Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
No local degrees of freedom when connection is flat I was studying Chern-Simons theory and variation of action gives us the flatness conditions $\mathrm{d} A + A \wedge A = 0$. I am wondering how to see that this implies there are no local degrees of freedom.
And what precisely does it mean that a degree of freedom is ... | Recall that $\mathrm{d}A + A\wedge A = F = 0$ means that the field strength is vanishing, i.e. the gauge field is always pure gauge locally.
Local degrees of freedom would mean that the equation of motion ($F = 0$) has more than one local solutions that are not related by a symmetry of the theory. But the field being ... | {
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How can interstellar space have a temperature of 2-3K? Several different sources online state that the average temperature of interstellar space (or the universe in general) is around 2-3K.
I learned that temperature is basically the wiggling of matter, and I find it somewhat counterintuitive that the wiggling of so fe... | Too long to be a comment, this is an extension to Chris's answer.
Suppose a macroscopic object, a thermometer, for example, was placed in that hot intracluster medium (ICM) Chris mentioned in his answer. Even though that thermometer is surrounded by this hot gas, the thermometer will not get hot. It will instead cool t... | {
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Does the speed at which sound travel depend on the volume (amplitude) of the sound? Lets say you have a plank is you hit it once and get t time if you hit is 2x as hard will it travel t/2? will it be the same or will it travel only slightly faster?
| Volume is not strict word for describing sound (look, how many meanings in acoustics it has)
http://en.wikipedia.org/wiki/Volume_%28disambiguation%29
"Loud" sounds are basically those of big amplitude. And amplitude of wave and its speed are two different things that have not much in common. Hitting a plank harder will... | {
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Scalar Particles, Flavor Changing Processes and Gauge Symmetries Let's consider an extended version of the Standard Model (SM) with a new Yukawa operator of the form
$$ \sum_\ell g_\ell\bar{\ell}\ell \phi ,$$
where $\ell$ is any lepton of the SM and $\phi$ is a new real spin-0 particle, which is assumed to be a singlet... | If you are considering massless neutrinos there is no such a diagram since all interactions would preserve flavor.
If you take instead massive neutrinos, you are probing lepton flavor violation within the SM since the new interaction with $\varphi$ respects flavor. It is thus very very small, being controlled by the ne... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/134155",
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Special Relativity, 2nd Postulate -- Why? As a lowly physics undergrad who has been chewing on this 2nd postulate of special relativity for a year or more, I simply can't wrap my head around reasons why it is true or how Einstein might have been convinced enough to propose this postulate.
Consider Alfred who is riding ... | Einstein did not prove this postulate ; he simply asked "what if it is true?". He had very good reasons for asking that question.
His efforts to answer the question challenged a whole raft of "beliefs" about time and space, none of which were based on proof either ; they were (up until then) assumed true by so-called "... | {
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Topological insulator vs. topological superconductors in any dimension My question today is simple. What is the difference between a topological insulator and a topological superconductor? How that difference depends on the dimensionality of space(time)? What is the difference in terms of the so-called edge states?
PS:... | *
*The topological insulator has the charge $U(1)$ symmetry, while the topological superconductor does not. In fact, this difference is general, and applies to all insulators and superconductors. In condensed matter physics, insulators and superconductors are both gapped fermion systems. Depending on whether the charg... | {
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Difference between heat capacity and entropy? Heat capacity $C$ of an object is the proportionality constant between the heat $Q$ that the object absorbs or loses & the resulting temperature change $\delta T$ of the object. Entropy change is the amount of energy dispersed reversibly at a specific temperature. But they ... | Taking the first law of thermodynamics you have
$$dU = \delta Q + dE_{work}$$
I.e. the change of internal energy is the change of energy expended by work of the macroscopic parameters plus heat. It is a non-trivial result in thermodynamics that there exists a function $S$, entropy, and absolute temperature $T$ which c... | {
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From Paris to ... London (Excuse the pun in the title, couldn't resist)
Paris and London are connected by a straight underground tunnel, as shown in the diagram below. A train travels between the two cities powered only by the gravitational force of the Earth. Find the maximum speed of the train and the time taken to ... | You are massively overthinking the problem. The question was "find the maximum speed".
If you know the greatest depth of the tunnel, then compute the difference in potential energy between the surface and that point.
Now convert that to kinetic energy, and then to velocity.
Evaluating
$$U = \frac{mgr^2}{2R}$$
at both ... | {
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What are washers for? When you attach a bolt to something using a nut, it is clear what the roles of the nut and bold are.
The more you tighten the bolt the more secure your fastening. However, you are often also told to use a washer as well.
I know this somehow prevents the bolt from loosening but from a physics/ma... | Interesting evolution of the nut-and-washer: nut with built in washer. It helps understand some of the above answers:
Built in lockwasher
this one will "bite" evenly over a large radius - improving the torque with which the nut is held once it is tightened.
If you imagine the contact surface being smooth instead (sorr... | {
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Is the fine structure constant a rational number? Since the fine structure constant (denoted alpha) is a pure real number, it just occured to me to ask if it is a rational number or not.
| The fine structure constant is given as: $$\alpha = \frac {k_{e} e^2} {\hbar c^2}$$
Immediately we have a problem in determining the rationality or otherwise of $\alpha$. The Coloumb constant, Planck constant (maybe not?) and speed of light are all either exact numbers or pre-defined. Since the elementary charge $e$ is... | {
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Units of Hubble Time and Hubble Constant? When we have Hubble constant and it's inverse Hubble Time (1/H) what units are they measured in? I know Hubble constant is in "km/s per Mpc" but is there any other units which are popular used with it, and if so is there any conversion between the two?
