Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How do bicycle spokes work? As you know, it is quite obvious that bicycle spokes attach the hub in the center to the rim. What else do they do? If you compare the wheels today with the ones from ancient times, there are more spokes now on motor bikes and bicycles than of a wheel of a chariot. Why is that? What effect d... | If there is weight on the axle the rim gets pushed down into the ground and tries to deform by flattening on the bottom and bulging right besides the ground. Properly tensioned spokes will counteract this bulging and lessen the deformation allowing for an easier and smoother ride.
This means that the rim does not have ... | {
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"url": "https://physics.stackexchange.com/questions/94001",
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Why are two independent sources incoherent? Coherent sources are produced from a single parent source. But, why are two independent sources always incoherent? Two sources can produce light of the same frequency. Then, I guess the problem is with phase. Are two independent sources always out of phase? why so?
| We usually generate light by black body radiation, that is by heating something until it glows. There are many sources for black body radiation, but the dominant one is usually random thermal motion causing random transient electric dipoles within the black body. The changes in these dipoles generates eletromagnetic ra... | {
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Do neutrinos travel faster than light in air? I read in wiki that the speed of light is 88km/s slower in air than it is in a vacuum.
Do neutrinos travel faster than light in air?
| The answer is yes. Neutrinos will travel faster than light in a medium with a refractive index ($n$) greater than one (which is the case of air). Indeed the speed of light in that medium will be $v_{\text{medium}}=c/n$ where $c=2.998\times10^8$ m/s and $n>1$.
Then, because neutrinos interacts only very weakly (only thr... | {
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If a material is built to handle tension, would removing the tension damage it? If an object is designed to cope with large forces such as tension, would removing these forces risk damaging the object?
For example: The neck of a guitar is built to handle the tension of steel strings (~800 Newtons),
if you removed/reduc... | A piston-cylinder pressure vessel is an example of an object that commonly fails in this way. Carbide has a high compressive strength but lower strength in tension. Pressure vessels may be pre-loaded externally, with a confinement ring and interference fit, so that they can withstand higher internal pressure. If the ca... | {
"language": "en",
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Alternative Critical Dimensions in String Theory Is it possible to write down a Lagrangian for a string theory with a critical dimension different than the familiar 10 or 26? How could one find a string theory Lagrangian for a given dimension? Could you prove that no string theory exists for a given critical dimension?... | You write a lagrangian for a string propogating in $(d,1)$ dimensional spacetime and notice that it is equivalently a 2d conformal field theory on the worldsheet of the string. Consistency conditions (vanishing of the conformal anomaly) then imply that necessary dimension of spacetime be 26 for bosonic string theory an... | {
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During reflection does the emitted photon have same properties? When light (photon) is reflected the the original photon is absorbed by an electron and then emitted again. Does this "new" photon have the same wavelength, frequency etc. as the original?
| Your title and question has some in-consistency, for example, photon can also have spin, and the field itself can also carry orbital angular momentum et.al. But from fresnel equation:
r=(n-1+ik)/(n+1+ik)
where n and k are the real part and imaginary part for the materials, you can see the reflection spectra already c... | {
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Angular speed of the planets Do all the planets in our solar system have the same angular speed? Physics teacher says yes, my research is not crystal clear. I want to make sure I have the right information for future reference.
| No, planets in the solar system do not have the same angular speed.
Firstly, strictly speaking, the angular speeds of the planets change with time because they travel along ellipses, not circles. But the eccentricities of many of the planets are close to $0$ (see e.g. Why are the orbits of planets in the Solar System ... | {
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Chiral Spin Liquid(CSL), Chern number, and the ground state degeneracy(GSD) Consider a 2D gapped CSL with a nonzero Chern number $m$, then is the GSD of the system on a torus directly related to the Chern number $m$?
For example, see this article, in the last paragraph on page 7, the authors give the 4-fold GSD from th... | Based on the paper, the answer is $|m|^2$. They suggest in their p.8, Eq.36, the effective theory is a Chern-Simons theory
$$
\frac{1}{4\pi}\int K_{IJ} a_I \wedge d a_J
$$
with the $ K_{IJ}$ bilinear K matrix as
$$K_{IJ}={\begin{pmatrix}m & 0\\ 0 & -m\end{pmatrix}}$$.
The up $m$ labels one sector and the lower $m$ lab... | {
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Gravitational collapse and free fall time (spherical, pressure-free)
A very large number of small particles forms a spherical cloud. Initially they are at rest, have uniform mass density per unit volume $\rho_0$, and occupy a region of radius $r_0$. The cloud collapses due to gravitation; the particles do not
intera... | There is a well-known solution for this problem which is as follows : Gravitationally the outer layer of the cloud is influenced by the rest just as the rest were compressed into a point mass. Therefore we have Keplerian motion: the fall of any part of the outer layer consists in a halfperiod of an ultra-elliptical orb... | {
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How is dark energy calculated This should be a very simple question. What would be the proper way to calculate Dark Energy in Joules at any point in history and that is consistent with the Standard Model? I'm thinking that knowing the mass-energy of matter (after estimating the Mass of the Universe):
$$E_m = mc^2\tag... | It's simpler, since the amount of dark energy remains the same while de Universe evolves. Then, you can calculate it today: $\rho_\Lambda(t)=\rho_\Lambda(today)$.
Today $\Omega_\Lambda=\rho_\Lambda/\rho_c$, where $\rho_c$ is the critical energy density today, and $\Omega_\Lambda$ is measured to have a value $\Omega_\La... | {
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White light diffraction I have a hard time understanding why light waves of different wavelengths diffract in a different manner. According to Huygens' principle, every point on the wavefront is a source of a secondary wave. So if we have a white light going through, say, a single slit (light rays parallel to each othe... | I do not think Huyghens principle can be applied to white light, only to simple harmonic waves. Waves of light with different color have different wavelength, which will affect the radius of sphere drawn in the Huyghens construction. Around obstacles, waves with different wavelengths will move differently.
| {
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Does a moving object curve space-time as its velocity increases? We always hear how gravity bends space-time; why shouldn't velocity?
Consider a spaceship traveling through space at a reasonable fraction of the speed of light. If this spaceship, according to special relativity, gains mass as a factor of y as it approac... | Equivalence requires acceleration to curve space the same way gravitating mass does. Hermann Weyl, Zur Gravitationstheorie, Annalen der Physik, 54, 117, (1917) argued that kinetic energy should curve space just as gravity and electromagnetic fields do by entering into the stress energy tensor.
The concept of inertial ... | {
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Measuring background radiation We tried to measure background radiation using a geiger counter for a experiment at school. The meter showed $0.12$-$0.21$ microSv/h during the day averaging at about $0.14$ mcSv/h.
As we tried to see ways how to shield incoming radiotion nothing seemed to work. Taking a cue from nuclear... | Thans for all answers and links.
If cosmic radiation is about 0.03 mcSv/h we should still see a drop.
There is no power plant in the area.
The pool was an indoor pool and it is in the basement which has a solid concrete ceiling.
