poles. Field just consists of field-lines. What we call pole is usually the place when field-lines collide with the physical source of the field (eg. earth, magnet).
But in fact these lines continue even through t... | {
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When should angles be expressed in degrees vs. radians? I am trying to calculate the albedo of a given latitude by following the methods of Brutsaert (1982), I have copied the formula below:
3.6 Shortwave and long-wave radiative fluxes
Albedo $\alpha$, the reflectance of solar radiation, increases as the solar angle d... | Argument reduction may slow down the calculation of trigonometric functions when using radians. The Java VM was notorious for having problems with speed when arguments don't fall within a specific narrow range. When using cycles instead of radians, it would be an efficient and trivial operation of ignoring the integer ... | {
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How do you determine the heat transfer from a $p$-$V$ diagram? I doubt this question has been addressed properly before, but if there are similar answers, do direct them to me.
I am currently studying the First Law of Thermodynamics, which includes the p-V diagram and of course, $\Delta U = Q_{to} + W_{on}$.
My questio... |
In the following, $R$ is the gas constant, $\gamma$ is the heat capacity ratio of the gas ($and $n$ is the number of moles.
*
*AB-isothermal.
$\Delta W=Q$; area under the curve depicts the work done , ie. heat intake.
$\Delta W=nRT\ln\dfrac{V_2}{V_1}$
*BC-isobaric.
Can't be calculated directly from curve.
otherwis... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$SU(N)$ Neel manifold I have seen multiple papers talking about the manifold, $M$ of the Neel order for an $SU(N)$ magnet is
$$M~=~\frac{U(N)}{U(m)\times U(N-m)}.$$
So for instance, a $SU(2)$ magnet has manifold
$$M ~=~ \frac{U(2)}{U(1)\times U(1)} ~\sim~ S^2.$$
Could someone explain to me physically how I could see wh... | I am going to try and reinterpret some of that paper, see if we can get some kind of answer started. Please comment and contribute, I think this is an interesting physical system.
It would seem that they are using the fact that there are both boson (symmetric) and fermion (antisymmetric) representations of $SU(N)$ to g... | {
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Is there any correlation between mass-energy equivalence and Maxwell's 4th equation? I wonder, how came in both equations proportionality constant is exactly $c^2$?
$$c^2(\nabla \times B) = \partial E/\partial t$$
where $E$ - electric field
$$c^2m = E$$
where $E$ - energy
I am just curious can we relate these two equa... | Electromagnetic field has energy density, and has momentum. According to relativity, rest inertial mass has a strict relationship with energy and momentum magnitudes as a conic-shell invariant
$$ m^2 c^4 = E^2 - c^2 P^2$$
So this allows the possibility of defining inertia for an electromagnetic field. Look at eq.(5) in... | {
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What's the criteria for black hole thermodynamically stability? (And dynamical?) It looks like usual criteria (positivity of Hessian; what geometrically means a cancave of entropy) is no useful, becouse entropy is not additive and not extensive for black hole. Then what is the right criteria?
| One can see that a black hole in thermodynamic equilibrium with its environment is always unstable by looking at the Hawking temperature of a black hole, given by
$$
T = \frac{\hbar c^3}{8\pi k_B M G}
$$
The temperature being inversely proportional to the mass means that big black holes are cold, a black hole with the... | {
"language": "en",
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"source": "stackexchange",
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How does an earthen pot keep water cool? I understand that evaporative cooling takes place thanks to small pores contained in the pot and that allow some water to go through and evaporate. However I couldn't understand clearly whether water inside the pot stays at its original temperature or would it cool further?
If i... | Evaporative cooling works by removing the high-velocity tail of the kinetic energy distribution. That is, only the fastest molecules escape the liquid, leaving the rest to thermalize at a lower temperature. If there is capillary action taking water to the outside of the pot and that is evaporating, then the pot cools d... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Can a current carrying loop or wire produces no magnetic field? A current carrying wire produces magnetic field around it. We can find the direction by Fleming's Right hand rule. We know change in electric filed produces magnetic field and change on magnetic field produces electric field. It is mutual relationship.
My ... | If you making a solenoid you want to also wind all the loops in a certain direction. If you instead wound it in a short loop with an equal number of turns both clockwise and counter clockwise then there would be practically zero magnetic field. Brandon's examples apply because again we have opposing counter currents ... | {
"language": "en",
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Momentum, Impulse and Newton's Second Law of Motion Newton formulated his Second Law as such:
$$\sum{\vec{F}} = \frac{\delta \vec{p}}{\delta t}$$
and of course, $\vec{p} = m \vec{v}$.
