Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Evaluating the Ricci tensor effectively If given a metric of the form $$ds^2=\alpha^2(dr^2+r^2d\theta^2)$$ where $\alpha=\alpha(r)$, then can one immediately conclude that $$R_{\theta\theta}=r^2R_{rr}$$ where $R_{ab}$ is the Ricci tensor, without doing any explicit calculations? I can show that this is true by the long... | I'm not sure what OP exactly is requesting, but OP's equation follows e.g. from the general fact that for an arbitrary 2D surface, the Ricci tensor
$$ R_{\mu\nu} ~\propto~g_{\mu\nu} $$
is always proportional to the metric tensor $g_{\mu\nu}$. This is basically a consequence of that in 2D the Riemann curvature tensor i... | {
"language": "en",
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Motivation for Wigner phase space distribution Most sources say that Wigner distribution acts like a joint phase-space distribution in quantum mechanics and this is justified by the formula
$$\int_{\mathbb{R}^6}w(x,p)a(x,p)dxdp= \langle \psi|\hat{a}\psi\rangle$$
as this allows us to compute the expectation value for t... | I want to add an interesting detail to the excellent answer by Qmechanic which I recently learned from Cosmas Zachos, here.
Royer showed in his 1977 paper, that the Wigner function $w$ is given by the expectation value of the parity operator $\Pi_{\mathbf{r}\mathbf{p}}$ which inverts the phase space around the point $\... | {
"language": "en",
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"source": "stackexchange",
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Row of pivoted magnets and energy scale This question is about a system involving a horizontal row of length L of equally spaced pivotable magnets, each with a pole at either end. These magnets will often be referred to as units.
So each unit when rotated causes it's neighbours to rotate in the opposite direction to it... | $\def\vm{{\bf m}}$If the magnets are free to rotate they will find the lowest energy state.
This occurs when all the magnets are aligned head to tail along the line.
$$\begin{equation*}
\rightarrow\rightarrow\rightarrow\cdots\rightarrow\tag{1}
\end{equation*}$$
If now the magnet at one end is rotated a quarter turn an... | {
"language": "en",
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Maxwell's Equations-Relativity How did Maxwell develop the magnetic field without relativity? Was it purely experimental? I don't see how else he would have developed any understanding for the magnetic field.
| A long time before Maxwell wrote down a unified theory, Oersted discovered a connection between electricity and magnetism. In the development of electromagnetism, there were many bits and pieces of partial knowledge that were discovered and formulated by many different scientists. The popular ones (after whom we've nam... | {
"language": "en",
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Why is there no UV catastrophe (divergence) in turbulence? I have just read that as the Reynolds number is increased, the separation of macroscopic and microscopic scales increases and that this also means that there is no UV catastrophy (or equivalently UV devergence?) in turbulence.
I do not understand what this mean... | The fluid equivalent to the Ultraviolet Catastrophe is the transition to turbulence. Just as Maxwell's Equations break down at high frequencies and are replaced by a stochastic model (Quantum Mechanics), the Navier-Stokes Equations break down at high Reynolds Number and are replaced by a stochastic model (Kolmogorov Ca... | {
"language": "en",
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"source": "stackexchange",
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Does inertia increase with speed? I have heard that when the speed of the object increase, the mass of the object also increase. (Why does an object with higher speed gain more (relativistic) mass?)
So inertia which is related to mass, increase with speed?
So, if I accelerate on a bus, my mass will increase and my iner... | For both interpretations, the answer is 'yes' since force still acts in an opposite force on anything which has mass. As you accelerate, your velocity increases and therefore mass will increase. The increase in mass will bring about an opposite force. The greater the mass, the greater the inertia.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/64661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How does an earthen pot keep water cool? I understand that evaporative cooling takes place thanks to small pores contained in the pot and that allow some water to go through and evaporate. However I couldn't understand clearly whether water inside the pot stays at its original temperature or would it cool further?
If i... | Latent heat of evaporation cools the pot in the same way it cools your skin. If the pot is then at a lower temperature than the water in it, heat energy will be transfered to the pot from the water in it by conduction.
Using plant xylem water for evaporative cooling, the desert cicada Diceroprocta
apache can maintain ... | {
"language": "en",
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"question_score": "11",
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Comparing Static Frictions In this figure, which of the static frictional forces will be more?
My aim isn't to solve this particular problem but to learn how is static friction distributed . Since each of the rough-surfaces are perfectly capable of providing the $-1N$ horizontal frictional force but why don't they ? T... | In the cases where you have static friction, the forces will always be defined by the looking at the system and applying the constraints(in other words $F_s\le \mu N$ will only give an upper bound). On the other hand when you are dealing with kinetic friction, it can be easily derived from the famous $F_k=\mu N$.
As an... | {
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Optics alignment of a confocal scanning microscope I am facing a challenge in my project regarding optical alignment.
See the figure:
The challenge is with the vertical optical system alignment. I considered placing a mirror and check back if the image and source coincide. But since the light is too low on power (less... | The basic concept of optical alignment is to perfectly align the beam step-by-step on all optical elements. Enshure your beam has the right height and angle before alignment. Your aim to align the beam on microscope optical axis. Using point sources like (e.g. Flourescent beads, microspheres, Quantum dots, quantum wel... | {
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Ducted or open fluid flow, which is best for aerodynamics and lift I'm designing a copter and trying to decide if the propellers should be ducted or open axial flow. I've read some theory on ducted and open air flow but I can't find any where that compares the two. I would prefer to use ducted over open for safety (ie ... | Actually, ducted fan propeller are much less efficient than open blade propeller soif efficiency is your priority, open propeller is your option
| {
"language": "en",
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Can a current carrying loop or wire produces no magnetic field? A current carrying wire produces magnetic field around it. We can find the direction by Fleming's Right hand rule. We know change in electric filed produces magnetic field and change on magnetic field produces electric field. It is mutual relationship.
My ... | A current carrying loop arranged in the form of a twisted pair of cables
with same diameter, will produce a nearly zero magnetic field.