How would I set this experiment to show that the counter goes down preferably to zero.
| {
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How do electrons get the energy to jump from one orbital to the next when in stationary orbits the electron does not radiate energy My Question is how do electrons get energy/lose energy to jump up or down an orbital as in a stationary orbit they do not gain energy and their energy is finite?
| Caution: simplified but hopefully not over-simplified answer approaches!
If you take a hydrogen atom, that is the system of one electron and one proton, and you solve the Schrodinger equation for it then you get a number of eigenfunctions. These are the well known hydrogenic orbitals; 1s, 2s, 2p, etc. These solutions a... | {
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Centre of instantaneous rotation problem Is there a point of Centre of Instantaneous Rotation (CIR) for every type of motion or only for cases of rolling?
| The fact you are stating is quite general in fact and even extends in a related form to 3 dimensions also.
It is known as Chasles's rotation theorem:
Any general displacement of a rigid body can be represented by a translation plus a rotation.
In the case of motion of a body in a plane,the axis intersects the g... | {
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Electrons on Stern–Gerlach experiment My questions about spin and negative charge of electrons. Stern-Gerlach experiment is very famous in order to find spin of electron. this video created by paris-sud university really well-explained on this experiment except one thing.watch the video first.
http://en.wikipedia.org/w... | Neat video.
That experimental setup is not measuring the charge of the particles. If it were, you would be correct; you'd get the same result each time for all of the identical particles. Instead, that setup measures some other property of the particles. Physicists gave that property the name spin. Spin is not charge, ... | {
"language": "en",
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Can a liquid insulator be electrically charged by touching a charged conductor?
Can a liquid insulator be electrically charged by touching a charged
conductor?
I understand that solid insulator will only be charge on the surface where it is touch, but the case is different from liquid which it circulates, so I imag... | Whether charge is transferred between an insulator and a conductor depends on the field strength. If it is high enough charge will move over due to corona discharge. That even works when the two objects are not touching because ions can move through the air.
So yes, if the field strength is sufficient you can charge a ... | {
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Why does a flat clover-like shape fall slower when it is rotating? The plastic flat clover-like shape pictured below falls noticeably slower when it is rotating fast. I wonder why. Its three edges are flat, so I don't think that the rotation makes it act like a propeller. An ideal explanation would include an analog... | There is an amount of lift that each blade gets, and it depends on the angle at which the air hits the blade.
When you just drop the toy, the air hits each blade directly at 90 degrees.
In that case, all you get is drag.
In airplane terms, the blade acts like a stalled wing.
It's more like a parachute than a wing, and ... | {
"language": "en",
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Why is $U(\Lambda)^{-1} = U(\Lambda^{-1})$ for a unitary representation? This is from the beginning of Srednicki's QFT textbook, where he writes (approximately):
In QM we associate a unitary operator $U(\Lambda)$ to each proper orthochronous Lorentz transformation $\Lambda$. These operators must obey the composition r... | Indeed, it is also necessary to assume that $U(I)= I$ where the former $I$ is the identity matrix in the group and the latter is the identity operator in the Hilbert space.
So, taking advantage of your first identity you have: $$U(\Lambda^{-1})U(\Lambda) =
U(\Lambda)U(\Lambda^{-1}) = U(I)=I$$ and it implies $U(\Lamb... | {
"language": "en",
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Faster than light galaxies/clusters? A few years ago in an astronomy course, we calculated some (transverse?) velocity of a moving object and got super luminal results. The answer was apparent and not physical velocity of the object. Hence no problem. But at the moment, I don't recall the solution to this apparent issu... | This question already has two good answers, but I needed an excuse to learn tikz, so here is my answer.
This "transverse" speed can be faster than the speed of light $c$ if the object is coming towards you sufficiently fast. To see how this works, look at the diagram below.
Here the object is moving at a speed $\beta ... | {
"language": "en",
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If one is travelling at a significant fraction of $c$, will the length of the trip be shortened? Assuming two stars are 1 light year apart and a traveler is travelling at 0.75 of $c$, from the point of view of the traveler what would be the observed time en route? Also, if a vehicle is constantly accelerating, will it ... | The time measured on board the ship during such a trip would be 0.88 years (as calculated by Alex A). If you factor in constant acceleration during the whole trip (starting from zero and passing by the target at maximum speed), the time on board would be 1.9 years and you would be passing by the target at 0.75 c (coinc... | {
"language": "en",
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Proof of minimum distance between object and image of a convex lens image being 4*focal length I can prove that the minimum distance between an object and its image, through a convex lens is 4*focal length, if I assume that the distance between the object and the lens is the same as the distance between the image and t... | Although the other answers are good, I will add my slightly different one. Rearranging the image equation gives;
$$
u+v=\frac{1}{f}uv
$$
On the left hand side you have the perimeter of a rectangle (well half of the perimeter, but it doesn't matter for the proof), and on the right hand side you have the area of the rec... | {
"language": "en",
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Energy of an inductor I know that for an inductor having self inductance $L$ energy stored in its steady state when a current $I$ has been established is given by $U = \frac{LI^2}{2}$.
But after this current has been established, if we suddenly cut the wires attaching the inductor to the potential source or short the c... | Satwik's answer is correct but I want to add a practical example. When we switch off some electric device(say an DC motor) there is a spark is produced ,this is where the energy is getting lost. This spark is so high that can be seen even with naked eyes at the switch.
PS: Do this experiment in vaccum then there b... | {
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Why am I not burned by a strong wind? So I was thinking... If heat I feel is just lots of particles going wild and transferring their energy to other bodies, why am I not burned by the wind?
When I thought about it more I figured out that wind usually carries some humidity, and since particles of liquid are moving same... | The other answers address your question quite well. Just as a reminder of the ability to be burned by a strong enough wind, the image below shows the Chelyabinsk meteor during entry into Earth's atmosphere last year over Russia. :)
| {
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Weinberg dimension 5 operator How to prove that the $\Delta L=2,$ dimension=5 Weinberg operator $LLHH$ is the unique operator which violates lepton number by two units, without derivative couplings, etc.??
| This $LLHH$ is not a unique operator. It just the operator with a lowest dimension using only Standard Model particles and giving Majorana neutrino masses. The way to understand this is as follows. Majorana neutrino mass must be of the form
\begin{equation}
\nu _{ L }^{\,\, T } m C \nu _L
\end{equation}
This term b... | {
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Euler-Lagrange equations and friction forces We can derive Lagrange equations supposing that the virtual work of a system is zero.
$$\delta W=\sum_i (\mathbf{F}_i-\dot {\mathbf{p}_i})\delta \mathbf{r}_i=\sum_i (\mathbf{F}^{(a)}_i+\mathbf{f}_i-\dot {\mathbf{p}_i})\delta \mathbf{r}_i=0$$
Where $\mathbf{f}_i$ are the cons... | The main point is that Goldstein is not saying we must exclude friction forces in our treatment, but we must place them in the tally of applied forces (that we keep track of in D'Alembert's principle) and not in the other bin of the remaining forces, see this and this Phys.SE posts.