And would Hubble Time uni... | Although in theory we should all be using SI units, for things that are very large or very small these units are an inconvenient size and it's common to invent new units that are more convenient. So, for example particle physicists measure mass in GeV (strictly speaking GeV/$c^2$) and cosmologists measure distance in l... | {
"language": "en",
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Hydrogen Bomb Mass to Energy? How much mass is converted to energy when a hydrogen bomb explodes? I remember an eighth grade chemistry class where, by going through the nuclear processes, my teacher estimated that roughly 2g of matter was converted in a fission bomb.This is a surprisingly small amount of mass! I have... | While it does take $2.33\:\mathrm{kg}$ of any mass to be converted to 50MT of energy using the calculations stated before ( $ m = \frac{E}{c^2} = \frac{2.092 \cdot 10^{17}}{8.987 \cdot 10^{16}} = 2.33\:\mathrm{kg} $ ), be aware that just a small fraction of the total hydrogen in a thermonuclear device is converted to ... | {
"language": "en",
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Physical meaning of imaginary part of Electric field? As far as I know (or I thought I knew), if we have an electric field
$$\mathbf{E}=\mathbf{E_0}\cos(\omega t - kx),$$
we can define it as the real part of
$$\mathbf{E}=Re(\mathbf{E_0}e^{i(\omega t - kx)}).$$
Introducing imaginary components to the electric field i... | I agree with Chris, the imaginary part here is just the mathematical way of showing a rotation. In this case, it happens to be that the second ket is 90 degrees out of phase with the first. You can think of the imaginary axis as a direction orthogonal to any real axis, thus it is similar to saying the imaginary part ... | {
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Strange definition of microcanonical partition function I always thought that the microcanonical partition function would measure the number of states that correspond to some fixed energy. Despite, I found in this paper (equation 3.4) that we integrate over all configurations that have an equal or lower energy than som... | Boltzmann entropy is the upper limit of Gibbs entropy. I myself never saw any usage of Gibbs entropy. Gibbs entropy is misused as the sum of pLnp over the states instead of over the microstates as it should. The sum over the states is an approximation that yields the canonical distribution. I recommend reading my post ... | {
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Charge outside Gaussian Surface doesn't contribute to Flux? I roughly understand the explanation for this: any electric field line that enters the surface, must leave it, since field lines can't terminate abruptly in space. My question is, what if you have a charge $+2q$ outside your gaussian surface, and a charge $-q$... | I think the correct answer for your question is that the electric charge outside of a closed Gaussian surface does not enter inside because the surface is enclosed. Take a look at the Faraday's cage and you can also take into account why we are not affected when driving in a car and get struck by a lightening.
| {
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Can we measure the depth of water by scattering water? Suppose we release an object and make it fall on the surface of water, then the scatter of water is recorded,
Taking some observations, can we calculate the depth of the water by analyzing the scatter of water pattern?
| In my answer, I assume that "scatter" can be generalized to wave motion.
In general, we recognize different kinds of surface waves: gravity waves (main driving force is gravity), and capillary waves (surface tension dominates).
Now gravity waves have a velocity that is a function of both wavelength and depth:
$$v=\sqrt... | {
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Why work $W$ and heat $Q$ are different concepts? I understand heat as the flow of energy (through radiation, convection or conduction) from one body to another. When I think about conduction (for example) I visualize particles that jiggle a lot bouncing against particles that jiggle less and transferring heat to them ... | A recent paper by the physicist Carlo Rovelli (Where was past low-entropy?), puts the type of answer given by @Jonas quite succinctly:
Heat versus work. Thermodynamics has developed as the science
describing the exchanges of heat and work between a system and its
environment. Both are exchanges of energy: what det... | {
"language": "en",
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$\rm Lux$ and $W/m^2$ relationship? I am reading a bit about solar energy, and for my own curiosity, I would really like to know the insolation on my balcony. That could tell me how much a solar panel could produce.
Now, I don't have any equipment, but I do have a smartphone, and an app called Light Meter, which tells... | I'm afraid that it is not easily possible to take the luminous flux and obtain the insolation (as radiant flux). Here's why:
The luminous flux $F$ is calculated from the radiant spectral power distribution $J(\lambda)$ by weighting each wavelength with a luminosity function $y(\lambda)$ as per
$$ F = c \int J(\lambda)y... | {
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Buoyancy / Drag Problem Buoyancy / Drag Problem
Just a little bit of help would be nice.
I have a spherical particle of radius $R$ and density $\rho$, surrounded in a fluid of density $\phi$ and viscosity $\eta$. I'm suppose to consider the forces of gravity, buoyancy, and drag.
I'm them told to consider the net forc... | 1)Yes sedimentation velocity is terminal velocity.
2)No, Not necessarily. Buoyancy Depends on the Volume of water displace by the object where as gravitational force is weight of object.
Let us look at 2 cases here:
Constant values:
g, Gravitational acceleration = 10 m/s2
d, Density of water = 103 Kg m3
CASE I: V, Vo... | {
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What technology can result from such expensive experiment as undertaken in CERN? I wonder what technology can be obtained from such very expensive experiments/institutes as e.g. undertaken in CERN?
I understand that e.g. the discovery of the Higgs Boson confirms our understanding matter. However, what can result form t... | You are thinking too much physics.
Instead, think about how you process the physics data that is generated from the experiments.
The answer is that there are not only physicists at CERN but also - and actually quite a lot - hardware and software engineers.
The detectors are one huge combination of custom built hardware... | {
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Does angular speed of the Earth around the Sun remain constant? I presume that angular speed is nothing but angular velocity without direction. So, since the earth always takes nearly 365 days to complete one revolution around the sun, can't we conclude that it's angular speed is constant ($19$ x $10^{-7}$ RPMs) ? Most... | Ignoring the (minor) effects due to the other planets, the angular momentum of the Earth-Sun system must be conserved, and the angular momentum is given by (making the approximation that the Sun is fixed):
$$ L = \omega m_e r_e^2 $$
where $m_e$ is the mass of the Earth and $r_e^2$ is the Earth-Sun distance. A quick rea... | {
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Does a pulled rubber band contain as much energy as a twisted rubber band? lets say I take two similar rubber bands. One of them I pull until it almost reaches its breaking point. The other I twist until it almost reached its breaking point. Do both of these rubber bands contain (roughly) the same amount of energy? ... | The band that is pulled contains much more elastic energy. This is because that band is stretched until breaking everywhere. The band that is twisted, only reaches it breaking point at the outer points of the cross section. The strain varies linearly with the distance from the center of the cross section and is zero at... | {
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Relationship between two viewpoints on Thermodynamics I've always seem the standard viewpoint on Thermodynamics that it is all about studying phenomena related to a property of systems called temperature. Then we have the zeroth law which allows us to give means to measure temperature and so on. Every basic Physics cou... | If two systems $X$ and $Y$ are in contact such that they can exchange energy, then the statement
$X$ and $Y$ are in equilibrium.
is equivalent to the statement
$X$ and $Y$ are at the same temperature.
Here is a proof.
The total entropy of the combined system is
$$S = S_X + S_Y \qquad (1)$$
where $S_X$ is the entrop... | {
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Does air conduct heat better than saturated steam? This engineering toolbox table shows thermal conductivity of steam at 0.016. I understand that water is better in conducting heat than air, but if I read this correctly, steam is worse in conducting heat than water?
Would there be any difference of steam in a vacuum? ... | From a heat transfer perspective, steam is a lot more like air that it is like liquid water. The chemical makeup doesn't matter as much as the state.
As a gas, steam is a bunch of $H_2O$ molecules flying around at random, bashing into each other occasionally. The mechanism of conduction in that case is that the hot m... | {
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How should I quote errors when measurements are asymetrically clustered? Suppose five people measure the length of a stick and report the following values
4.90cm 4.92cm 4.93cm 4.94cm 4.94cm
In high school science we are told that in cases like this we should report the average as 4.92cm +/- 0.02cm because the range of... | The most accepted way to do so is use the standard deviation, and there are several reasons for it.