Why is it that if the net force $\sum \vec{F}$ is constant (which implies that the rate of change of momentum is constant), then
$$\frac... | A constant net force means:
$$\Sigma\vec{F}=\frac{d\vec{p}}{dt}=C$$
where $C$ is some constant. This means that
$$\int \ dp=p=C\int\ dt=Ct+p_0$$
where $p_0$ is the initial momentum. Now, you can easily verify that
$$p_2-p_1=\Delta p=Ct_2+p_0-Ct_1-p_0=C(t_2-t_1)=C\Delta t$$
In particular, you see that $\Delta p \neq \fr... | {
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How does tension apply torque on a pulley? How is tension in a string able to apply torque on a pulley? How does string itself able to apply a force on pulley? What is happening inside the pulley?
The pulley has a mass $m$ and is a disc.
| Consider the pulley together with the segment of the string that touches it as a single object with
the same moment of inertia as that of the pulley, since the string is massless. As far as the angular
acceleration of this object is concerned, one is allowed to disregard the internal forces between
the aforesaid segmen... | {
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Interpretation of de Broglie wave Until what point can the de Broglie wave be thought as a real wave?
I mean, is it made of something?
What amplitude does it have? Is it a sine wave?
How can it be related to the wavefunction of the particle?
| De Broglie s' matter wave, is, in some sense, the prehistory of the wavefunction, but its original interpretation is false.
De Broglies relations makes a link between particles characterics (momentum, energy) and wave characteristics (frequency, wavenumber). De Broglie thought (like Einstein and Schrodinger), that it w... | {
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Relativistic Lagrangian transformations I need to study the relativistic lagrangian of a free particle.
It's
$\ L= - m c^2 \sqrt[2]{1- \frac{|u|^2}{c^2}} $ I need to study the translation, boost and rotation symmetry. I say it doesn't depend of the position, so it has translation symmetry and the momentum will conserv... | Time and space translation invariance imply conservation of energy and momentum.
Rotation invariance implies conservation of angular momentum.
As I recall: Boost (a 4-d rotation) invariance tells you that all inertial reference frames have constant velocity wrt each other.
Caveat: I haven't thought about the last one... | {
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Phase Space dimension of Lorenz Strange Attractor It is often discussed in 3 spatial dimensions and the need for third dimension to prevent self intersection is mentioned. But shouldn't the phase space of the Lorenz system be 6 dimensional, i.e., the 3 momenta also?
| The dimensions mentionned here are phase space dimensions.
The dynamics of the Lorenz system are defined by its 3 coordinates X(t),Y(y),Z(t) solving a system of 3 non linear ODEs.
This has of course nothing to do with ordinary space because the Lorenz variables X(t) etc are actually functions of temperatures.
The Loren... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/65575",
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"source": "stackexchange",
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Can a single molecule have a temperature? A show on the weather channel said that as a water molecule ascends in the atmosphere it cools. Does it make sense to talk about the temperature of a single molecule?
| I will explain in simple....
We can measure temperature changes when a Source release energy or extract energy.
We know that every thing want to attain equilibrium. Therefore there should be an equality in temperature in all the atoms or molecules of a system.
The temperature of an atom will be same as that of temperat... | {
"language": "en",
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What's wrong with this application of Thomas Precession to circular motion velocity measurements? If you happen to have the Third Edition of Classical Electrodynamics by John David Jackson, turn to section 11.8, as that's where I'm getting all this from. If not, you should still be able to follow along.
In said section... | The thing is that the derivation of this formula implies that $\bf{\omega}_T$ describes additional precession due to relativistic effects:
Note that Jackson in the next equation adds $\bf{\omega}_T$ to $\frac{e\bf{B}}{mc}$ - precession due to magnetic field.
| {
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"source": "stackexchange",
"question_score": "11",
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Why are the magnetic moment and the angular moment related? Why are the magnetic moment and the angular moment related? I've always read everywhere that they are related but found nowhere a satisfactory explanation of the cause
| Let's first look at the classical situation. A charged particle moving round a circular loop had an angular momentum and because it is also a current, it produces a magnetic moment. Therefore, it can be considered to be a magnetic dipole with a moment $\vec{m}$. The magnetic moment and the angular momentum are proporti... | {
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"source": "stackexchange",
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Find E and B from vector potential I have a vector potential given by:
$\mathbf{A}(x,t) = \mathbf{e}_{y}\frac{1}{2} e^{-(x-ct)^{2}/{4a^{2}}}$
Now, the question is "Determine the E and B under the condition that the scalar potential vanishes $V = 0$."