Of course, there does not exist a ge... | {
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Why does a star collapse under its own gravity when the gravity at its centre is zero? The gravity at the centre of a star is zero as in the case of any uniform solid sphere with some mass. When a massive star dies, why does it give rise to a black hole at it's centre?
I know how to derive the field equations for grav... | Since every particle attracts all other particles, there is a net force directed towards the center of the star (or any object), for any particle not at the center. Therefore, the particles will move towards the center (collapse), unless some opposing force prevents it. In the case of a star, the kinetic energy of th... | {
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Non-Locality of Space - QFT (Srednicki's book) I was going through Mark Srednicki's book on QFT. It says in the relativistic limit the Schrodinger equation becomes something like :
$$ i\hbar\frac{\partial}{\partial t} \psi(\vec x,t) = \sqrt{-\hbar^2c^2\nabla^2+m^2c^4}\psi(\vec x,t) $$
Now he says that if I expand the ... | If you consider a standard differential operator $B$ working on functions defined in $\mathbb R^n$, like $\partial/\partial x_i$ or a polynomial of partial derivatives, and pick out a sufficiently smooth function $f$ vanishing in a neighbourhood $\Omega$, you see that also $Bf$ vanishes therein. This is the relevant no... | {
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Which electron gets which energy level? Electrons sit in different energy levels of an atom, the farther the higher energy is. Every electrons have the same structure, they can gain energy from environment, electrons which gained energy could jump to a higher energy level and will finally fall back again.
I'm wonderin... | I'm surprised that no one has mentioned that there is really no such thing as "this electron" or "that electron" in an atom. Those are useful approximations that help us visualize energy levels; but the actual quantum-mechanical theory of, for example, a carbon atom with six electrons, is based on a single electron wav... | {
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Propagator of Maxwell-Chern-Simons theory I need to compute the "topologically massive photon" propagator.
I've started with:
$$
\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \frac{\mu}{4}\epsilon^{\mu\nu\lambda}A_\mu\partial_\nu A_\lambda
$$
$$
=A_\mu\underbrace{[\frac{1}{2}g^{\mu\lambda}\partial^2+\frac{\mu}{2}\epsi... | You might try it by the standard way of calculating Green's function (which is what the propagator is) of linear differential operators. For this purpose you have to derive the equations of motion from the given Lagrangian, which will be some operator acting on the gauge field. The crucial observation is now that the G... | {
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Admixtures of longitudinal and timelike photons! In the quantization of electromagnetic field the physical states $|\psi\rangle$ are found to obey the following relation:
$[a^{(0)}(k)-a^{(3)}(k)]|\psi\rangle=0$
It is explained as the physical states are admixtures of longitudinal and timelike photons. What do longitudi... | The total field consists of the "near" field like the Coulomb one and more generally (and loosely) a retarded Coulomb field, which are always "attached" to the charge, and the photon (radiated) field with different polarization orientations. The near field is always present, its "photons" are not created and annihilate... | {
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Frequency of rotating coil Given a coil initially in the x-y plane, rotating at angular frequency $ \omega $ about the x-axis in a magnetic field in the z-direction. This uniform time varying magnetic field is given by $B_z (t)=B(0)cos(\omega t) $ I am required to show that there is a voltage of frequency $2\omega $ ac... | User31782 gave the right answer, but it's quite hard to read because of formatting. Let me repeat the argument for you:
The coil rotates at $\omega$, and the field is also changing at $\omega$.
At any moment in time, the area of the coil normal to the direction of the field is
$$A = A_0 \cos(\omega t)$$
and the field i... | {
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If Earth was the size of an orange, what consistency would it be? I understand that the Earth is composed of a thin crust "floating" over a semi-solid layer of mantle.
I was wondering how it would feel at a human scale, say the size of an orange in your hand. Could you somewhat squeeze it or would it rather be totally... | At first you'd burn your hand, then it would feel like a normal rock.
An orange sized Earth would cool very rapidly.
If an object gets twice as big, its volume increases by $2^3$, but its surface increases only by $2^2$. You can only lose heat at the surface but you 'hold' all your heat in your interior. Simply said, t... | {
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Why the speed of light is represented by $c$? In almost every textbook, I've found that the speed of light is $c \approx 3 \times 10^8\: \mathrm{m/s}$. I wonder why it's just $c$ ?
| It's c for constant or celeritas, which means speed in Latin. Everyone uses it because it's convention. You could use $\xi$ or $\zeta$ or $\gamma$ or any other symbol you wanted, but then you'd have to explain what it meant, and people would have to go through the trouble to remember this every time they read your pape... | {
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In what way are the Mathematical universe hypothesis and A New Kind of Science connected The Mathematical universe hypothesis, mainly by Max Tegmark and A new Kind of Science, mainly by Stephen Wolfram both claim (as least as I understand it) that at its innermost core reality is mathematics.
Can this statement be made... | I think that Wolfram is arguing that the study of cellular automata and perhaps similar computational systems could serve as an organizational principle, providing a coherent framework to look at different problem (just like the more familiar frameworks provided by physics and chemistry). This explains the title of his... | {
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Does there exist a single plate capacitor(conductor)? Does there exist a single plate capacitor(conductor)? if yes
How will you define the capacitance and potential(difference)
of such conductor?
| Yes. Capacitance is very well defined for a sphere in vacuum, and can be extended to other media and shapes. The following is a simple abd correct definition of capacitance. Adding other conductors disturb the capacitance of a single conducting body, giving rise to mutual capacitance.
The following is exerpted and ... | {
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What's the dimensionality of a solid angle? I haven't seen this explained clearly anywhere. Solid angles are described usually as a fraction of the surface area of a unit sphere, similar to how angles are the fraction of the circumference of a unit circle. However, I don't know how solid angles are actually quantified.... | The solid angle is defined as the area on the unit sphere subtended by the angle divided by one unit area. It's a ratio so it's a single dimensionless number.
I see why you think it should be a 2D quantity, because the surface of a sphere, and any patch on it, is a 2D manifold and you need two quantities (traditionally... | {
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Where is the energy lost in a spring? Thinking about springs, and their extensions, I recently came to a confusion which I hope this wonderful community can help me solve.
The question is this. When the block is initially attached to the spring, the spring has some extension $x_0$. Now the spring gets extended to some... | You're missing a somewhat subtle point in your analysis. The block on the left in your diagram, where the spring is at its equilibrium position, is moving, so it has kinetic energy (which you're currently ignoring). I'll leave it to you to sort out what the speed needs to be and check that CoE holds.
It needs to be mov... | {
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Relativity of simultaneity and cause-effect Relativity says that two events simultaneous in one reference frame might not be simultaneous in another reference frame. Can we extend this idea a little and say that the order of two events might be be reversed in going from one reference frame to another? And if so, what i... | No, ordering of timelike-separated events is always preserved in special relativity.