The use of a coaxial cable will produce a still lower magnetic field.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/64993",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Volume element $\mathrm{d}^4k =\mathrm{d}k^0 \,|\mathbf{k}|^2\,\mathrm{d}|\mathbf{k}| \,\mathrm{d}(\cos\theta) \,\mathrm{d}\phi$ in Minkowski space? Suppose we have an integral
$$\int \mathrm{d}^4k \,\ f(k)$$
we want to evaluate and that we're in Minkowski space with some metric $(+,-,-,-)$.
Is it true that: $$\mathrm... | Yes it is.
The volume form on any (pseudo-)Riemannian manifold $(M,g)$ of dimension $n$, where $g$ is the metric, is given in local coordinates $(x^1, \dots, x^n)$
$$
\sqrt{|\det (g_{\mu\nu})|}dx^1\wedge \cdots \wedge dx^n
$$
where $\det(g_{\mu\nu})$ is the determinant of the metric in these coordinates. In cartes... | {
"language": "en",
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"source": "stackexchange",
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Momentum, Impulse and Newton's Second Law of Motion Newton formulated his Second Law as such:
$$\sum{\vec{F}} = \frac{\delta \vec{p}}{\delta t}$$
and of course, $\vec{p} = m \vec{v}$.
Why is it that if the net force $\sum \vec{F}$ is constant (which implies that the rate of change of momentum is constant), then
$$\frac... | I think I have figured out the answer, hopefully.
Firstly, let us begin by stating the First Law:
$$\sum \vec{F} = \frac{\delta \vec{p}}{\delta t}$$
When the net force is constant, we it means that there is no change of momentum, in other words,
$$ \frac{\delta \vec{p}}{\delta t} = 0$$
In this case, we know that the fu... | {
"language": "en",
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Acceleration due to gravity? I was looking into orbitals and found something I haven't been able to understand.
http://www.math.ubc.ca/~cass/courses/m309-01a/hunter/satelliteOrbits.html
There is a part on the page which states the following:
$$\vec a = \frac{G(M+m)\vec{r}}{r^3}\approx GM\vec{r} / r^3$$ since $m << M$
I... | We have
$${\bf F} = \mu {\bf a}$$
where $\mu = M m/(M+m)$ is the reduced mass.
Thus,
$$\frac{G M m}{r^3}{\bf r} = \frac{M m}{M+m}{\bf a}$$
or
$$\begin{eqnarray*}
{\bf a} &=& \frac{G(M+m)}{r^3}{\bf r} \\
&=& \frac{G M}{r^3}{\bf r}\left(1+\frac{m}{M}\right) \\
&\simeq& \frac{G M}{r^3}{\bf r}.
\end{eqnarray*}$$
Notice... | {
"language": "en",
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How does tension apply torque on a pulley? How is tension in a string able to apply torque on a pulley? How does string itself able to apply a force on pulley? What is happening inside the pulley?
The pulley has a mass $m$ and is a disc.
| Tasos' answer above is the simplest answer to the question. With this approach you do not have to address the details of the interaction between the string and the pulley. This is a great example of how picking the appropriate system (here the pulley with the string) leads to a simpler evaluation. This assumes no sli... | {
"language": "en",
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"source": "stackexchange",
"question_score": "3",
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How does relativity explain gravity, without assuming gravity I have seen the "objects pull down on space-time" explanations, but they assume a "pull down" force themselves. Could anyone explain the space-time explanation without assuming gravity in the first place?
| Why don't you want to assume gravity? Gravity it is an experimental fact, a starting point for doing physics. General Relativy is a geometrical theory of gravity, built on the basis of Special Relativity and always having in mind that it should recover the non-relativistic Newtonian theory of the gravitational field.
T... | {
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Relativistic Lagrangian transformations I need to study the relativistic lagrangian of a free particle.
It's
$\ L= - m c^2 \sqrt[2]{1- \frac{|u|^2}{c^2}} $ I need to study the translation, boost and rotation symmetry. I say it doesn't depend of the position, so it has translation symmetry and the momentum will conserv... | Only the action S is a relativist invariant : invariant under translations, rotations, and boosts. The Lagrangian itself is not invariant under boosts.
The action $S$ is : $S = -mc \int ds = - mc \int \sqrt {ds^2} = - mc \int \sqrt {c^2 dt^2 - \vec {dx}^2} = - mc^2 \int \sqrt {1 - (\frac{\vec {dx}}{c \, dt})^2} \,\,... | {
"language": "en",
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Why does sound move faster in solids? I know that the molecules are closer together in solids, and I know thicker springs also respond carry waves faster than thinner springs, but for some reasons I can't understand why.
The molecules will have a larger distance to move before colliding with another molecule, but in a... | I assume "faster in solids" means faster than in gases.
The speed of a mechanical wave is in general proportional to $\sqrt{k/m}$, where $k$ is some measure of the restoring force (e.g., the tension in a string, or a Young's modulus), and $m$ is some measure of inertia (e.g., the mass per unit length of a string, or th... | {
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Can a single molecule have a temperature? A show on the weather channel said that as a water molecule ascends in the atmosphere it cools. Does it make sense to talk about the temperature of a single molecule?
| As stated in the other answers, in the specific scenario depicted in your question, it makes no sense to talk about the temperature of a molecule. But I can't help to widen the picture because there is a very interesting case here: what about the internal degrees of freedom of the molecule? So far everybody has only co... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/65690",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "32",
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Why are the magnetic moment and the angular moment related? Why are the magnetic moment and the angular moment related? I've always read everywhere that they are related but found nowhere a satisfactory explanation of the cause
| You can understand, it through Einstein-de Hass effect or other way Barnett effect;
In classical mechanics, when an object of mass $m$ moves circularly, it gives rise to angular momentum. Similarly, when a charge particle moves circularly it gives rise to magnetic moment. So charge particles are not massless, they do h... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/65822",
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"source": "stackexchange",
"question_score": "4",
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Find E and B from vector potential I have a vector potential given by:
$\mathbf{A}(x,t) = \mathbf{e}_{y}\frac{1}{2} e^{-(x-ct)^{2}/{4a^{2}}}$
Now, the question is "Determine the E and B under the condition that the scalar potential vanishes $V = 0$."
But I'm not quite sure what it means when $V=0$ ?
As far as I can see... | The expression for the electric field includes derivatives of the vector potential with respect to time.
| {
"language": "en",
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A Musical Pathway Using a small number of sound emitters, could you create a room where certain nodes emitted particular tones, but no meaningful sound was heard anywhere else.