But I'm not quite sure what it means when $V=0$ ?
As far as I can see... | The relation $\mathbf{E} = -\nabla V$ holds only in the absence of vector potential, otherwise the electric field changes to
$$
\mathbf{E} = -\nabla V-\frac{\partial\mathbf{A}}{\partial t}.
$$
The reason for this is that when you introduce vector potential by $\mathbf{B} = \nabla\times\mathbf{A}$, Faraday's law reads
$... | {
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Boundary conditions for Laplace's equation Given a grounded conducting sphere, $V=0$ and $\text{radius} = R$, centered at the origin with a pure electric dipole (dipole moment $\vec p$) situated at the origin and pointing along the positive $z$ axis, I should be able to solve Laplace's equation in spherical coordinates... | As $r \to 0$, we know that the potential approaches $\frac{1}{4 \pi \epsilon _0} \frac{|\vec p|\cos{\theta}}{r^2}$.
Therefore(just by comparing the terms, which is usually the easiest method for solving these equations):
$$n \neq 1 \Rightarrow B_n = 0\\
B_1\ P_1 \left(\cos(\theta)\right)=B_1\ \cos\left(\theta \right)=\... | {
"language": "en",
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"source": "stackexchange",
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A Musical Pathway Using a small number of sound emitters, could you create a room where certain nodes emitted particular tones, but no meaningful sound was heard anywhere else.
So, for example, by walking down a certain path, you could hear the tones for "Mary Had a Little Lamb." Is there a generalized algorithm to ma... | You could use audio spotlights to make the sounds audible only along certain beams. This should be close enough to let people walk around in the room and only hear certain notes at certain places.
These audio spotlights use intense ultrasound that makes the air along its path a nonlinear acoustic medium, so modulation ... | {
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"source": "stackexchange",
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Anomalous dimension for bare actions with a standard kinetic term In this paper on p42, it is explained that when starting with a bare action that contains a standard kinetic term, this kinetic term attains a correction in the course of the RG flow which can be denoted by $1/Z_{\Lambda}$, such that the effective kineti... | I will give some hints:
It is anomalous scaling dimension. scaling dimension is defined as
$$x \rightarrow \lambda x,\\
\phi(x) \rightarrow \lambda^\Delta \phi(\lambda x) $$
From the formula (3.45) in the reference (maybe it is better to be $\phi(x) \rightarrow \Lambda^\frac{d-2}{2}\phi(\Lambda^{-1 }x)$), we know that ... | {
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"source": "stackexchange",
"question_score": "5",
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Can open, unsafe nuclear fusion reaction burn the atmosphere? I happened to hear people saying that the nuclear fusion bomb tests could set the atmosphere on fire. I have some serious doubts about that - but I have no facts.
Nuclear fusion reaction requires $15*10^{6}$ kelvins to start. If we produce such temperature i... | From what I have read in "American Prometheus: The Triumph and Tragedy of J. Robert Oppenheimer" Teller was the first one to express this concern before the Trinity test.
Also quoting from: http://www.sciencemusings.com/2005/10/what-didnt-happen.html
Physicist Edward Teller considered another possibility. The huge tem... | {
"language": "en",
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"source": "stackexchange",
"question_score": "14",
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Minimization of energy and maximization of entropy Are maximization of entropy and minimization of energy equivalent? Or are they contrary?
Why should the thermodynamic potentials such as $G$, $A$, etc, be minimum at equilibrium?
I am confused. Because I read in one place that both these compete and result is minimiz... | In the canonical situation, you have a total system , which is made of a heat reservoir "HR" (with constant temperature $T$) and the systeme to be observed "OBS".