First, let's agree on some definitions. Say event $O$ occurs at the origin of our 1D coordinate system. Consider another event $A$ in spacetime, at time $t$ and position $x$ in some pre-fixed coordinates. There possibilities for how $x... | {
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Capacitors: why is the energy not stored in a magnetic field? When a capacitor is charging, the rate of change $dE/dt$ of the electric field between the plates is non-zero, and from the Maxwell-Ampère equation this causes a circulating magnetic field.
Now, since a magnetic field exists, why is the energy of a capacitor... | For a constant potential on the capacitor, there is no B-field and that is the case usually considered for this calculation. When charging a capacitor, the currents will generate a B-field and there is stored energy in that field (same as for an inductor). But once the charging stops, the B-field will "collapse" and ca... | {
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Can zinc nitride be used as light emitting material? I cannot manage to find any journal papers about the applicability of zinc nitride as active layer of an light emitting diode (LED). But certain papers got mention that zinc nitride with a direct bandgap can be fabricated with potential applications in optoelectronic... | I have recently published a paper on Zn3N2 nanocrystals (http://pubs.rsc.org/en/content/articlelanding/2014/tc/c4tc00403e#!divAbstract). The material appears to have a direct band gap around 1eV and makes nice nanophosphors so if you can work out how to p and n dope it I am sure you could make an LED.
| {
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$(\frac{1}{2},\frac{1}{2})$ representation of $SU(2)\otimes SU(2)$ The representation $(\frac{1}{2},\frac{1}{2})$ of the Lorentz group correspond to a four- vector or a spin-one object. Right? Does it imply that any four-vector is identical to a spin-one object or any scalar is identical to a spin-0 object? This can't... | The problem here is with the identification of the $(A,B)$ values of a representation with spin. $A$ and $B$ do not correspond to spin (they are not even Hermitian!), they just happen to obey $SU(2)$ Lie algebras, and as such they add up in the same way that spins do. When we say that $A_\mu,J_\mu,p_\mu,...$ are all in... | {
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Parity violating Dirac particle We normally write down the Dirac Lagrangian as
\begin{equation}
{\cal L} _D = \bar{\psi} ( i \partial _\mu \gamma ^\mu - m ) \psi
\end{equation}
but are the Lagrangian's,
\begin{equation}
\bar{\psi} ( i \partial _\mu \gamma ^\mu \gamma ^5 - m ) \psi , \quad \bar{\psi} ( i \partia... | All the alternatives to the Dirac Lagrangian are actually forbidden by the requirement of requiring the hamiltonian to be well behaved (bounded from below and unbounded from above) and hermiticity of the action. To see this most simply we write the Lagrangian in terms of the fundamental left and right handed fields, $ ... | {
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Eigenvalue spectrum of $L_x+iL_y$ Is it possible to find out the generic eigenvalue spectrum of the non-Hermitian operator $L_x+iL_y$, without using any representation?
| I am assuming that $(L_x,L_y,L_z)$ satisfy the usual angular momentum algebra. Then, one knows that all states can be labelled by the eigenvalues of $(L^2,L_z)$. Pick such a state and label it as $|j,m\rangle$ where the $L^2$ eigenvalue is $j(j+1)$ and the $L_z$ eigenvalue is $m$. Note that we have not made any assumpt... | {
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Range Of An Interaction Why is the Compton wavelength $\lambda_c=\frac{\hbar}{mc}$ used as a sensible measure for the range of an interaction, where $m$ is the mass of the corresponding mediator?
| For massive force carriers, one finds (in natural units) an exponential dependence $\mathrm{e}^{-mr}$ that prevents long-range forces with massive mediators. Restoring SI units, one sees that the compton wavelength is the length at which the damping is exactly $\mathrm{e}^{-1}$.
| {
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Magnetic field due to electron in Hydrogen? We can calculate the current density $\mathbf{j}$ of the electron in Hydrogen, and it is given by:
$$
j_\phi=-e\frac{\hbar m}{\mu r\sin\theta}\left|\psi_{nlm}\left(r,\theta,\phi\right)\right|^2
$$
(derivation found here on page 6)
How can I calculate the magnetic field produc... | Actually I think it is explained fairly well in these lecture notes that you linked. Since the current j only has a φ component, you have to integrate over circles like the ones shown in the figure. These circles have a fixed radius and θ coordinate. Only φ varies.
The radius of that circle is $r \sin\theta$ and there... | {
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Number of Parameters of Lorentz Group We embed the rotation group, $SO(3)$ into the Lorentz group, $O(1,3)$ : $SO(3) \hookrightarrow O(1,3)$ and then determine the six generators of Lorentz group: $J_x, J_y, J_z, K_x, K_y, K_z$ from the rotation and boost matrices.
From the number of the generators we realize that $O(1... | You've got two very good answers from Hunter and NowIGetToLearnWhatAHeadIs. However, it's probably useful to know that this beast $O(1,3)$ is isomorphic or locally isomorphic (i.e. has the same Lie algebra) to a surprising number of other interesting groups, which each give you a slightly different way to think about ... | {
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why galaxies look like discs rather than spheres? Black holes have much gravitation to hold all stars and nebulas
but why they are aligned in disc type shapes rather than spheres
because gravitation is everywhere around black holes. Even on upper and lower sides.
Even if rotation is the cause why cant they move above ... | TL;DR They can orbit in any direction but they usually stick to the disk.
This is an excellent question.
It happens to be for the same reason that the earth and all of the planets all orbit the sun in the same direction on the same plane, I.E. the solar system is a disk as well.
First imagine a large ball of gas in spa... | {
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Molecules and electrons energy types What are the types of energy that an atom or a molecule could have?
For example they have kinetic energy, could they also have other types?
| The Lagrangian $\mathcal L$ and Hamiltonian $\mathcal H$ are mathematical objects that can be used to describe the behaviour of dynamical systems.
In classical systems the Lagrangian is the Kenetic energy minus the potential energy, whereas the Hamiltonian is the the Kenetic energy plus the potential energy.
Most syste... | {
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The shape of speaker cones This is related to another question I just asked, but they are different enough I thought it deserved its own spot. Speaker elements seem to always be shaped like a cone with a portion of a sphere at the center. What is the physics reasoning behind this shape being so standard? Is it a precis... | I believe it's to do with the fact that the speaker's function is to propagate pressure waves through the medium (air).
So, it's mainly a mechanical concern: you want something to push air, and you do not wish to expend much energy. So it has to be light and rigid, which the cone manages to fulfil due to its shape. A p... | {
"language": "en",
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If an object moved 5 meters in a second, how can its velocity be 10 m/s? On physicsclassroom.com, an assignment gives data for an object experiencing acceleration:
This is the corresponding chart for the object's velocity:
At $0$ seconds the position of the object is $0$ meters. And the velocity is $0$ m/s which ma... | One must distinguish between instantaneous velocity and average velocity.
| {
"language": "en",
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Vertex operator - state mapping in Polchinski's book In Polchinski's textbook String Theory, section 2.8, the author argues that the unit operator $1$ corresponds to the vacuum state, and $\partial X^\mu$ is holomorphic inside couture $Q$ in figure 2.6(b), so operators $\alpha_m^\mu$ with $m\ge0$ vanishes.