So, for example, by walking down a certain path, you could hear the tones for "Mary Had a Little Lamb." Is there a generalized algorithm to ma... | Wavefield Synthesis can do this but not with a small number of emitters, uses a massive array of speakers create a field effect, and phase alteration can reposition sounds within that field.
http://en.wikipedia.org/wiki/Wave_field_synthesis
Downside is it's not perfect, it takes a big array of speakers placed very accu... | {
"language": "en",
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Harmonic oscillator - wavefunctions I understand now how I can derive the lowest energy state $W_0 = \tfrac{1}{2}\hbar \omega$ of the quantum harmonic oscillator (HO) using the ladder operators. What is the easiest way to now derive possible wavefunctions - the ones with Hermite polynomials?
I need some guidance first ... | I think the easiest way to do this is to avoid solving differential equations to the greatest extent possible. There is, in fact, a way to use ladder operators and only requires you to solve one, fairly easy differential equation;
First, we note that the ladder operator technique can be used to derive the entire spect... | {
"language": "en",
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Momentum of particle in a box Take a unit box, the energy eigenfunctions are $\sin(n\pi x)$ (ignoring normalization constant) inside the box and 0 outside. I have read that there is no momentum operator for a particle in a box, since $\frac{\hbar}{i}\frac{d}{dx}\sin(n\pi x)=\frac{\hbar}{i}n\pi\cos(n\pi x)$ and this isn... | There are two different issues. One of them is the sign of the momentum; the other one is whether the momentum is spread (it's not because of the unnatural boundary conditions).
Concerning the first point, the standing wave (sine) is a real function and every real wave function has the same probability to carry momentu... | {
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Discovery of $E=hf$? How was the equation $E=hf$ discovered?
Was the proportional expression between energy and frequency of light $E\propto f$ discovered only by experiment? Or is there some logical(theoretical) senses affected?
| The relationship $E = h f$ was proposed by Max Plank in 1899 or 1900 as a way to "fix" a major problem in the existing understanding of the how light was emitted by hot bodies (the so called "ultraviolet catastrophe"). The conventional story holds that Plank did not consider this as fundamental.1
Later Albert Einstein ... | {
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Which causal structures are absent from any "nice" patch of Minkowski space? Which "causal separation structures" (or "interval structures") can not be found among the events in "any nice patch ($P$) of Minkowski space"?,
where "causal separation structure" ($s$) should be understood as a function from $n (n - 1) / 2$ ... | It seems you don't us to differentiate between past time-like (null) and future time-like (null). Perhaps you have in mind more general spacetimes, but I'm having a hard time imagining a situation where one could distinguish time-like and space-like separated points but not past time-like and future time-like. If one w... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/66636",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Interaction of an electromagnetic wave with a two level system in the domain of quantum field theory Suppose I shine an electromagnetic wave on a two-level system. I need to describe how the system evolves in context of quantum field theory i.e. using a quantized EM field in the problem. The first step would be to writ... | A single free-space field mode of the quantized electric field in the dipole approximation can be written as:
$$
\vec{\hat{E}}(t) = i\left(\frac{\hbar \omega}{2\epsilon_0 V}\right)^{1/2}\vec{e}(\hat{a}e^{-i\omega t} - \hat{a}^{\dagger}e^{i\omega t}),
$$
with $\omega$ the frequency of the field, V the volume in which th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Human power on treadmill On an elliptical treadmill a regular person can easily burn 1000 calories in one hour (treadmill reports calories burnt). This translates into:
$$(1\times 10^3\mathrm{cal/hr}\times 4.2\times10^3\mathrm{J/cal})/3.6\times 10^3\mathrm{s/hr} \approx 1.2 \; \mathrm{kW} \approx 1.5 \; \mathrm{hp}$$
O... | This almost a duplicate of How efficient is the human body?. Humans are about 20% efficient at turning food into mechanical energy, so your cyclist generating 400W is metabolising food at about 2kW or about half a Calorie per second or 1800 Calories per hour.
| {
"language": "en",
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"source": "stackexchange",
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Can a hybrid vehicle ever be more efficient than a hydrocarbon-only vehicle built with the same parts? Based on the laws of thermodynamics, shouldn't it be theoretically impossible for a non plug-in hybrid vehicle to ever be more fuel-efficient than a vehicle that connects the same engine directly to the wheels without... | Regenerative braking is one thing, another thing is that in practice large powerful engines are generally less efficient than small engines (because a substantial fraction of driving time the engine is not fully loaded, see detailed explanation in Wikipedia on Active Fuel Management), and having an auxiliary electric m... | {
"language": "en",
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"source": "stackexchange",
"question_score": "6",
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Fermion annihilation amplitude Is there a physical reason why annihilation of 2 identical fermions with mass m to 2 scalars amplitude for $s=4m^2$ (fermions at rest) is zero? For example we can have 2 scalars annihilating in 2 scalars and the amplitude is non-zero if we have a quadrilinear term in the largangian $\phi... | For generally valid reasons, the amplitudes analytically depend on the momenta and on $s$ in particular. There can't be any "jumps", the simplest discontinuities. Because the processes you enumerated with the total energy $E_{cm}\lt 2m$ i.e. $s\lt 4m^2$ are strictly prohibited by energy conservation – $2m$ is the minim... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/67085",
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On Einstein notation with multiple indices On Einstein notation with multiple indices: For example, consider the expression:
$$a^{ij} b_{ij}.$$
Does the notation signify,
$$a^{00} b_{00} + a^{01} b_{01} + a^{02} b_{02} + ... $$
i.e. you sum over every combination of the indices? Or do you sum over the indices at the sa... | You would sum over every combination of indices that match. So the i should match, and the j should match. For instance, if each is from 1 to 3, you would get:
$a^{11}b_{11}+a^{12}b_{12}+a^{13}b_{13}+a^{21}b_{21}+a^{22}b_{22}+a^{23}b_{23}+a^{31}b_{31}+a^{31}b_{32}+a^{33}b_{33}$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Composition of squeeze operators? I'm wondering if it exists a composition law for the squeezing operation ? I guess so for geometric reason, since they are (generalized, and the phase is annoying of course) hyperbolic rotations of the annihilation $a$ and creation $a^{\dagger}$ operators of some bosonic modes.