But to put it simply, whenever you are measuring a magnitude that "ideally" should have only one value, it is expected that the measurements will behave as a gaussian distribution. That is, the probability of getting a ... | {
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A ball is dropped from the roof of a building. How fast is it moving after 4.9 seconds? (Absolute value) i really need help. A ball is dropped from the roof of a building. How fast is it moving after 4.9 seconds? (Absolute value)
in m/s
A water balloon is thrown downward from a tall building with an initial velocity ... | Hint 1: Everything accelerates (the velocity increases) downwards by 9.81 m/s, every second - on earth.
Hint 2: The relationship between height and velocity can be calculated using kinetic and potential energy equations:
0.5*m*v^2 = m*g*h
m = mass, v = velocity on the ground, g = gravity (9.81) and h = initial height.
... | {
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Why does Energy-Momentum have a special case? I was reading Energy-momentum, and I came across this simplified equation:
$$E^2 = (mc^2)^2 + (pc)^2$$
where $m$ is the mass and $p$ is momentum of the object. That said, the equation is pretty fundamental and nothing is wrong when looked upon, I similarly also believed thi... |
If the body's speed $v$ is much less than $c$, then the equation reduces to
That is an imprecise wording, and the cause of your confusion, I believe.
The correct description is that the second equation is an approximation of the first, and it will be closer to accurate the lower $v$ is.
The approximation is important... | {
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How can I calculate the speed of an object knowing its horizontal and vertical velocity components? Let's say a ball is thrown and it experiences typical projectile motion (moves in a parabolic arc etc.) and the only information we know are the equations for the horizontal and vertical components of its velocity for it... | The formula you have written is correct; but they are functions of time. Hence, by inserting the particular instant , say $t$ on the function ,you get the instantaneous components of velocity. Then using phythagoras theorem you will get the total instantaneous velocity. Taking your example, at time $T$ s , the X-comp. ... | {
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Loads in decelerating a falling object I remove trees in confined spaces setting up rigging systems, I am keen to look at the science behind it in more depth so far the formulas that I have found don't take into effect the ongoing effect of gravity once the object or log begins decelerating.
So far,
If a $300\:kg$ log... | To obtain the fall velocity we have:
$$mgh=\frac{1}{2}mv^2$$
From this you can extract a value for the velocity, as a function of the altitude.
Subsequently, we need to slow the tree down. If we have a distance to stop, we know
$$\Delta s= \Delta v \cdot \Delta t$$
If we know the impulse, and the time available to stop... | {
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What will happen in principle if one tries to push a neutron star? When we push an object it moves due to the mutual repulsion between the electrons in our hand and the electrons in that object. Since a neutron star contains only neutrons, what will happen in principle if one tries to push it?
| Despite pop-science descriptions, neutron stars do not contain only neutrons. They contain both protons and electrons as well, and though there are many fewer of them the electrons are also degenerate.
Accordingly it would be electron interactions (not those of neutrons) that transmitted that vast majority of the force... | {
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Importance of Powers of Velocity in Classical Mechanics Is there any general significance to calculated quantities that depend purely on general powers of the velocity of a particle/system/etc? The first power being momentum and the second being kinetic energy.
I know that in relativistic mechanics the momentum and en... | You are probably thinking about the moments of a velocity distribution function.
Suppose you had some probability distribution that was a function of space, time, and velocity or $f$ = $f\left( \mathbf{x}, \mathbf{v}, t \right)$, where $f$ has the units of # per unit volume per cubic velocity. From this we can define ... | {
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The effect of windspeed on a car I've worked problems in the past in trig class concerning the effect of wind on the speed of a plane and it's flight path and was wondering if a similar thing occurs with a car.
First off, I'm pretty sure that if the speedometer reads 60 mph, even if the wind is blowing 15 mph in the s... | No, while the work the engine does would be reduced by a tailwind, it would not be reduced to be equivalent to the relative speed travel.
Work $\ne$ Force
The work that the engine must exert to maintain speed is equal to the drag times the velocity. So let's look at the ideal situation where the only drag on the car wa... | {
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What is the highest frequency directly detected? The Feynman Lectures of Physics states:
In fact, although we mentioned many frequencies, no phenomenon directly involving a frequency has yet been detected above approximately $10^{12}$ cycles per second. We only deduce the higher frequencies from the energy of the part... | This kind of depends on what you mean by "directly detected", but there is a strong case to be made that optical frequencies have indeed been directly detected.
This has a large overlap with my answer to Have we directly observed the electric component to EM waves?, so I'll leave the details there, but the short answer... | {
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Spaceship Doppler frequency A spaceship starts falling under gravity with an acceleration $g$ as measured by an observer Barry at rest on Earth. At the instant that the ship starts to fall, an astronaut Harry at the base of the rocket ship sends a light signal of frequency $w$ vertically upward to another astronaut Sal... | I sure hope my assumption is correct that you're just needing help understanding where an equation that you encountered in your book came from, rather than this having been a homework problem that you were supposed to do. I fully support the policy here of not doing people's homework problems for them.
From the starti... | {
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How can the Big Bang singularity exist in spacetime if it has zero volume? How can the Big Bang singularity exist if it has zero volume? I tried googling to find the answer - no help.
Can someone give a general idea how can the big bang singularity exist even if it has zero volume. Please try to avoid post graduate co... | Have a read through Did the Big Bang happen at a point? and the answers to it.
The singularity at the Big Bang is the zero time limit of the equation (the FLRW metric) that describes the expansion of the universe. Most physicists believe that this is a mathematical artefact and does not describe what actually happened.... | {
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When referring to weights and mass of weights in a physics laboratory, do we use the term mass or weights? What terminology is used to refer to weights/ mass/ weight of mass/ mass of weights when referring to the mass of weights in a physics report? My question is more of the weights that we use in the physics laborato... | We use the term mass, when we mean the mass of a weight, and we use the term weight, when we mean the weight of a mass. :-)
The important thing to remember is, that the mass is the same everywhere, while the weight varies with the local gravity. So if you are referring to the constant mass of an object, you use mass ex... | {
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Where does the $\partial \vec{E}/\partial t$ term from Maxwell's equation go in Ampere's Law? One of Maxwell's Equations (ME) is:
$$\nabla\times\vec B = \mu_0\vec J+\epsilon_0\mu_0 \frac{\partial \vec E}{\partial t}.$$
While Ampere's Law (AL) is:
$$\nabla\times\vec B = \mu_0\vec J.$$
Griffiths E&M book derives that for... | You are right. These two equations cannot be valid at the same time. In particular, AL is only valid for steady current situation. Indeed, current induces a circulating magnetic field, as indicated by AL. However, current is not the only thing which can produce circulating magnetic field. The time-varying electric fiel... | {
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Normal modes of two parallel $LC$ oscillators coupled via mutual inductance
Consider the circuit shown below. The two LC circuits are arranged in such a way that their mutual inductance M results in a coupling between the currents flowing in the two circuits.