I am a bit c... | The main point is that the operator-state correspondence maps all the annihilation operators to zero, so that an operator-valued Laurent series in $z$ and $\bar{z}$ maps to a ket-state-valued power series in $z$ and $\bar{z}$.
| {
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Why is the moment of inertia (wrt. the center) for a hollow sphere higher than a solid sphere (with same radius and mass)? Why is the moment of inertia (wrt. the center) for a hollow sphere higher than a solid sphere (with same radius and mass)? I have completely no idea and I am inquiring about this as it is an intere... | The key is... the closer the mass to the axis of rotation, the easier it is to add angular velocity to the body.
For instance a figure skater rotates faster when she puts her limbs closer to her body.
Let's see how it works in a more intuitive fashion:
For instance, in the figure bellow, trying to lift up table (A) wo... | {
"language": "en",
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Maximum Weight Sustained by wire / Breaking limit of wire I know that breaking stress depends on material of wire, not on the diameter or the length of wire. and also that Breaking load depends on the area of cross section of wire. But still i have a confusion in this question.
a steel wire can sustain at the most 100... | You are correct: both halves of the wire will still support the same weight. Look at it this way: the tension in each bit of the wire is the same regardless of whether the weight is hanging directly from it or suspended by another bit of wire in the middle. To put it in terms of stress, the force remains the same and t... | {
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Eigenfunction associated with the $\hat{x}$ operator Consider the following operator $\hat{x}=i\hbar \frac{\partial}{\partial p}$.
I am trying to show that the eigenfunctions of $\hat{x}$ are not square-normalizable. I am interested in doing so since theoretically, we notice that the eigenfunction of the momentum opera... | The eigenvalue equation
$$\tag{1} \hat{x}\psi(x)~=~x_0\psi(x)$$
in the standard Schrödinger position representation
$$\tag{2} \hat{x}~=~x, \qquad \hat{p}~=~-i\hbar\frac{\partial}{\partial x},$$
reads
$$\tag{3} (x-x_0)\psi(x)~=~0,$$
which has general solution
$$\tag{4} \psi(x) ~\propto~ \delta(x-x_0). $$
| {
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What is decay associated spectra? What is decay associated spectra?
Suppose we measure the fluorescence intensity over different wavelengths and over time, we get:
$$I(\lambda,t) = \sum_i^n \alpha_i(\lambda) \exp(\frac{-t}{\tau_i}).$$
The assumption is that there are n component,species, in the $I(\lambda,t)$. If we fi... | Originally the term DAS (rather than LAS) was meant to also include multiexponential decay functions (e.g. the rise then fall of an acceptor species) -- but it was used so often for diagonal (heterogeneous mixture) work that the term SAS was subsequently proposed to distinguish the off-diagonal dynamics spectra from th... | {
"language": "en",
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Explain the microscopic nature of Electric current? Explain the microscopic nature of Electric current?i.e
What is is average current and Instantaneous current? A microscopic view what really happens?
| Electrical current can be carried by conduction electrons, or by 'holes'.
For ordinary matter, there is roughly one electron per two daltons of matter, which is to say, $\frac 12 N_Ae$. This roughly works out at $480*10^6$ coulombs per kilogram. Many of these electrons are bound in the inner orbitals, but there are... | {
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Eigenenergies and eigenkets given the Hamiltonian For a two level system the Hamiltonian is:
$$
H=a(|1\rangle \langle1|-|2\rangle\langle2|+|1\rangle\langle2|+|2\rangle\langle1|)
$$
where $a$ is a number with the dimension of an energy.
I need to find the energy eigenvalues and the corresponding eigenkets (as a combina... | Finding eigenvalues of matrices is a straightforward process, so to solve this problem we'll begin by writing the Hamiltonian in a matrix form in the basis of $|1\rangle$ and $|2\rangle$.
To find the matrix form of any linear transformation in linear algebra, we can apply the transformation to the basis vectors. In our... | {
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Is the uncertainty principle axiomatic or derived? To take an example,
Feynman Lectures Vol 3 13-1
Let's think of an electron which ban be in either one of two positions [...] There are two possible states of definite energy for the electron. Each state can be described by the amplitude for the electron to be in each... | There are several mathematical proofs for the Uncertainty Principle, although it is also based out of intuition. A good, fundamental proof of the mathematical sort is found here: http://www.tjhsst.edu/~2011akessler/notes/hup.pdf.
| {
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How do greenhouse gases trap heat? I am looking for a molecular-level understanding of the greenhouse effect.
What is it about the carbon-dioxide molecule (and methane, and water, etc) that is different from other gasses (particularly, N2 and O2) such that it works in the atmosphere to trap heat?
Is it, say, the dista... | To absorb infrared light, a stretching or bending vibration of the molecule must change the molecule's dipole moment. In $N_2$ and $O_2$ there is no dipole moment regardless of how you stretch the bond. On the other hand, O=C=O can change dipole moment by the C moving toward one O and away from the other O, or by bend... | {
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Why does foam dull knives? I have recently taken up prop making and just started my first foam-built costume from a video game. These kinds of costume armour builds are often built out of the various foam floor mats you can buy in say Home Depot or BJ's or Five Below for around $1 for 4 sq. foot tile.
The one thing tha... | Having previously done academic research on various aspects of knives, I agree with @MSAlters - the actual polymer you're cutting is quite tough. The practical solution is to do what professional knife-users, like butchers, carpet layers, whatever, do: run the blade through a hand-held sharpener after every few stro... | {
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Which angle should it be? in the formula
$$dB = \frac{\mu_0l ~|dl \times r|}{4 \pi r^3} $$
and the image
where dl is in y-z plane and dB is in x-y plane. the ring conductor is in y-z plane carrying current I in direction as mentioned
EDIT: also point p can move in the circular ring
EDIT 2:To clear the confusion...T... | the angle between $dl$ and $r$ is $\pi/2$, which is not $\theta$.
$\theta$ is the angle between $r$ and y-z plane. if you know what is $x$ and $r$, then $\sin\theta=x/r$, where $x$ is the distance from the origin to the point of intersection of $r$ and x-axis, and $r$ is the distance from p point to the same place
UPDA... | {
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Divergent path integral What does it mean to have a divergent path integral in a QFT?
More specifically, if
$$\int e^{i S[\phi]/\hbar} D\phi (t)=\infty $$
What does this mean for the QFT of the field $\phi $?
The field $\phi$ has action
$$S[\phi]=\int\left(\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-V(\phi)\right)\mbo... | If the path integral itself diverges, it means that the v.e.v. diverges. That by itself is bad, because then any arbitrary $n$-point function vanishes. Recall that to compute correlation functions, we append a $J(x)\phi(x)$ to the action and calculate
$$
\frac{\delta^n}{\delta J(x_1)\ldots \delta J(x_n)} \int e^{i S[\p... | {
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Why is a wine glass shaped the way it is? Why is a wine glass shaped the way it is? And why are there different shapes for different wines? Is this a tradition, or is there any scientific reason behind it?
| A little bit of physics - but mostly just fashion.