If we call $E_{total}$ the total energy of the system (HR + OBS), it can be shown that the total entropy of the system (HR + OBS) $S_{total}$ could be writte... | {
"language": "en",
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Energy stored in a clothespin spring (non linear spring) I was wondering how one would go about figuring out the energy stored in a spring in a clothespin. When opened the spring is deformed, but I don't have any intuition for why it gets stronger with more loops, how it works, or how to calculate the energy or force ... | I would use the well known formula of
$$ U = \int \frac{T^2 }{2 G J} + \frac{F^2 }{2 A G}\;{\rm d}l $$
where $T$ is the torque on the coils, $F$ is the axial force, $A$ is cross section area, $G$ is the modulus of rigidity and $J$ is the polar moment of area. The axial length $l$ is only that if the active coils that c... | {
"language": "en",
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How do you build a nuclear reactor in your garage? When I came across this link earlier today, I was astounded. Nuclear reactors are immensely complex devices that took scientists years to develop, and are usually housed in huge facilities. The link says that 15 high school students in the world have built nuclear fusi... | Note that this is not a nuclear reactor, but a Fusor. Fusors work by using a voltage drop to accelerate ions to high velocities and a few may fuse. Due to inefficiencies, they cannot be used for power generation. They are relatively simple to build (they are basically beefed-up plasma globes).
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Canonical / Grand-Canonical average annihilation operator Does anyone knows a simple way to understand why the average value of the creation (or annihilation) operator should be equal to zero in the Canonical Ensemble? Why instead if I'm dealing with a Grand-Canonical ensemble the same averages can be different from ze... | I'm unconvinced that the ensemble average value of an annihilation operator doesn't vanish in the grand canonical ensemble. Here's my argument for why it does vanish:
Let $H$ be the Hamiltonian and $\mathscr N = \sum_{i=0}^\infty a_i^\dagger a_i$ be the number operator. Let
$$
|\mathbf n \rangle = |n_0, n_1, \dots\... | {
"language": "en",
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On Einstein notation with multiple indices On Einstein notation with multiple indices: For example, consider the expression:
$$a^{ij} b_{ij}.$$
Does the notation signify,
$$a^{00} b_{00} + a^{01} b_{01} + a^{02} b_{02} + ... $$
i.e. you sum over every combination of the indices? Or do you sum over the indices at the sa... | Written explicitly, (assuming summation over indices from 0 to 3)
$$a^{ij}b_{ij} = \sum_{i=0}^3 \sum_{j=0}^3 a^{ij}b_{ij}$$
You can expand this to
$$a^{ij}b_{ij} = \sum_{i=0}^3 \left( a^{i0}b_{i0} + a^{i1}b_{i1} + a^{i2}b_{i2} + a^{i3}b_{i3} \right) $$
$$\implies a^{ij}b_{ij} = a^{00}b_{00} + a^{01}b_{01} + a^{02}b_{02... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is Biot-Savart law obtained empirically or can it be derived? There's already a question like this here so that my question could be considered duplicate, but I'll try to make my point clear that this is a different question.
Is there a way to derive Biot-Savart law from the Lorentz' Force law or just from Maxwell's E... | The problem with Biot-Savart's Law is that theoretically it is formulated in terms of current elements $Idl$ and then integrated. But in most textbooks it is formulated also for POINT charges, in terms of $qv$. The problem here is that when a point charge $q$ moves with velocity $v$ the magnetic field in nearby spaces ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/67445",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "59",
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Calculate magnitude of force acting on some area by falling object I have simple question: is it possible to calculate magnitude of force acting on some area by falling object?
Let's say I have an object with mass $5\text{ kg}$. I drop that object in height 1 meter. It potential energy was
$$\begin{align}
E_p &= hmg \... | Your data is not enough. You need some data on the ground material (such as modulus of elasticity), body material and shape, etc. For example, if the ground is not hard, the force will be small.
EDIT (06/10/2013): As I said, you need more input data. I don't have time to give you a detailed analysis. You may wish to lo... | {
"language": "en",
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Is a proton collision (collisions like in the LHC) visible to the human eye? I was curious if a proton collision is visible to the human eye.
(This might sound like a really basic question and forgive me if it is. I am very inexperienced in Physics and just wanted an answer to my curiosity)
| It all depends on your definition of visible.
Elementary particle collisions have been made visible since the time of cloud chambers and bubble chambers. A good site for bubble chamber pictures exists . Unfortunately proton scattering is not as photogenic as scattering by other particles so I was unable to find a phot... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why do current-carrying wires heat up? Obviously wires heat up too, but why do they heat up? And for the same reason, why do we get electrical burns?
| When electricity moves through anything -- wires or bodily tissues -- there are actual electrons (typically) moving. These electrons are being pulled along by an electric field, but they're also bumping into the atoms that make up the wire or bodily tissue. When an electron bumps into an atom, it transfers some of it... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Correct terminology for combined kinematic and dynamic state The kinematic state is defined as the position and orientation in space. The dynamic state is defined as the associated velocities.