I defin... | You will not have easily a closed general formula, because, first, the commutator of $aa$ and $a^+a^+$, so $4(a^+a + 1/2)$ does not commute with $aa$ and $a^+a^+$, and secondly,worse, because the same thing happens for all the high-order commutators of the Baker-Campbell-Hausdorff relations
(see more precisely Chapter... | {
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Can deterministic world view be denied by anything other than quantum mechanics If we ignored quantum mechanics and looked at the world with a deterministic Newtonian view. Does not that mean that there is no randomness and that if all the information of the state of the universe during the big bang is accessible one c... | From the theory of highly dynamic systems and chaos, perfectly classical and in principle deterministic systems can exhibit a behavior, where minute perturbations of the initial conditions are exponentially enhanced over time, and thus arbitrarily small perturbations can, after a finite time span, lead to a state that ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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Why do current-carrying wires heat up? Obviously wires heat up too, but why do they heat up? And for the same reason, why do we get electrical burns?
| The current in the conductor is due to the drifting of electrons inside a conductor in a direction opposite to the flow of electrons
During their drifting they collide with their atoms vibrating about their mean position and lose some of kinetic energy to the vibrating atoms which increases the amplitude of the vibrati... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/67639",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Navier-Stokes equations: conservation of momentum The first Navier-Stokes equation (conservation of mass) says:
$\vec \nabla \cdot \vec v=0$
For a stationary flow, the l.h.s of the second equation is (conservation of momentum):
$\rho \frac{D\vec v}{Dt}=\rho (\underbrace{\frac{\partial \vec v}{\partial t}}_{=0} + (\vec ... | Your mistake here is to assume that the multiplication $\vec v\cdot \vec \nabla$ is commutative. It is not; the dot product here is just a convenient mathematical notation. This part of the Wikipedia article on Navier-Stokes equations explains how to interpret this term.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/67699",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Dissolving oxygen into water I was wondering how one would actually calculate how much oxygen would dissolve into water given the necessary initial conditions, and what those initial conditions would need to be. I assume they would be pressure, and initial concentration, but I really don't know where I would go from th... | In water the gas is in aqueous state, because gas molecules are interacting with molecules. At equilibrium the concentration of the gas in water and above water are different, with the ratio being a constant of Henry's Law.
Check out http://en.wikipedia.org/wiki/Henry's_law for a list of constants.
The most anomalous... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Why do metal objects in microwaves spark? I heard that electrons accumulate at points on metals, and this clearly explains the arcing phenomenon, but how does a microwave make an electron imbalance on the fork?
| The energy produced in a microwave oven is at a macro-scale wavelength (on the order of centimeters), and leaves the emitter as a coherent wave - one wavelength, phase, and polarization. Despite the "stirring fan" and other measures, objects in the oven will still experience (relatively) orderly macro-scale radio waves... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/67880",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 3,
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Ball jumping from water Few days ago I played with ball(filled with air) in swimming pool. I observed interesting phenomenon.
When I released a ball from 3 meters depth the ball barely jumped above the water surface but when I released it from 50 cm depth it shoot out of the water like nothing.
I observed when released... | The answer is quite simple. The pressure exerted by the water column on the ball is more at 3m depth as compared to the pressure exerted on the ball at 50cm depth. As a result, due to the buoyancy of the ball it shoots out of the water with a greater force when it is released from a depth of 50cm.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/68055",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 4,
"answer_id": 2
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Fusion vs. Fission I understand why fission generates large amounts of energy when the nucleus is split, but then why does fusion generate such large amounts of energy. If fission releases energy when some mass is lost as energy, then shouldn't the fusion process absorb energy to fuse nuclei together?
I also am curious... | The strong force is attractive at short range and wins over the electromagnetic repulsion between protons. Pull the protons apart a little and you will get fission because the electromagnetic force wins. Conversely, push protons together and you will eventually get fusion when the strong force takes over from electroma... | {
"language": "en",
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"source": "stackexchange",
"question_score": "4",
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Mechanics question (a block on top of a block) This question is very similar to this one here.
A block of mass $m_1$ is placed on another block of mass $m_2$ lying on a smooth horizontal surface.
The coefficient of friction (static and kinetic) between $m_1$ and $m_2$ is $\mu$.
Find the acceleration of the blocks ... | Because you are not pulling with the critical force, $6N$, then static friction $F_f<\mu m_1g$, where $\mu=0.2$.
The equations you get from Newton's second law are:
$$F-F_f=m_1a$$
$$F_f=m_2a$$
Substitute $m_2a$ into $F_f$ in the first equation:
$$F-m_2a=m_1a$$
$$F=a(m_1+m_2)$$
$$\frac{F}{m_1+m_2}=a$$
$$\frac{5 \mathrm... | {
"language": "en",
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"source": "stackexchange",
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No well-defined frequency for a wave packet? There are similar questions to mine on this site, but not quite what I am asking (I think). The de Broglie relations for energy and momentum
$$ \lambda = \frac{h}{p},
\\
\nu = E/h .$$
equate a specific frequency and wavelength to a particle, yet we know that a wave packet
i... | Let's say an isolated atom emits a photon. The excited state in the atom has some lifetime $\tau$. Through the energy-time uncertainty relation, that gives the excited state some uncertainty in energy $\delta E\sim h/\tau$ (not the same as $\Delta E$, which is a difference in energy between atomic states). The photon t... | {
"language": "en",
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"source": "stackexchange",
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Neutrinos: how can they carry information about universe? I know that neutrinos are particles with a very small mass and no electric charge. They infrequently interact with matter and so they can give us information about the "old" universe. But how can they do it?
| Note: I am not an expert on neutrinos, so if I have missed anything, someone please let me know.
For one thing, the knowledge of the mere existence of neutrinos is important for a full understanding of our universe. Also you may have heard that for a long time neutrinos were thought to be massless, but the observation ... | {
"language": "en",
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If the multiverse theory is true, can there be a Universe where there are different laws of physics? This is probably a very difficult question. But my question is essentially this, if there are other Universes can different laws of physics exist in those Universes and if so, can't there be a Universe where the laws of... | this is of course, entirely philosophical, and cannot be answered from physics considerations. But it is a fair question nonetheless, so i'll take a shot.