Find the frequencies of normal modes as a function of L, C ... | After solving the eigenvalue problem, I got $\omega_{A}$ to be $1/\sqrt{C(L+M)}$ and $\omega_{B}$ to be $1/\sqrt{C(L-M)}$. These frequencies make sense, because as you bring these circuits far apart from one another $M\rightarrow0$, and you get $\omega = 1/\sqrt{LC}$, the natural frequency of the LC circuit!
| {
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What is the difference between diffraction and interference of light? I know these two phenomena but I want to know a little deep explanation. What type of fringes are obtained in these phenomena?
| *
*Two separate wave fronts originating from two coherent sources
produce interference. Secondary wavelets originating from different
parts of the same wave front constitute diffraction. Thus the two
are entirely different in nature.
*The region of minimum intensity is perfectly dark in interference.
In diffraction t... | {
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Will an un-scattered photon go to the edge of the universe? Will an unhindered (un-scattered) photon go to the edge of the universe?
| The answer probably depends on how that question is interpreted. The universe is expanding. The ultimate fate of the universe isn't known for sure, but the growing consensus among cosmologists is that the universe will probably continue to expand forever. If that's the case, then a photon that leaves the Earth now w... | {
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Why is cooling much harder than heating? I'm trying to invent a distillation apparatus that runs solely on electricity. Suddenly, I realized that cooling things is really hard, while heating them up is so easy.
Actually, it seems that there are just three ways to cool something down:
*
*Peltier modules (incredibly i... | The rate of heat transfer depends on a temperature difference between objects, and heat transfer is always from high temperature to low temperature. When heating something with a flame, the temperature difference between the flame and the low temperature object can easily be 1000 deg C, so the rate of heat transfer is... | {
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Homemade Spectrometer Recently I have had ideas of how to build a spectroscope, but I'm not sure if it will work. As can be seen in the diagram, the experiment is simple: it consists of a laser that generates the light that passes over a gas in a closed tube sample and the light goes straight into a prism which, accord... | The experiment could work if only the gas inside the spectre would be thin enough to let the light through. If the gas is thick then it wold block some of the light making it flicker.
| {
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Can't understand the meaning of a formula describing a stable equilibrium? I've found this formula in a book and I can't understand its meaning:
$$\frac{df}{dx}\biggr|_{x_0} < 0$$
It describes the condition of an object to being in a $\textit{stable}$ equilibrium.
At first glance, it looks like an equivalent form for ... | Isn't it just a test of what kind of potential minimum you are sitting in (interpreting $f$ as a force)?
So if $d f/d x < 0$, and $f = -d U/d x$, where $U$ is the potential, then your condition reduces to $d^2 U/dx^2 > 0$ at $x=x_0$.
If $f$ only depends on $x$ then that clearly marks a local potential minimum (assuming... | {
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Why does electron move in an elliptical path? According to Sommerfeld's atomic model, an electron moving around a central positively charged nucleus is influenced by the nuclear charge. As a result of which, the electron moves in an elliptical path with the nucleus situated at one of the foci. But how the path of elect... | The Sommerfeld model, and the Bohr model from which it is derived, are toy models developed in an attempt to describe spectral lines in the era before modern quantum mechanics. You might be interested to look at the question Is it possible to recover the old Bohr-Sommerfeld model from the QM description of the atom by ... | {
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Simple estimation of the critical temperature of water I'm trying to develop fermi estimation skills and I came up with a question for which I don't even know where to start from. Here goes:
Is it possible to estimate the critical temperature (say in Kelvin degrees) of water in a simple way using fermi estimation?
By c... | Estimation: I want the two densities of vater and vapour to become approximately equal.
*
*the density of water is nearly constant
*the vapour pressure (you can derive this from the above mentioned Clausius-Clapeiron-equation) is approximately exponential in $1/T$. This means, that if you increase pressure by a fac... | {
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If an antenna must be $\frac{1}{4}$ of the wavelength, how can car antennas be so small? If the transmission antenna has to be $\frac{1}{4}$ of the wavelength, how can the car antennas' size be much less than that and properly receive the radio signal?
| Antennas don't have to be sized to the wavelength of the signal. It just happens, that if they are significantly smaller than a quarter wavelength, the sensitivity drops very quickly with antenna size. That, by the way, is also an enormous advantage for electronics design of switched mode power supplies. If the transmi... | {
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Why rubber is incompressible material? Why rubber is incompressible material? I know its Poisson's ratio is nearing to 0.5. So I don't understand physically, what it means by 0.5 Poisson's ratio and incompressibility. When I tried searching it, I found that rubber (or similar polymers) conserve volume after deformation... | Incompressibility implies Poisson's ratio $\nu = 1/2$, and vice versa.
A cylinder of length $L$ and radius $r$ has volume
$$
V = \pi r^2 L
$$
For an incompressible material (constant $V$) differentiation gives
$$
dV = 2\pi rL dr + \pi r^2 dL = 0
\\\nu \equiv -\frac{L}{r}\frac{dr}{dL} = \frac{1}{2}
$$
Thus a material ... | {
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constant speed elevator light beam paradox I can't get my head around this problem. Suppose there is an elevator moving up at a constant speed relative to an observer outside of the elevator, far away from any gravitational effects. Suppose a laser gun with a 1 meter long narrow barrel with a diameter of .10 cm emits ... | If the elevator is going up rather fast, the light from the back of the elevator takes longer to reach the observer than the light from the front.
This means that I see the back of the elevator at an "earlier time" - put differently, it looks to be a bit lower. The angle by which it appears to be lower is given by the ... | {
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Kernel normalization in Smoothed Particle Hydrodynamcs I am trying to write some code for a fluid simulation. I have read about Smoothed Particle Hydrodynamics (SPH) and my question is related to the properties of the smoothing kernel.
*
*How do I set the support $h$ for a kernel?
I found the following kernel func... | The parameter $h$ is the maximum distance that two smoothed particles, $a$ and $b$, can be before the distance between them is negligible for SPH purposes. If the distance, $\vert r_a-r_b\vert>h$ then the weight is zero.
For any kernel, the integral over the particular region, e.g. $r\in(-h,\,h)$, is necessarily 1. Sin... | {
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Gibbs free energy and maximum work I'm getting confused between two important results.
The Gibbs free energy is $G = H-TS$
where $H$ is the enthalpy and $S$ is the entropy.
When the temperature and pressure are constant the change in the Gibbs energy represents maximum net work available from the given change in system... | There is no contradiction here. The change in Gibbs free energy tells you the maximum extractable work and at constant teperature and pressure $dG = 0$. The conclusion here is that you can extract no work while maintaining a fixed temperature and pressure.