Red wines (supposedly) need to oxidize to release the flavo(u)r so red wine glasses have wider necks to allow more air. White wines don't and so the glasses have narrower necks.
If there really was a significant difference then you would drink white wines from a sealed... | {
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Does a mirror help a near-sighted persion see at a distance clearer? A near-sighted person without eye-glasses can not clearly see things at distance.
If he takes a photo of the things at distance, he can see the things from the photo much clearer, because he can place the photo much closer to his eyes.
If he turns his... | Not necessarily. It depends on where the image is formed from the mirror. Depending on the radius of curvature (assuming spherical curvature) the image ould form anywhere, but the person would want it to form on their retina. You can calculate this using the mirror equation
$\frac{1}{d_{0}}+\frac{1}{d_{i}}=\frac{1}{f}$... | {
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Coupling constant is turned off adiabatically? To me, adiabatic processes are idealisation. What do people mean with statements such as: "turning off the coupling constant (in QED say) adiabatically"?
| In general it means varying (or turning off an interaction in your particular question) a parameter on a time scale that is much larger than the smallest energy separation of your Hamiltonian. More explicitly : Suppose you have a Hamiltonian $H$ with energy levels $E_n$ and suppose that $\left|E_a-E_b\right|$ ($a\neq ... | {
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Why isn't the electrical field between two parallel conducting plates quadrupled? For the sake of clarity and brevity, here is the description from my textbook (Halliday & Resnick 9th):
This description states that the electric field is doubled. I am seriously confused. When the two plates are brought together, should... | Since the plate is a conductor the surface charge is spread across both sides of plate (see fig 23.16 a or b). Hence this gives that the electric field in between the two plates is actually
$$E=\frac{\sigma}{2\epsilon_{0}}$$
This comes from Gauss' Law $\oint \vec{E}\cdot d\vec{A}=\frac{Q_{enclosed}}{\epsilon_{0}}$
Now... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Focused beam of light I'm trying to understand what happens to photons when a beam of light is focused down to its waist. In the image attached, do photons take the path 1 or 2. That is, do the photons cross or just get 'deflected'.
|
do photons take the path 1 or 2. That is, do the photons cross or just get 'deflected'.
They take path 1 and travel in straight lines.
However due to the finite effective size of the photon and Heisenberg Uncertainty Principle it is not possible for all of the lights straight lines to pass through the centre point.
... | {
"language": "en",
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How does a half-life work? Carbon-14 has a half-life of 5,730 years. That means that after 5,730 years, half of that sample decays. After another 5,730 years, a quarter of the original sample decays (and the cycle goes on and on, and one could use virtually any radioactive isotope). Why is this so? Logically, shoul... | Imagine a sample of 1000 atoms with a half-life of 1 hour.
That means every hour, the sample is reduced to 50% of its size.
After one hour, you are left with 500 atoms.
How much time for that new sample (500 atoms) to be reduced to 50% (250 atoms) ?
In your interpretation :
For the new sample to be reduced to 50%, it ... | {
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How to differentiate between ferri-magnetic and para-magnetic substances? Both are feebly attracted by a magnetic field. I know the difference between these substances(on why these get attracted to magnetic field) but as both get feebly attracted how to find out whether a given substance is ferri-magnetic or para-magne... | Refer NCERT class 12 book 1 chemistry...Solids chapter!
It's that ferri magnetic have some moment..but the net moment of all particles cancels out!
In paramagnetic...it has unpaired electrons..so there is a moment!
| {
"language": "en",
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In calculating resistivity of skin do I use the body's surface area?
Most of the resistance of the human body comes from the skin, as the interior of the body contains aqueous solutions that are good electrical conductors. For dry skin, the resistance between a person’s hands is measured at typically $500\: \mathrm{k ... | In general, when formulae call for "area" they are talking about the surface area on which the force or whatever is occurring. Note that this is not always the entire surface area, it might just be the curved part of a cylinder for example. This is one such example, because we actually aren't interested in the end effe... | {
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Naive visualization of space-time curvature With only a limited knowledge of general relativity, I usually explain space-time curvature (to myself and others) thus:
"If you throw a ball, it will move along a parabola. Initially its vertical speed will be high, then it will slow down, and then speed up again as it appro... | Yes, that's a fair description of what happens though of course from the ball's perspective it isn't moving - the rest of the universe is moving around it.
However statements like this, while true, give little feel for what's going on. Actually it's extraordinarily difficult to get an intuitive feel for the way spaceti... | {
"language": "en",
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Renormalizing IR and UV divergences In lectures on effective field theory the professor wanted to find the correction to the four point vertex in massless $\phi^4$ theory by calculating the diagram,
$\hspace{6cm}$
We consider the zero external momentum limit and denote $p$ as the momentum in the loop. Then we get,
\beg... | I think you misunderstood what the professor wanted to say. To understand this, let us evaluate the integral more thoroughly (your expressions contain some mistakes). If we use the dimensional regularization prescription $d\rightarrow d-2\epsilon$ and an additional mass scale $\mu$, we get for the integral in question ... | {
"language": "en",
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electron levels in a high voltage conductor what is the electron energy level in a 300,000 volt power line? This voltage is way above the ionisation potential but electrons are not emitted from the wire.
| (As far as I know, 300kV is the exclusive domain of HVDC so I will not mention AC here.)
Ionization potential is the energy difference between a free electron at rest outside an atom, and a particular electron energy state inside the atom. If you zap a bound electron with enough energy, it can assume a new, unbound wav... | {
"language": "en",
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Schrödinger's Equation and its complex conjugate I would like to know why there is a minus sign on the right-hand side of the Schrödinger's complex conjugate equation, whereas in the Schrödinger's equation there isn't. I know it is a simple question, but I don't know where this comes from.
$$
-\frac{\hbar^2 }{2m}\frac{... | Just because $\psi$ is a solution to the Schrodinger equation doesn't mean its complex conjugate is. After all, the Schrodinger equation is a pretty strong restriction, and not any random function is a valid wave function. So while it may be tempting to just substitute $\psi^*$ for $\psi$ in the Schrodinger equation, t... | {
"language": "en",
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Bogoliubov transformation with a slight twist Given a Hamiltonian of the form
$$H=\sum_k \begin{pmatrix}a_k^\dagger & b_k^\dagger \end{pmatrix}
\begin{pmatrix}\omega_0 & \Omega f_k \\ \Omega f_k^* & \omega_0\end{pmatrix} \begin{pmatrix}a_k \\ b_k\end{pmatrix}, $$
where $a_k$ and $b_k$ are bosonic annihilation operators... | Hamiltonian is already diagonalized by momentum. You need to define new Bose-operators
$c_k = u_k a_k + v_k b_k \\
d_k = w_k a_k+x_k b_k $
This is general form, with some complex constants $u_k, v_k, w_k, x_k$ for each $k$ independently. There are also $c^+_k$ and $d^+_k$, conjugated with previous one. Now you need $... | {
"language": "en",
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Field from non-conducting plate?