The only notion of metaverse that makes sense to me is Tegmark's vision of mathematically consistent universes. This merges with the anthropic principle in order to ... | {
"language": "en",
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"source": "stackexchange",
"question_score": "3",
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About the microscopic form of magnetocrystalline anisotropy Currently people write magnetocrystalline anisotropy as
$H_{an}=-K s_x^2$
from its classical counterpart:
$H_{an}=-K ( \sin \theta)^2$
where $K$ is the anisotropy constant, but for spin 1/2, $s_x^2$ is just the identity matrix, which shows no anisotropy, so ho... | A couple of points:
*
*For $S=\frac{1}{2}$, the expectation value for the "single-ion" magnetocrystalline anisotropy is indeed zero, and it should be! That is, if we consider a lattice of sites labelled by $i$, then $(\sigma_i^x)^2=\mathbb{I}$, the identity matrix, and there is indeed no anisotropy as you say. There... | {
"language": "en",
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"source": "stackexchange",
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Why is there a line in the middle of the Planck full sky map? Why there is a straight line (or perhaps a flat surface) in the middle of the Planck full sky map?
(source: esa.int)
and zoom
(source: handshakemag.com)
(it makes it seem that the big bang was a collision between two unknown surfaces)
| What you're seeing is the galactic plane of the Milky Way. There are several processes emitting at a variety of wavelengths, and because the emission is so nearby it comes out quite bright on the all sky map. Planck observes at about $\mathrm{mm}$ wavelengths, so the most prominent emission within the galaxy is thermal... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/69127",
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"source": "stackexchange",
"question_score": "5",
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Gravitational field strength and Horizon in Rindler coordinates I came across the following statements in 't Hooft's black holes notes, but not being able to justify them.
The metric in the Rindler coordinates $x=\tilde{x}, y=\tilde{y}, z= \rho \cosh{\tau}, t= \rho \sinh{\tau}$ is
$$ds^2 = -\rho^2 dt^2 + d\rho^2 + d\t... | 1)Gravitational Field Strength
You have to consider the analogy with a uniform accelerated move, in special relativity, that is :
$z' = \frac{1}{a} ch (a \tau'), t' = \frac{1}{a} sh (a \tau')$
Here $a$ is the acceleration and $\tau'$ is the proper time.
You get : $dz'^2 - dt'^2 = -d\tau'^2$,as wished.
Now, make the ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/69198",
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"source": "stackexchange",
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Doubt over Kelvin-Planck statement of thermodynamics' second law I have a doubt over the Kelvin and Planck's statement of thermodynamics' second law, in particular applied to a cycle. Let's take a Carnot cycle as an example, and let's call the first two transformations (the isotherm and the adiabatic) done. Now, isn't ... | Every thermodynamic system satisfies the first law during a given process;
$$
\Delta E = Q-W
$$
Here $\Delta E$ is the change in its internal energy, $Q$ is the heat transferred to the system, and $W$ is the work done by the system. For a system undergoing a cyclic process, namely one for which it starts and ends in... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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is it possible to load a transversal wave with a longitudinal wave like in communication engineering for sending information what we do,
we take a high frequency carrier and modulate it with the message signal
so can we do the same thing like take a high frequency carrier (transversal wave)
but the message is not tr... | That is what FM Radio works on.
So lets say we have an entirely analogue setup - a mic as input (RJ talking for example) and that being broadcast over a 91 MHz Signal real-time = Audio wave over a High frequency carrier. There is no digitization as you are not sending 0s and 1s here.
However Digital audio over FM ... | {
"language": "en",
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"source": "stackexchange",
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Operator on Function of Momentum (QM) I have exactly 0 clue on how to start this problem, but I would be forever grateful for a hint in the right direction.
Given the operators $\hat x=x$ and $\hat p=-i\hbar \frac{d}{dx}$, prove the following relation:
$$
[\hat x, g(\hat p)]=i\hbar\frac{dg}{d\hat p}.
$$
| Deduce the general form of the commutator
$$[\hat{x},\hat{p}^n] $$
write your function as a power series of $\hat{p}$
$$g(\hat{p})=\sum_{n=0}^{\infty}g_{n}\hat{p}^n $$
apply linearity of the commutator and then you should get your result
| {
"language": "en",
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"source": "stackexchange",
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Static Spherical symmetric solution of Einstein's equations with a perfect fluid I am reading Wald for the interior solutions of a static spherical metric. Assume it to be of the form
$$ds^2 = -f(r)dt^2 + h(r)dr^2 + r^2 ( d{\theta^2} \sin^2{\theta}d{\phi^2})$$
Wald states: For a perfect fluid tensor $T_{ab}= \rho u_a ... |
*
*First, why is the the static killing vector $\frac{\partial}{\partial t}$ equal to $-f^{\frac{1}{2}} dt$?
The vectors $\partial/\partial t$ and $-f^{\frac{1}{2}} dt$ are not equal to each other. They're parallel to each other, and the factor of $-f^{1/2}$ is just so that the four-velocity is properly norm... | {
"language": "en",
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"source": "stackexchange",
"question_score": "2",
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Vacuum polarization in QCD and gluon bubbles In analogy to QED, the following Feynman diagram is a diagram contributing to the vacuum polarization effect, leading to anti-screening, asymptotic freedom and running of the strong coupling constant. It can also be interpreted as a correction to the propagator of the gluon.... | Yes, this is another diagram that contributes to "vacuum polarization", also called self-energy. If you want to make a consistent calculation of the self-energy in perturbation theory, you need to include the two diagrams that you mention, plus the diagram where the gluon splits into $\bar{q}q$, plus, depending on your... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The General Relativity from String Theory point of view I have a hard time understand the statement that
When you only look at the classical limit or classical physics, string
theory exactly agrees with general relativity
Because from what I know, String Theory assumes a fixed space time background (ie, all the st... |
UPDATE: I have written a more complete answer here: How do Einstein's field equations come out of string theory?