Why is this? Well you have 5 variables in your thermodynamic s... | {
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If light is electromagnetic then can light produce electricity or attract metals? what I mean to say is that if light is electromagnetic in nature then shouldn't it show electric or magnetic properties on matters? Like if light falls on a metal it should produce current due to its electric nature but it doesn't. Seco... | Other people wrote about photoelectric effect. Light can also cause mechanical motion of both metals and non-metals ( radiation pressure - https://en.wikipedia.org/wiki/Radiation_pressure )
| {
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Do weather conditions change after sunset? Maybe I haven't been searching well enough, but I can't find the answer to this question anywhere. The reason I'm asking this is because to me it seems as though it rains and storms less after dark. It's almost as if the weather calms as the sun sets.
Do weather conditions ch... | The weather does change after sunset, although it's still possible to have very significant weather events at night. In fact, these tend to be far more dangerous due to the lack of visibility and people sleeping. It's hard to avoid a tornado when you're asleep when it hits and it's hard to visually see coming.
The pri... | {
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In a column of fluid does density vary? In a column of fluid does density vary?
| It depends on the fluid.
Consider, for example, an ideal gas at fixed temperature near the surface of the Earth. Does the density vary in such a column? Yes.
Let's investigate as follows. Imagine that the column is in the $z$-direction and has cross-sectional area $A$. Let $z=0$ at the ground. Consider a small, ... | {
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How to show circular orbit as a precessing ellipse
A uniform distribution of dust in the solar system adds to the gravitational attraction of the sun on the planet an additional force linear in $r$ where m is the mass of the planet, k is a constant (proportional to the gravitational constant and the density of the dus... |
I am unsure how this is the precession of the ellipse and how the
precessing ellipse can be proved to give a nearly circular orbit. Will
showing that the angular velocity $w_0$ being nearly constant work?
Consider the figure below showing the effective potential (continuous line) of a particle of mass $m$ and ang... | {
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What are spin and valley symmetries in graphene? I have been assigned a presentation on a part of a paper (http://arxiv.org/abs/1303.6942). My task is to present on the spin and valley symmetries in graphene, and relate it back to the paper above.
However, after looking at many papers detailed spin and valley degenerac... | The hexagonal Graphene lattice can be considered as a superposition of two identical sub-lattices set off by one one carbon-carbon bond length. As a result, it has two sets of wavevectors k,that are picked out by the lattice, inequivalent (since the two sublattices really are distinct) but otherwise identical (since it... | {
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Explain why we use this formula to calculate total resistance Why do we use this formula to find the total resistance? Lets say we have three resistors in a parallel circuit
$$R_t = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}}$$
Where does it come from?
| Think about current flow.
If we take each individual resistor and determine the current for the applied voltage, we get: $$I_T=\frac {V}{R_1} +\frac {V}{R_2} + ...$$
Dividing everything by the voltage give us:
$$\frac {I_T}{V}=\frac {1}{R_1} +\frac {1}{R_2} + ...$$
Which is the same as:
$$\frac {1}{R_{eq}}=\frac {1}{R... | {
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Why is force a localized vector and not a free vector?
A vector which is drawn parallel to a given vector through a specified point unlike free vector in space is called a localised vector. The effect of a force acting on a body depends not only on the magnitude & direction but also on its point of application & line ... | For a force, you need the following bits of knowledge:
*
*The direction and magnitude of the force.
*The point it acts on.
Some people call that pair of specifications a "localized vector", because it consists of both a location (#2) and a vector (#1).
| {
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Pulling on a weakened rope - where will it tear? Let's say I have a rope of 10m length and it is weakened in 3 spots:
at 2.5m, at 5m and at 7.5m. Weakened means that if enough tension is applied it will tear at these points (all points are equally weakened).
This rope is connected to an immovable wall. There is a perso... | I suspect the result depends on how fast you increase the pulling force. If you pull abruptly, I would expect the rope will tear at the weakened point that is the farthest from you, as the tension wave from the wall will first arrive at that point, whereas, if the pulling force is increased very slowly, I would guess t... | {
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Benefits of rear spoiler in cars What would be the benefits of rear spoilers in cars, like this one:
| What would be the benefits of rear spoilers in cars, like this one?
On cars like that one, there are no significant objective benefits. There are some disadvantages
*
*increased weight, so higher fuel consumption
*increased drag, so higher fuel consumption
On racing cars on racing tracks at high speeds, there can... | {
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The Delta-Function Potential I'm reading through Griffiths Intro to QM 2nd Ed. and when it comes to bound/scattering states (2.5) they say:
$E<0 \implies$ bound state
$E>0 \implies$ scattering state
Why doesn't this change depending on whether you have a positive or negative delta-function potential?
| If $E < V\left(-\infty\right)$, $E<V\left(+\infty\right)$, and $E > V_{min}$ (necessary for $\Psi$ to be normalizable), then it is a bound state, and the spectrum will be discrete:
$$
\Psi\left(x,t\right) = \sum_n c_n \Psi_n\left(x,t\right).
$$
Otherwise -- if $E > V\left(-\infty\right)$ or $E > V\left(+\infty\right)$ ... | {
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How does water help extinguish fire? How does water extinguish fire? Heat energy from the fire is transferred to the water, isn't that how it works? How does water deprive oxygen and stop combustion? How is the specific heat of water connected to this? If we use hot water instead cold water, does that make a difference... | Water does not, in general, help extinguish a fire.
Typical fires, however, can be successfully attacked using water alone, as it can cool the fuel at the base of the fire or generate a vapor barrier between atmospheric oxygen and the hot fuel.
Water can accelerate liquid hydrocarbon fires by dispersing fuel.
Water can... | {
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What is the conceptual difference between Gibbs and Boltzmann entropies? In simple words what is the conceptual difference between Gibbs and Boltzmann entropies?
Gibbs entropy: $S = -k_B \sum p_i \ln p_i$
Boltzmann entropy: $S = k_B \ln\Omega$
| The expression
$$
I=-\sum_i p_i\ln p_i
$$
is a function of probabilities $p_i$, also called information entropy. It can be calculated for any values of $p_i$, even those that are not appropriate for an equilibrium state. In such case it has nothing to do with equilibrium thermodynamic entropy $S$. When some equilibrium... | {
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Relation between conformal and topological field theories The Chern-Simons (CS) theory is a topological quantum field theory (TQFT).
*
*The question is, is a conformal field theory (CFT) a topological quantum theory?
*Or the reverse, topological quantum field theory is a CFT?
*What is a conformal field theory (CF... | Roughly, every chiral part of a rational CFT gives a TFT theory.
For example for WZW models the chiral parts are current algebras. The corresponding TFT is Chern-Simons theory.