For a non-conducting sheet, the electric field is given by:
$$E = \frac{\sigma}{2\epsilon_0}$$
where $\sigma$ is the surface charge density.
This equation holds well for a finite nonconducting sheet as long as we are dealing with points close to the sheet and not too near its edges.
... | Consider a square sheet with edges located at $(a,0)$, $(-a,0)$, $(0,a)$ and $(0,-a)$. Suppose, we wish to find the electric field at a point $(0,0,z)$. By symmetry, this electric field will point solely in the $z$-direction. To find the electric field, consider a small element on the sheet located at $(x,y)$ of area $... | {
"language": "en",
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Superconducting wire in a Magnetic Field? A superconducting wire($SC$) is moved rapidly in a magnetic field( $1$ $Tesla$), what would happen to the wire? Are there any forces induced of attraction or repulsion?
In a typical conductor, we know that if it is moved around a magnetic field $-V$ is induced within the wire ... | I suppose that the moving wire is a closed circuit and that the magnetic flux enclosed is time-dependent. The Faraday’s law is of course always applicable. The current will not be infinite. Yes R=0 but, what about the self inductance L? It is never zero, in such a manner that the total E field will be null. If you make... | {
"language": "en",
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Killing vector contractions along isometric curves Imagine $\xi_{\nu}$ is a Killing vector field on a manifold. Does $\xi_{\nu}\xi^{\nu}$ remain constant along any isometric curve defined by the Killing vector field?
My guess is that yes since as you move along an isometric curve every point "around" you looks pretty... | Let $\lambda$ be an affine parameter of the integral curves of $\xi^{\nu}$ then you question translates as
$$
\frac{d}{d\lambda}(\xi_{\nu}\xi^{\nu}) = \xi^\mu \nabla_\mu(\xi_{\nu}\xi^{\nu}) = (\xi^\mu \nabla_\mu\xi_{\nu})\xi^{\nu} + \xi_{\nu}(\xi^\mu \nabla_\mu\xi^{\nu})
$$
if the connection is Levi-Civita (i.e metric ... | {
"language": "en",
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Measurements for thermal diffusivity of graphene? We have known for a long time that graphene has in-plane thermal conductivity ranging between 2000 and 4000 $W m^{-1} K^{-1}$. But in order to model heat transport on a sheet of graphene, we need more than the conductivity: we also need specific heat in order to obtain ... | According to this article, the molar heat capacities of graphite and graphene should be identical above roughly 100 Kelvin. The heat capacity of graphene is dominated by phonon contributions above roughly 1 Kelvin (below that, free electron heat capacity becomes a significant contributor).
The molar heat capacity of gr... | {
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How does one exert greater force on the ground by jumping? When one jumps, how does he/she manage to exert greater force on their ground than their weight?
Also, what is normal force and the reaction force (are they the same thing?) and by newton's third law, shouldn't the reaction(weight) when we are standing on the ... | When you jump, your muscles are using their chemical energy to contract or extend appropriately to create an additional force on the ground; by Newton's third law, the ground exerts additional force on you, overcoming your weight and pushing you off the ground.
In this case, the normal force is indeed the reaction forc... | {
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When I stretch a rubber band, it breaks. When I hold the broken ends together, why doesn't it join again? The question is simple. When we join the two broken surfaces, what is it that keeps the surfaces from connecting with each other, while earlier they were attached to each other? Also, would the two sides join again... | Your question is perfectly valid despite most people may think it's odd as reason looks obvious.
There's a property of system called entropy which must not decrease for any process (do some research on web; You may have problem in understanding it as you're in high school). Only those processes in the universe happen i... | {
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Magnetic field inside and outside cylinder with varying current density I am reading through Introduction to Electrodynamics by David J. Griffiths and came across the following problem:
A steady current $I$ flows down a long cylindrical wire of radius $a$. Find the magnetic field, both inside and outside the wire if t... | Those answers are correct. A first check is to see if the units match. You can always check direction by the right hand rule. The part for outside the wire is the same as if the current were uniform, because the enclosed current is all that matters when you have enough symmetry for Ampére's Law. For the part inside... | {
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Action and reaction pair problem If I sit on a chair, the first action and reaction pair force is my weight and the force acting on the earth by me. And the 2nd pair is the force acting on the chair by me and the opposite force.
According to newton third law, the action and reaction force must be equalI. 'm confused be... | Where does the 'pairing' in your first pair come into place? I.e. what is the force counteracting the earth's gravity pull on you (in your theory)?
The relevant force pair in your example is the attractive force between your body and the earth (gravitational pull) and the repulsive force between your body and the chair... | {
"language": "en",
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Creating electricity from mains water pressure. Could someone cleverer than me help me out?
I had a crazy thought going through my head the other day and I can't lay my mind to rest until I get an answer.
Q. How much energy could be produced by using mains water pressure to turn a generator? And would it be feasible ... | We pay 3 dollars per cubic meter for water where I live. At 400 kPa (60 psi) that's 400 kJoules per cubic meter maximum theoretical power. But that's only about a tenth of a kilowatt-hour, which costs about a penny at 10 cents/kW-hr. So it's not that good a proposition.
| {
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How jet engine works in vacuum Its not about availability of oxygen for combustion of gases. I want to know since there is no air or friction providing things in vaccum, how the force applied by a jet engine get can make thrust to move the vechile
| Think about the recoil of a gun. The gun shoots some mass (the projectile) with very high momentum into negative direction. Because of the balance of momentum the gun gets a momentum into positive direction.
The jet engine does the same with highly accelerated gas which has also mass and therefore also a high momentum ... | {
"language": "en",
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How is it possible for light to be a wave and a particle? I have always been interested in Physics, and lots of people say that light is a particle and a wave.
How is it possible? How can a photon (a light particle) be a wave as well, when its a particle?
I have tried looking this up, but I do not really understand the... | First, we need to ask ourselves what exactly do we mean when we say that something is a wave or a particle. Something is a wave when it oscillates through a medium. Something is a particle when it has a definite size and position at a given time in space.
Now, when photons interact with anything (say to excite an elec... | {
"language": "en",
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In the Dirac equation, do $\alpha$ and $p$ commute? The Dirac Hamiltonian is given as $H = \vec \alpha·\vec pc + \beta mc^2$ , Do the alpha and beta operators commute with the momentum operator? If yes then how?
| Yes, they do. All three of the $\alpha$ matrices, as well as the beta matrix, are operators on the 'internal' degree of freedom of the electron (that is, its spin and the electron-positron distinction), while the momentum operator operates on its spatial degree of freedom. These are independent degrees of freedom, and ... | {
"language": "en",
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Without seeing the lightning, can you tell how far away it struck by how the thunder sounds? Is there any way to tell how far away a lightning strike is by how its thunder sounds? I thought one way might be by using the fact that higher frequencies travel faster than lower frequencies. Would you have to correct for the... | I presume you did not see the flash, so cannot use that as a timing mark. From Wikipedia "The dependence on frequency and pressure are normally insignificant in practical applications. In dry air, the speed of sound increases by about 0.1 m/s as the frequency rises from 10 Hz to 100 Hz. For audible frequencies above ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/104304",
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How does particles gain electrical charges and repel each others? (electrostatic stabilization) When I study electrostatic stabilization, I understand that the particles have same charge and thus repel others, this is how colloid is stabilize. But how does particles gain electrical charges and repel each other in the f... | There are two mechanisms. Which one is more important depends on the colloid.