The effective gravitational terms of the spacetime action, which can be derived from the Polykov action (gravitons are bosons) are --
$$S_{G}=\lambda\int\left(R+\ell_s^2R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\si... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/70060",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How is energy extracted from fusion? I understand that combining deuterium and tritium will form helium and a neutron. There are three methods to do this (1) tokamak (2) lasers and (3) cold fusion. I would like to know after helium is formed. How is that energy extracted from tokamak and stored?
| If you search the ITER site, ITER being the international prototype fusion reactor which will demonstrated the possibility of getting megawat useful energy from fusion, one sees that their main aim is to demonstrate this feasibility:
The main carrier of energy out of the plasma is the neutron, and methods to efficientl... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Full time-derivative of a function and Schrodinger equation From Hamiltonian formalism there is well known equation,
$$
\frac{d F}{dt} = \frac{\partial F}{\partial t} + \{F, H\}_{PB},
$$
where $ \{H, F\}_{PB}$ is the Poisson bracket.
After using Hamiltonian formalism in quantum mechanics it transforms into
$$
\frac{d ... | In the Heisenberg representation, the "equivalent" of the Schrodinger equation is :
$$\hat H(t) = \frac{\hat P^2(t)}{2m} + V(\hat X(t))$$
with $[\hat P(t), \hat X(t)] = i\hbar$
If you are looking at eigenstates and eigenvalues of the hamiltonian, you will look for a constant Hamiltonian.
For instance, for the harmonic... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/70466",
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"source": "stackexchange",
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Movie airplanes and suction Having watched a recent action movie (with zombies in it) I wondered whether the suction from a hole in the airplane's hull would really be able to rip out luggage, persons and even seating benches.
To my understanding, "suction" is nothing but the lack of pressure, i.e. the suction in the a... | I guess this depends on the size of the hole and the altitude of the plane. You are right that the suction effect will last only as long as there exists a pressure differential between the cabin and the outside. The hole's size determines the rate of equilibration. For example, in the James Bond movie Goldfinger, a fir... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/70562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Biggest experimental validations of postulates of Quantum Mechanics What are some experimental results that validate postulates of Quantum mechanics completely beyond any doubt ? Since there are alternate theories being used by various physicists to describe the same phenomena, I wonder there are some results that are ... | There's no such thing as verifying a theory "to complete exclusion." One of the key principles of the scientific method is that theories can't be proven by experiment, only supported (or disproven as the case may be). The scientific reality is that quantum mechanics, like any other theory of its caliber, has been suppo... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/70623",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Divergence theorem in complex coordinates This question is related to Stokes' theorem in complex coordinates (CFT)
but, I still don't understand :(
Namely how to prove the divergence theorem in complex coordinate in Eq (2.1.9) in Polchinski's string theory
$$\int_R d^2 z (\partial_z v^z + \partial_{\bar{z}} v^{\bar{z}... | Let $\sigma^1$ and $\sigma^2$ be real coordinates on $\mathbb R^2$. Using the results on page 33, we find that
\begin{align}
\partial_zv^z
&= \frac{1}{2}(\partial_1 -i\partial_2)(v^1 + iv^2)
= \frac{1}{2}(\partial_1v^1 + i\partial_1v^2 - i\partial_2v^1 + \partial_2v^2) \\
\partial_{\bar z}v^{\bar z}
... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/71168",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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How does adding electrons break the angular momentum degeneracy? In the hydrogen atom, the energy does not depend on l. This degeneracy is sometimes called "accidental" (because it does not correspond to some symmetry?). However, there is l dependence in the energy multi-electron atoms. Since the original degeneracy wa... | You could say that the degeneracy for the H-atom is accidential, but this unique property that the energy does not depend on l is because of the fact that you have a pure Coulomb-potential. For this pure Coulomb potential there exists an operator:
$\textbf{R} = \frac{1}{2\mu}(\textbf{p}\times\textbf{l}-\textbf{l}\times... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/71231",
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Hydrogen cloud at the universe's beginning? What prevented all of the hydrogen at the universe's start from coalescing into one gigantic star?
| Pulsar and Chris White have given nice explanations of why the dynamics of the early universe would not lead to the formation of one big starlike object. There is also another, much more generic argument against such a process, which boils down to the existence of cosmological horizons. At any given time in the evoluti... | {
"language": "en",
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Why does the event horizon of a black hole not look like a bright sphere? All infalling matter-energy appears to an external observer as frozen in time at the event horizon. Why then is this horizon not extremely bright due to radiation that is able to escape radially? So should all black holes not appear as a bright s... | As Ben says, black holes that have accretion disks shine very brightly indeed due to heating of the matter in the disk as it falls inwards. However not all black holes have accretion disks. For example the black hole at the centre of our galaxy does not, and I suspect you're asking why it is no longer shining brightly.... | {
"language": "en",
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Is it possible for a photon to be at rest? I know it doesn't really make sense if looking at the photon from the wave point of view, but is there any law of physics which prohibits a photon from stopping completely? Thanks.
| Those laws of physics would be Maxwell's Equations. I won't go into too much detail but from those equations you can get a wave equation for light. The speed of the wave is determined by two fundamental constants, $\epsilon_{0}$ and $\mu_{0}$. If the speed of light was variable in anyway then those constants would also... | {
"language": "en",
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Why does Hawking radiation cause black holes to die? If a particle is being expelled from a black hole and an antiparticle is being driven into it, shouldn't the opposite occur as well and in the same frequencies?
I mean, black holes should emit antiparticle radiation as well and gather energy from it and the black hol... | The following answer is not "rigourous", but it may gives a simple explanation.
Suppose you have a quantum fluctuation just near the horizon, but outside.
This quantum fluctuation create 2 particles, one with a negative energy -E, the other with a positive energy +E.
If the 2 particles stay outside the black hole, th... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/71718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Why neutrons in nucleus don't decay? In this question it is explained that neutrons in nucleus don't decay because the next state would not be lower in energy than the previous.
How come neutrons in a nucleus don't decay?
But it doesn't say what causes the neutron to know that the next stage is not lower in energy?
| hdhondt is right. A neutron doesn't need to "know" that another state isn't lower in energy to stay in the state is it at. It stays in the state of lowest energy because it is the state of lowest energy (just like a ball at the bottom of a pit).
Furthering hdhondt's ball example: the ball in the pit moves to the bo... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/71818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why does positronium decay into 2 photons more often than into 3 photons? I cannot find the answer to the above question.
I know that para-positronium is created with a probability of $25\%$ and decays into 2 photons, while ortho-positronium is created with a probability of $75\%$ and decays into 3 photons.