The point is the representation of a chiral rational CFT is a modular tensor category. From a modular tensor category one can construct a 3D ... | {
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What happens to waves when they hit smaller apertures than their wavelenghts? I was wondering this for quite a long time now. Let's say you have a water wave (like ripples, not the ones you see during tsunamis) with wavelength 10 m. Imagine you put a boundary with an opening of 1 m. Will diffraction be observed? Accord... | Well - it depends what you mean by "diffraction".
Any point along a wave front will act as a point source of wavefronts travelling radially outwards - so while a plane wave will be arriving at the aperture, the waves that come out from the other side will almost look circular.
To me, that's diffraction. Although there ... | {
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What is the meaning of pre-tension for a stiff membrane? On one hand, I know that the tighter a drum head is stretched, the higher its natural frequencies. This relation is given by:
$$f_{ij}=\frac{k_{ij}}{2\pi R}\sqrt{\frac{T_0}{h\rho}}$$
where $k_{ij}$ are roots of the Bessel function of the first kind, $R$ is the me... | One should differentiate between ideal membrane vibration, in which the vibration is described by a wave equation, and plate/shell theory (in which the vibration is described by more complex approaches, such as Kirchhoff–Love plate theory and Mindlin–Reissner plate theory).
As @wetsavannaanimal-aka-rod-vance pointed ou... | {
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How is antimatter made? How is antimatter made in laboratory? Can anyone explain, at the particle level, specifically how anti-protons and anti-electrons are made?
| Since John is not addressing positrons one should know that positrons are easily created once a photon has more energy than twice the mass of the electron, in electron positron pairs.
This can be seen clearly in this bubble chamber picture:
where the positron is shown in purple on the right. One knows they are elect... | {
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Why do propellers look like they are moving really slow when they are moving really fast? I was just looking at an airplane video and was wondering why it looks like the propellers are moving relatively slowly when they infact have a RPM in the thousands probably.
| Videos typically have a frame rate of ~25 frames per second. The 'movie' you see is a result of these snapshots. If you assume the propellers were moving at 9k RPM, that is 150 revolutions per second. If the camera recorded at 150 FPS, the propellers would appear stationary as they would be in the same position at each... | {
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Why Nitrides have internal polarization? Why GaN has internal polarization? I know that in wurtzite crystal structure, the atomistic bonds are not equivalent and as a result there appears a net dipole and consequently a polarization. But what is special about the Nitrides like InN and GaN which atoms prefer to arrange ... | It's because there are alternating layers of + and - ions in the c-direction, yielding a dipole in the crystal. See Tasker -- http://www.surface.tulane.edu/teaching/classnotessurface/TaskerJPhysC79.pdf
| {
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How much entropy is produced in evaporating water due to irreversible evaporation towards equilibrium (humidity=100%)? $dS=dH/T_{boil}$ for the increase in entropy by changing a phase at saturation ($T=373\text K$ for $p=1\text {atm}$ for water). However, water also obviously evaporates below boiling point when equilib... | $T$ should be the actual temperature at which the water evaporates. That is, the temperature at the interface between the air and the water, not the boiling point.
This is simply because $dU = TdS + pdV - \sum_i \mu_i dN$ (where $T$ is most definitely the temperature of the system), or by rearranging,
$$
dS = \frac{1}{... | {
"language": "en",
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Reconciling Units in Classical System Analogies: Why Does Torque Have Units of Energy? In classical physics we often cast an analogy between translational and rotational systems
Force < > Torque
Energy < > Rotational Energy
Momentum < > Angular Momentum
and considering SI units we have [Force] = N, [Torque] = N-m, [Ene... | We could standardise Torque at 1 meter and it would then only be a force. It is in fact only a force, but one that can be varied with r (radius). As a vector it has direction, but we do not use this by convention. As engineers, we want to divide this force, or multiply it. Gears, pulleys, chains. The units are the unit... | {
"language": "en",
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What make us fall when we move on ice? This is a common phenomenon and most of the people have encountered this while moving on ice:falling down! Why does this occur?
According to me, it is due to lack of static friction. But why will it be absent here? What is the cause? I have heard that there is a thin layer of wate... | Actually, the surface of the ice is frozen, until you step on it, then it melts. When pressure on the ice is increased, it can melt if it is not too cold. This is why ice skates work so well on the ice. They create a lot of pressure; creating a water layer between the ice and blade. This water layer decreases the frict... | {
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Born's rule and Schrödinger's equation In non-relativistic quantum mechanics, the equation of evolution of the quantum state is given by Schrödinger's equation and measurement of a state of particle is itself a physical process. Thus, should be governed by the Schrödinger's equation.
But we predict probabilities using... | According to the Many Worlds Interpretation, the measurement device, and by extension everyone who reads it, gets entangled with the particle in exactly the way the Schrödinger equation predicts. The Born rule tells you how subjectively likely you are to be in a given universe. It's basically just a really bizarre anth... | {
"language": "en",
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How can nuclear decay both annihilate and create? This'll be back to basics for many of you, but here's something I still don't get.
How can nuclear decay of an unstable atom both create and annihilate positron-electron pairs?
You have an unstable atom, say Mg-22 (nucleon: 22, proton: 12), which decays and emits positr... | Gamma decay is the emission of photons. You are thinking of $\beta$ decay.
When the particle is decaying, if it emits a $W^{-}$boson, it will subsequently decay and create an electron ($e^-$), and an electron antineutrino ($\overline{v}_e$), the antimatter particle to an electron neutrino. It will also flip a neutron ... | {
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Black hole "no hair" theorem The "no hair" theorem (or conjecture), suggests that black holes can be entirely described by their mass, angular momentum and charge. All other details of the BH formation are lost.
Is there a simple way of understanding why you would be able to tell the difference between a BH made from p... | The simplest answer lies in a combination of Gauss's law and Birkhoff's theorem. These say, alternately, that the electric field and gravitational field of a spherically symmetric charge distrubtion only depends on the charge and mass-energy enclosed in the spherical shell${}^{1}$.
Therefore, if I have a spherically... | {
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Why do vapour cones form around jet fighters? Apparently this phenomenon has nothing to do with jets breaking the sound barrier and has something to do with the Prandtl-Glauert singularity as described on Wikipedia. But, the Wikipedia article isn't very detailed and it doesn't explain why the cone arises.
Is there a re... | Condensation of atmospheric water vapour due to the shock wave. Air molecules can travel with the limit of speeed of sound (343 m/s), hence molecules hitting the body of an aircraft travelling at sppeds lower than that move away. Once the aircraft crosses the speed of sound, the molecules are pushed with a speed faster... | {
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How fast would someone have to run to travel vertically up a wall? I am currently doing a physics project on the effects of so-called 'super-speed'. I was wondering how fast you would have to run to vertically travel up a wall? That is, to negate the force of gravity. Is it even possible? Help would be appreciated!
| Assuming you have overcome the lack of traction, it would be more about the force required than the speed.