A large number of colloidal particles have ionisable groups on their surfaces. These are usually salts of carboxylic acids. The vast majority of organic colloids (e.g. milk) are in this class, as are colloids prepared from acidic monomers lik... | {
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Neutron-Antineutron creation/annihilation data What is the most precise data for neutron-antineutron production by one photon (hitting a target in the laboratory system)?
and/or
What is the most precise data for neutron-antineutron
annihilation to two photons?
Is this data available online?
| There is no process $ \gamma \to n $ at all, nor $ \gamma \to n \bar{n}$ with an on-shell photon. The first violated multiple quantum number conservation rules and the second conservation of four-momentum.
The two-photon process
$$ \gamma + \gamma \to n + \bar{n} \,,$$
has allowed quantum numbers but will be exceeding... | {
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What is sound in terms of acoustic sources? Sound is nothing more than small amplitude, unsteady pressure perturbations that propagate as a longitudinal wave from a region in space which created it (called the source region) into a quiescent (still) region where it is observed by hearing.
It is at this point where I g... | Hydrodynamic perturbations = change in pressure due to a flow velocity (particles don't return to equilibrium positions).
Acoustic perturbations = change in pressure due to the fact the particles undergo an elastic restoring force (for a compressible fluid) which causes perturbations to travel at the speed of sound.
An... | {
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Expectation Value of a Dynamical Variable In quantum mechanics, we generally take about "expectation values of dynamical variables". However, by the postulates of quantum mechanics, every dynamical variable in quantum theory is represented by its corresponding operator.
Is it therefore, incorrect to talk about "expecta... | Remember, operators are nothing but maps. Expectation value of an operator is pretty much defined (I guess) in general operator theory. It just turns out that in QM (Hermitian) operators correspond to dynamical variables. In general you can also calculate expectation values of operators like $L_+$ and $a^{\dagger}$ etc... | {
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How to show the invariant nature of some value by the group theory representations? Let's have Dirac spinor $\Psi (x)$. It transforms as $\left( \frac{1}{2}, 0 \right) \oplus \left( 0, \frac{1}{2} \right)$ representation of the Lorentz group:
$$
\Psi = \begin{pmatrix} \psi_{a} \\ \kappa^{\dot {a}}\end{pmatrix}, \quad ... | short answer if $ \hat {S}^{-1} S = \mathbb{I}$
I can give you a general example of $\psi^\dagger\psi$ not being invariant.
because for Dirac spinor $\psi$ whe have the following transformation rules
$$\psi(x) \rightarrow S[\Lambda] \psi(\Lambda^{-1}x)=S[\Lambda] \psi(x^\prime) \\
\psi^\dagger(x) \rightarrow \psi^\... | {
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What's this about kinetic energy increasing with the fifth power of length? I don't quite understand this quote from Stephen J. Gould's Ever since Darwin, where he talks about the compensating physical characteristics of organisms for their size.
Other essential features of organisms change even more rapidly
with i... | Model the child as a rod of mass $m$ and length $l$ standing on the ground vertically, with center of mass at height $l/2$, with feet glued to the ground but the rest of the body able to rotate.
When upright, the potential energy is $mgl/2$. When lying on the ground, the potential is 0. So when falling, the child hit... | {
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Why is the Gibbs Free Energy $F-HM$? With magnetism, the Gibbs Free Energy is $F-HM$, where $F$ is the Helmholtz Free Energy, $H$ is the auxiliary magnetic field, and $M$ is magnetization.
Why is this? Normally, in thermodynamics, we Legendre Transform the various free energies into each other to maximize the global e... | Without magnetism, you have $F(T, V, N)$ and $G(T, P, N)$. So you did a Legendre transform so that your potential depends on $P$ instead of $V$. It also means that you move from an extensive to an intensive quantity.
The free energy $F$ comes via $U = TS$ from the internal energy, which depends purely on extensive quan... | {
"language": "en",
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How exactly is the Poisson bracket of the modes of a classical string defined? In the theory of a classical bosonic string, we have expressions like:
$$ \{\alpha^\mu_m,\alpha^\nu_n \} = - i m \delta_{m,-n} \eta^{\mu \nu} $$
were $\alpha^\mu_n$ are the Fourier modes of the string. How is this Poisson bracket defined? Th... | Concerning OP's last sentence (v1), the Fourier modes $\alpha^{\mu}_{m}$ are (some of) the fundamental variables of the string. Phrased equivalently, the Poisson bracket reads
$$ \{F(\alpha),G(\alpha)\}~=~ \sum_{m\in\mathbb{Z}}
\frac{\partial F(\alpha)}{\partial \alpha^{\mu}_{m}} (-im \eta^{\mu\nu})
\frac{\partial G(\... | {
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Deriving $F = ma$ - Newton's Second Law of Motion Context:
In my textbook it is given: 'momentum' short for 'linear momentum':
Mass = $m$, momentum is $p=mv$. In time $\Delta t$, momentum changes by $\Delta p$, the rate of change of momentum is:
$$\frac{\Delta p}{\Delta t} = \frac{\Delta(mv)}{t} = m \frac{\Delta v}{\D... | You are right, there is a $\Delta$ missing in front of the $t$.
$\Delta v = v_2 - v_1$. If the mass is not changing, then $\Delta (mv) = mv_2 - mv_1 = m(v_2 - v_1) = m\Delta v$. Hope that helps.
The equation that includes $\frac{\Delta m}{\Delta t}$ is not Newton's second law. The second law is valid only for system... | {
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} |
What will happen to water at $0^\circ$ Celsius kept in large evacuated chamber Suppose some water is kept at $0^\circ$, in a glass (ideal container, if necessary). Now it is placed in a large evacuated chamber. What will happen?
I think some water will vaporize, some will freeze and rest will remain liquid. But then I ... | When one says the boiling point is x degrees at such a pressure, what it means is that it holds a partial vapour pressure up to this. Heat is fed in to make water into steam, even at room temperature, but it's so slight you don't notice it in the speciic heat.
When you have a large volume, a large amount of water is t... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/105568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Stern Gerlach with spin in opposite directions So for the Stern-Gerlach apparatus, we assume that we either have a particle spin up or spin down. We also have the varying field, $\partial B/\partial z$. This initial configuration results in the particle wither going to plus $\hbar$ or minus $\hbar$.
Suppose instead... | $\newcommand{\ket}[1]{\left| #1 \right>}$$\newcommand{\bk}[2]{\left< #1 | #2 \right>}$Notice that the eigenvectors of the operator $S_z$ spans the whole space, which means that you can write any state as a superposition, (if you prefer as a linear combination) of these states. The situation is akin to the basis vectors... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/105694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
} |
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