I also kn... | I'm afraid the responses so far are either misleading or do not answer the question.
In fact in a dense medium a positron when it forms the longer lived ortho-positronium can "pick-up" an electron from an adjacent atom and then decay into 2 x 511 keV photons long before the 3-photon ortho-positronium state would have d... | {
"language": "en",
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"source": "stackexchange",
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Electromagnetic Radiation of Charged particles This question is motivated by similar one.
If an accelerated point charge $q$ radiates with power $W$ then I assume the same particle with charge $-q$ will radiate with the same rate $W$. Now what if we make a dipole with these two charges and accelerate it with the same ... | According to this paper, New approach to the classical radiation fields of moving dipoles, the answer is:
$P = \dfrac{18d^2a^4}{35c^7} + \dfrac{2d^2\dot a^2}{15c^5}$
Here, $d$ is the fixed electric dipole moment and the acceleration, velocity, and dipole moment are along the $z$ axis.
From the paper:
This formula may ... | {
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First principle calculation of boiling point of water How can we theoretically calculate the boiling point of water at given pressure (other subtle parameters as well, if any)? What is the most accurate (minimum discrepancy with experimental value) computation that can analytically predict the
boiling point of water?
P... | The boiling point (i.e. the saturated vapor pressure as a function of temperature) is described by the Clausius–Clapeyron equation which is a consequence of classical thermodynamics. Of course if we start investigating why certain thermodynamic potentials used in that equation have to be such and such then eventually w... | {
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Any physical example of an "explosive" differential equation $ y' = ky^2$? I was told that in physics (and in chemistry as well) there are processes that may be described by a differential equation of the form $$ y' = ky^2. $$ That is, the variation of a variable depends from the number of pairs of the elements.
I und... | As described in Wikipedia, world population was growing in just that fashion (hyperbolic) for some time in the 20th century.
| {
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What are the advantages in using 2 identical capacitors? What advantage might there be in using two identical capacitors in parallel connected in series with another identical parallel pair, rather than using a single capacitor?
| Capacitors are sometimes connected in series to increase the working voltage range, but it is not a good design technique because variances in capacitors can cause a variance in voltage and thus exceed the voltage rating on one of the capacitors.
And if you can't achieve a value of capacitance with one capacitor, som... | {
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Minimum height of mirror required to view image I wanted to know the minimum height of mirror required to be able to view a complete image of a person. I considered the following setup:
$HF$ is the person in question. $H$ denotes the head, $F$ the feet, and $E$, the eyes. For the person to see his complete image, a ra... |
In image ,a person of height H is standing against a plane mirror of length AB.We will need the part AF of mirror so that the person will be able to see his full reflection .
In triangle CED we can say $\frac{CD}{CA}=\frac{DE}{AF}$
Or
$\frac{2\not{x}}{\not{x}}=\frac{H}{minimumlength}$
Minimum length =$\frac{H}{2}$
We... | {
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Overtaking with non-constant acceleration I have tried to solve this problem by adding the sum of the displacements during acceleration, constant velocity and deceleration, but it does not work out.
Question:
A car accelerates from rest to $20~\text{m/s}$ in $12$ seconds ($a =5/3~\text{ms}^{-2}$), it travels at $20~... | Since the acceleration changes discontinuously, there is no one way to solve it in one step.
One simplification is to switch to the frame of reference of the RC car. You can do this simply by making the initial velocity of the starting/stopping car $-14$ m/s. The question then reduces to: When is the displacement of... | {
"language": "en",
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What is the meaning of "CW" in LASER? I am reading a user's manual, and the word appears here.
At first, I think "CW" means "center wave".
But later, I find that the meaning of "CW" is "continuous wave".
It makes me confused.
Generally, Laser has a unique frequency. Properly speaking, it has a too small fluctuation ... | The choice between continous or pulsed can come down to the power output of the laser. For the same input power you can either output that power at some constant and therefore the average power is equal to the peak, or send the energy out in pulses. In pulsed mode the peak energy can be much higher while maintaining ... | {
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Can spheres leaking charge be assumed to be in equilibrium? I am struggling with the following problem (Irodov 3.3):
Two small equally charged spheres, each of mass $m$, are suspended from the same point by silk threads of length $l$. The distance between the spheres $x \ll l$. Find the rate $\frac{dq}{dt}$ with which... | If we continue with the suggestion you made, of obtaining $\ddot x$ from the equation of motion $v=a/\sqrt{x}$ which was provided, and substituting this into the equation $F=m\ddot x$, then we do indeed find that $\dot q$ is not constant. It is only by ignoring the $m\ddot x$ term - by assuming that $v\approx 0$ - that... | {
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Can an Incandescent Bulb be considered as a heater? Due to an experiment, I need a small heater (around 70 to 100 watts). I intend to use an incandescent bulb so it can act as a heater. What I wonder here is will a 70 watts Incandescent Bulb be equal to a 70 watts heater?
| Whatever bulb is used, basically all the power consumed will go into heating the environment, with the notable exception being any photons that escape through a window or some such thing.
Different styles of bulb have different visible light-heat ratios, but even the visible light emitted will simply bounce around the ... | {
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Sliding force less or equal? Why is the force required to slide a magnet off a steel plate A LOT less than the force required to directly pull it off?
The force required to pull the magnet can be: 20lb
While the force required to slide the magnet can be: 1lb more/less.
Why is that?
| When you slide a magnet off a plate, you are doing the same amount of work as when you pull it directly off (slightly more, in fact, because of the friction). However, while it's the same amount of work, it's spread out over a much larger distance, and so requires much less force. When you slide a magnet off a plate, t... | {
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The relation between Gauss's law and Coulomb law and why is it important that the electric field decrease proportionally to $\frac{1}{r^{2}}$? My question relates to the third MIT's video lecture about Electricity
and Magnetism, specifically from $21:18-22:00$ : http://youtu.be/XaaP1bWFjDA?t=21m18s
I have watched the d... | The $1/|r|^2$ dependence is fundamentally geometrical in nature, stemming from our 3d world.