The required force would also depend on your weight.
The force of what we see as gravity is 9.8 N/Kg.
Therefore to maintain your position without moving, if you weigh 200Kg, you would need to exert at least 1,960 ... | {
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Is a suit that hides a soldier's heat signature fundamentally possible? I recently played "Crysis", a game where the protagonist wears a suit that allows the player to hide both himself and his heat signature. Then I watched Iron Man 3, where a kid suggests that Tony Stark should have implemented retro reflection panel... | You could try to develop a material that acts like a fluorescent one, that is, transform infrared radiation into lower energy photons, such as microwave. So you will not glow on infrared but on some other wavelength of your choice. Now, if this is technologically feasible (using nano-engineered materials perhaps), I ha... | {
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Solving flow equations with less information Suppose we consider the flow of a fluid through a channel, considered to happen in a plane. We want to consider the region $D = [0,L]\times [-a,a]\subset \mathbb{R}^2$ representing the piece of the chanel we are looking at. Furthermore, considering constant density $\rho_0$ ... | About the $t$ independence part of your question, the answer is yes, you can find a lot more solutions which do depend on the time. Once you get to the PDE for
$u_1(t,y)$, you will find it is like the forced ODE's you studied, where you can add a solution of the homogeneous equation. Specifically here the PDE is
$$\fr... | {
"language": "en",
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How much effort would be required to fix the Earth's rotation? Given that the earth's rotation has been slowing down by very slight amounts over time, forcing us to introduce leap seconds and so forth into our clocks and calendars, I would like to ask if this could be fixed by "generating" more spin via some sort of po... | A day is currently about 86400.002 seconds long. If we could just increase the Earth's rotation rate by a mere 2 milliseconds per day we would get rid of the need for those pesky leap seconds. No problem! We only need something that rotates with an angular momentum of 1.4×1026 joule-seconds about an axis pointing due s... | {
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How is Kirchhoff's voltage law understood in the water flow analogy? I met the Kirchhoff circuit laws in the past, but now I'm trying to associate them with a practical representation to be sure to understand them.
Let's start with the Kirchhoff current law: If I say that the electrons are like water going though a pi... | As you said, current is like water flow, similarly voltage is like water level and voltage difference like difference of water level. We know that water flow from higher level to lower level like current flow from higher voltage to lower voltage. Voltage difference means there is a difference of charge, i.e., a differe... | {
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Poisson brackets of the Kepler Problem For the hamiltonian of a particle of unit mass in a kepler potential:
$$H = \frac{1}{2}\mathbf{p} \cdot \mathbf{p} - \frac{\mu}{r}$$
The angular momentum vector is given by: $\mathbf{L} = \mathbf{r} \times \mathbf{p}$
I know and can show that the poisson brackets of $\mathbf{r} \c... | The classical poisson bracket with the generator of any symmetry gives the infinitesimal evolution with respect to that symmetry. The most familiar statement of this is that the time-evolution of any observable $f$ on the phase space is given by
$$ \partial_t f = \{H,f\}$$
Similarily, for a rotation around the $i$-th ... | {
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What type of energy is Dark Energy? What type of energy is Dark Energy? Just as the title says. Is it kinetic/potential or some other type?
| In Newtonian mechanics, we have potential energy and kinetic energy. All types of energy can be classified as one or the other -- never both or neither.
But when you deal with fields, this distinction doesn't really work. For example, a light wave (electromagnetic field) carries energy, and this energy doesn't fit neat... | {
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Clausius statement of the 2nd Law I'm slightly messed up with the Clausius statement of the 2nd Law.
I've seen at least two versions, which seem to be conceptually different.
a) It is impossible to transfer heat from a colder body to a hotter body without any other effect.
b) It is impossible to transfer heat from a co... |
It is impossible to transfer heat from a colder body to a hotter body without any other effect...
in the surrounding bodies.
Every reservoir is a body too.
| {
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Negative frequency contributions for very short pulses? I am wondering if very short optical light pulses can have a Gaussian envelope?
When I describe the pulse shape with a Gaussian than the frequency distribution has also a Gaussian shape. But if the envelope of the short pulse will have a pulse width of $1\,\text{a... | We sometimes forget about negative frequencies but they are always there. All real signals are composed equally of negative frequency and positive frequency components.
If you take an oscillating pulse with Gaussian envelope,
$$E(t) = A \exp(-\tfrac 12 t^2/\tau^2)\cos(\omega_0 t)$$
and convert it into its spectral repr... | {
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Tesla Coils - Is there a risk that the discharge can create x-rays? I've built a Tesla coil that stands about 3 ft tall and uses a spark gap as the interrupter for the primary circuit. Judging by the size of the streamers it's reaching at least a million volts.
Someone once told me that you have to be careful with Tesl... | Generally Tesla coild are fairly safe from the point of view of X-ray generation. You could potentially test for it using a sealed can of camera film with a high ISO rating, placing it in the vicinity of the coil while it is operating, and then developing it to see if it is fogged.
However, a bigger problem might be ul... | {
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Rotational Speeds I was driving the other day and couldn't help but look at all the wind mills in the wind farms. I know I am not doing the math correctly, but I want to know what is "really" happening when a windmill spins.
Suppose there is a wind mill with one blade that is 1 foot long. Now lets say it takes 1 secon... | Different parts of the blades have different speeds, but all parts of the blades have the same instantaneous angular speed, that is all parts travel through the same angular displacement in a given amount of time. This is always true for a rigid body (when the angular speed is measured about the rotation axis).
As ACur... | {
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Is it possible to eliminate Van der Waals interactions? I came to know that the friction force actually depends on the surface contact area due to weak interactions (adhesion due to Van der Waals forces) between the atoms of both materials increasing in number with an area increase.
If this force could be over-ridden w... | I don't know what you have in mind, but friction can be mitigated using such things as low friction materials (such as PTFE), wheels, roller/ball bearings, lubrication, air lift, magnetic levitation, and what not...
| {
"language": "en",
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Renormalization in non-relativistic quantum mechanics I read many articles about renormalization in the Internet, but as I currently don't know much of QFT (currently just studying classical field theory and QM), and as all this looks quite interesting, I'd like to still get some bit of understanding and feeling of it ... | To understand the essence of perturbative renormalization you don't need any quantum field theory nor any quantum mechanics. A simple toy model suffices.