What is the correct terminology for the combined kinematic and dynamic state? Can I call it the kinetic state?
| After all kinematics + dynamics = Kinetics.
I agree wholeheartedly. Kinematics is a study of positions, velocities, accelerations, and orientations. In the F=MA and therefore, A=F/M, doesn't the A- accelerations include the implied force and mass required?
Please another stupid word for someone who doesn't understand... | {
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Dissolving oxygen into water I was wondering how one would actually calculate how much oxygen would dissolve into water given the necessary initial conditions, and what those initial conditions would need to be. I assume they would be pressure, and initial concentration, but I really don't know where I would go from th... | 1) Calculate the Gibbs free energy for (in principle, by quantum mechanics and statistical methanics)
$\mathrm{O}_2(\mathrm{g}) + \mathrm{H}_2\mathrm{O}(\mathrm{l}) \rightarrow \mathrm{O}_2 (\mathrm{aq}) $
(g, l, and aq stand for gas, liquid, and aqueous solution)
2) Using the the reaction isotherm equation (linked wit... | {
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Dirac magnetic monopoles and quark fractional electric charge quantization When applying the Dirac quantization rule for electric and magnetic charge, I assume one is considering unit electric charges such as electrons. How does the Dirac quantization rule apply for the fractional electric charges of quarks?
| The logic is the same: If $q$ is an electric charge and $g$ is a magnetic charge, one must have $gq \in 2\pi\mathbb{Z}$ (in lazy theorist units). So if there is a largest magnetic charge, then there is necessarily a smallest electric charge. You don't have to assume that you are dealing with unit charges to make thi... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/67812",
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Ball jumping from water Few days ago I played with ball(filled with air) in swimming pool. I observed interesting phenomenon.
When I released a ball from 3 meters depth the ball barely jumped above the water surface but when I released it from 50 cm depth it shoot out of the water like nothing.
I observed when released... | Try to eliminate your initial hypothesis by using dimensional analysis to make a quantitative estimate based on variables you can observe and measure easily.
You should be able to estimate a Strouhal number for your system from your observations of ball diameter, 'zig-zag' frequency, and the time for the ball to reach ... | {
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"source": "stackexchange",
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Fusion vs. Fission I understand why fission generates large amounts of energy when the nucleus is split, but then why does fusion generate such large amounts of energy. If fission releases energy when some mass is lost as energy, then shouldn't the fusion process absorb energy to fuse nuclei together?
I also am curious... | Fission is exothermic only for heavy elements, while fusion is exothermic only for light elements. Intermediate nuclei, in the iron/nickel range, are the most tightly bound, and so you generally release energy moving in that direction.
Fusing stable elements into uranium would consume energy, as would trying to break h... | {
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Why can't some light ever reach earth? Note: I'm a layman in physics. Excuse me if this question is a duplicate, I lack the knowledge in technical terms to find the answer to this by research (I've tried).
Let's suppose there's a star moving faster than the speed of light (relative to earth) due to the expansion of the... | Don't think of it as the other star moving away faster than light.
Think of the space between them expanding faster than light, so they get further away faster than the photon can travel
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/68198",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Decomposition of this wave function in eigenfunctions I have this wave function of a system on a central potential: $V(r)$:
$$\Phi(x,y,z)=C(x+y+z)e^{-\alpha r^2}.$$
And I'm asked a few things about probabilities. I don't have problems with that, because I understand how it's done. The main part is decomposing as a sum ... | In your case, the function on the LHS is very simple so you can more or less read off the coefficients (by inspection). The LHS is a linear combination of
$$ e^{i \phi} \sin \theta, \quad e^{-i \phi} \sin \theta \quad \mathrm{and} \quad \cos \theta.$$
Now the spherical harmonics are of the form $e^{i m \phi} P_{\ell}{}... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/68263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Can someone explain to me inertia? I'm a little new to physics and I'm not quite getting the concept of inertia. Can someone help me?
| The Wiki article is very much satisfactory. Without a proper definition of your question - like where exactly you require an explanation or confused about, all I can say is this. It's the resistance of an object to an applied force. Or in other words, the ability of an object (of course it should be massive) to maintai... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/68362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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