Suppose your theory makes a prediction for two distinct observables $F$ and $G$ in terms of a perturbative parameter $g$:
$$F = g + g^2 (S+1) + g^3 (S+1)^2 + g^4 (S+1... | {
"language": "en",
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Finding the total energy of satellite in orbit with the Earth I'm given a problem where a satellite with mass $m$ is orbiting the Earth with mass $M$ and radius $R$ ($m<<M$) in a elliptical orbit. The ellipse has semimajor axis $a$ and semiminor axis $b$. I'm asked to find the total energy of the sattellite in terms of... | It would be better to think in terms of conserved quantities (angular momentum and energy): for a central potential we have that $$E=\frac{1}{2}m\dot{r}^2+\frac{L^2}{2mr^2}+U(r)$$ In this case (gravitational potential) $U(r)=-\frac{GMm}{r}$. You could evaluate energy at $r=a$ and $r=b$ (where $\dot{r}=0$) then calculat... | {
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The supersonic bandwagon I happen to stumble upon this intriguing animation of a Tachyon, and immediately started to think about the same phenomenon, but with sound waves. I imagined a bandwagon B moving at a constant supersonic speed, and passing an stationary observer O.
... | It would appear to be correct. An example is a supersonic bullet fired over your head from a few hundred metres distance. You hear the crack of the bullet's shock wave as it passes, the sound of the round fired and (if your hearing is till up to it) the kind of sucking sound the bullet makes as it travels downrange. I ... | {
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Energy-Time Uncertainty Principle and Photons Heisenberg's uncertainty principle states that:
$$
\Delta E \cdot \Delta t \ge \frac{\hbar}{2}
$$
It is clear that this has nothing to do with the accuracy of our measurements, but rather is a fundamental 'law' in the quantum world. Now, we also know that photons do not exp... | The question is correct in that the relativistic (or lorentz-invariant) time-energy uncertainty relation is a bit different (but still there is)
For example here is a pre-print Lorentz-Invariant Time-Energy Uncertainty Relation for Relativistic Photon, arxiv
Abstract:
The time-energy uncertainty relation is discussed ... | {
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Distribution of gravitational force on a non-rotating oblate spheroid Suppose a person is standing on a non-rotating$^1$ oblate spheroid of uniform density. He first stands on one of the poles, then on the equator. In which case is the gravitational force greater?
In Case 1, the distance between the centre of the pers... | For when the guy stands on the pole, slice the planet into circular disks, with their centers on the axis, each with radius = r = f(h) and thickness = dh, where h is distance of the center of the disk from one pole, and f(h) is function that represents shape of oblate spheroid. Then slice each disk into multiple rings... | {
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What is the relation between kinetic energy and momentum? If kinetic energy is doubled, what happens to momentum? Is it also doubled?
I've tried working through the formulas for each but keep getting lost.
$$KE=\frac{mv^2}{2}$$
$$p=mv$$
so if $v=\frac{p}{m}$ then $KE= \frac{m}{2} \cdot (\frac{p}{m})^2$ so $KE=\frac{1}... | The relation between Kinetic Energy and momentum is derived in the following way. I hope this answers your question.
| {
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How to know if a vehicle is moving without any external source of information? The situation is the following:
I'm inside a vehicle (plane or a car, it doesn't matter) and I need to know if the vehicle is moving at a constant speed BUT I cannot perceive any external change like visual changes, vibration, etc.
How can I... | Assuming that you know the gravitational field in the area you're passing through, you can determine whether or not you're moving at a constant speed (which I take to mean "stationary in some inertial frame") by whether or not you're feeling gravitational forces other than what you'd expect from the field. For example... | {
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Bose Enhancement Factor How may one explain the fact that the probability of a boson transferring to a state with an occupation number n is 'enhanced' by a factor of (1+n), compared to the classical case? (In the classical case, the probability is supposed to be independent of the occupation of the final state.)
| I will start from rather far away, but will get to the point eventually. Of course, it is possible to rigorously derive kinetic equations, or consider some statistical field theory (as suggested in some of the answers to this question), but I want to take a simpler approach. I will discuss (effectively) non-interacting... | {
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Why the photon can't produce electron and positron in space or in vacuum? $$\frac{hc}{\lambda} = K_e + K_p + 2m_e c^2$$
could be the energy conservation equation for a photon of wavelength $\lambda$ decaying into a electron and positron with kinetic energies $K_e$ and $K_p$ and rest mass energy $m_e c^2$.
Why does this... | The uncertainty principle is an inequality so the momentum times c times cosine which is energy may be made to equal the energy of the momentum and rest mass of the combined electron and positron. energy and momentum conservation is satisfied using the uncertainty principle.
| {
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How does a single charge produce magnetic field? I have studied in Introduction to electrodynamics (Griffiths) that magnetic field is actually due to effects of relativity
unequal Lorentz contraction of the positive charge and negative lines, a current- carrying wire that is electrically neutral in one inertial system... | Related How do moving charges produce magnetic fields?
A moving charge $q$ with velocity $\vec{v}$ produces a current
$$\vec{j}=\rho(r) \vec{v}$$
($\rho$ charge density associated to charge $q$, $\vec{j}$ current density, ref)
and by Maxwell's equation for the magnetic field (Ampere Law):
$$\nabla \times \vec{B} = \... | {
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What is the entropy of a pure state? Well, zero of course. Because
$S = -\text{tr}(\rho \ln \rho)$ and $\rho$ for a pure state gives zero entropy.
But... all quantum states are really pure states right? A mixed state just describes our ignorance about a particular system. So how can properties like entropy and tempe... |
But... all quantum states are really pure states right?
There exists the density matrix formulation for the many body system quantum mechanically. What happens in a many body problem, order of 10^23 molecules per mole that are the appropriate numbers for a thermodynamic formulation, is that the off diagonal elements... | {
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Semiclassical quantization of bouncing ball Consider an elastically bouncing ball of mass $m$ and energy $E$. This has a triangular potential
$$ V(x)~=~\left\{\begin{array}{ll} mgx & \text{if } x>0, \\
\infty & \text{if } x<0, \end{array}\right. $$
where the $x$-axis points upwards. Let $\hbar = m = g = 1$, so that ... | I) Ignoring the metaplectic correction/Maslov index, the Bohr-Sommerfeld quantization rule reads
$$\begin{align} N ~\approx~&\int_a^b \!\frac{\mathrm{d}x}{\pi\hbar} p(x)\cr
~=~& \int_a^b \! \frac{\mathrm{d}x}{\pi\hbar} \sqrt{2m(E-V(x))},\end{align} \tag{1} $$
so that
$$\begin{align} \frac{dN}{dE} ~\stackrel{(1)}{\app... | {
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} |
Why does correlation length diverge at the percolation threshold? I'm reading a paper about electronic percolation. $p$ is the fraction of occupied bonds (or sites, depending on the model you're using, but I'll just use bonds), $p_c$ is the critical fraction of occupied bonds, and $G(x)\propto e^{-|x|/\xi}$ is the 2 po... | This is what the percolation threshold and phase transitions are all about, the emergence of diverging length scales with power laws - which makes the system scale invariant.
If you want physical intuition, play around with one of the many Ising model applets found on the web, like this one.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/145987",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
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