Differential equations, even PDEs, are sometimes said to have Green's functions. Consider the differential equation
$$\nabla \cdot K = \alpha$$
for some vector field $K$ and some scalar field $\alpha$. This is structurally id... | {
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Heating coffee by yelling? Is it a myth that yelling to a coffee mug will heat it? I have been hearing my friend saying that screaming will heat coffee or water.
| This probably comes from an advertisement campaign by Physics Central. Their poster is
It is a silly question; no amount of yelling will appreciably heat a coffee cup. We are very inefficient at turning energy into sound. The page from which I got the photo says that a loud shot puts 0.001 W of power into the air. Thi... | {
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Projectile Motion Question involving a ball and a ramp inclined at an angle The question is to finde the initial horizontal velocity of the ball at end of the ramp, where it is released.
I know how to do this using gravitational potential energy and kinetic energy ($v=\sqrt{2gh}$), assuming all potential energy is conv... | Vertically:
$\ddot{y}=a$
$\frac{d\dot{y}}{dt}=a\therefore \int d\dot{y}=\int adt\therefore \dot{y}=u\sin\theta +at$
$\frac{dy}{dt}=u\sin\theta +at\therefore \int dy=\int \left ( u\sin\theta +at \right )dt\therefore y=u\sin\theta t+\frac{a}{2}t^{2}$
Horizontally:
$\ddot{x}=0$
$\frac{d\dot{x}}{dt}=0\therefore \int d\dot{... | {
"language": "en",
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The Momentum Operator in QM I've seen the 'derivation' as to why momentum is an operator, but I still don't buy it. Momentum has always been just a product $m{\bf v}$. Why should it now be an operator. Why can't we just multiply the wave function by $\hbar{\bf k}$? Why should momentum be a derivative of a wave function... | The simplest possible solution to the Schrodinger equation is that of a plane wave: $$ \psi = e^{i(\boldsymbol{\mathbf{k}}\cdot \boldsymbol{\mathbf{r}}- \omega t )} $$ In this scenario, the position of the system is indeterminate, but the momentum is known. Next, take the gradient of this wavefunction. $$ \nabla \psi ... | {
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How does one prove that Energy = Voltage x Charge? We know
$$E = q V$$
where $E$ is the energy (in Joules), $V$ is the potential difference (in Volts), and $q$ is the charge. Why is this equation true and how we prove it?
| electric field strength is
$$E=\frac Fq=\frac Vd$$
with $V$=voltage, $d$=distance between charged plates
\begin{align}
\frac Fq&=\frac Vd \\
Fd&=qV
\end{align}
but $Fd$=energy
$$\therefore {\rm energy}=qV$$
| {
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Will a diamond helm be more protective against bullets? Will a helm made entirely of diamond be more protective against bullets than standard steel or kevlar helmlets?
| Diamond, though hard, is brittle. The crystalline planes within it can shear relatively easily.
"The toughness of natural diamond has been measured as 2.0 MPa m1/2, which is good compared to other gemstones, but poor compared to most engineering materials." - Wikipedia
The fracture toughness of steel is about 50 MPA.m1... | {
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How do we know photons have spin 1? Electrons have spin 1/2, and as they are charged, they also have an associated magnetic moment, which can be measured by an electron beam splitting up in an inhomogeneous magnetic field or through the interaction of the electrons's magnetic moment with an external magnetic field in s... | Spin of the photon is an ongoing theoretical research. For classical electromagnetic field the total angular momentum is $\vec{J}=\vec{r}\times <\vec{E}\times \vec{B}>$ (all vectors). In field theory this quantity is $\vec{J}=\vec{L}+\vec{S}$, it is gauge invariant and possible to observe. But! Nobody found a correct w... | {
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Compound microscope vs astronomical telescope In compound microscope, we take such an objective lens which has small focal length. While in astronomical telescope, we take such an objective lens which has large focal length.
Why don't we use same objective lens in both? The function of both the devices is to enlarge a... | Wikipedia's article on Focal length explains pretty clearly (emphasis mine):
In most photography and all telescopy, where the subject is essentially infinitely far away, longer focal length (lower optical power) leads to higher magnification and a narrower angle of view; conversely, shorter focal length or higher opti... | {
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Equations of motion of displacement field We have an action: $$S[\boldsymbol{u}] = \frac{1}{2} \int dt \int d^3x \left\{ \mu (\frac{\partial u_{i}}{\partial t})^{2} - \nu (u_{ii})^{2} - \rho(u_{ij})^{2}\right\} $$
Where $u_{ij} = (\partial_{i}u_{j} + \partial_{j} u_{i} )/2$.
Where index $i = 1,2,3$ denotes the $x$ $y$... | Partial answer :
Define functions $u_i(p) = u_i(p^0,p^1,p^2,p^3)$ as Fourier transform of the functions $u_i(x)$
Then take the Fourier transform of your three Euler-Lagrange equations.
You get three equations of kind :
$A_{11}(p)u_1(p) + A_{12}(p) u_2(p) + A_{13}(p)u_3(p) = 0$
(The functions $A_{ij}(p)$ are quadrati... | {
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Why is there a factor of $4\pi$ in certain force equations? I mean to ask why there is $4\pi$ present in force equations governing electricity? Though all objects in universe are not spherical and circular, the constant of proportionality in both equations contain $4\pi$. Why?
| Any differential equation of the form $\nabla \cdot A = \alpha$ and $\nabla \wedge A = 0$ in $n$-dimensions has as its Green's function (that is, the solution for a point source, for $\alpha = \delta$, the Dirac delta function) a field $G$ of the form
$$G(r) = \frac{1}{S_{n-1}} \frac{\hat r}{|r|^{n-1}}$$
where $S_{n-1}... | {
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Path of least resistance vs. short circuit Some sources on the web claim that "electricity follows the path of least resistance" is not true, e.g. this physics SE question. However, in every explanation of "short circuits", the author says that current flows through the short because it's following the path of least re... | It's more like "Electricity tries to follow the path of least resistance". The sentence is a non-rigorous intuitive tool that helps one quickly make sense of current paths. It's not a physical law.
This comes from the behavior of resistors in parallel --this is where current has a "choice" of which direction to take. ... | {
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Does alternating current (AC) require a complete circuit? This popular question about "whether an AC circuit with one end grounded to Earth and the other end grounded to Mars would work (ignoring resistance/inductance of the wire)" was recently asked on the Electronics SE.
(Picture edited from the one in the above lin... | Indeed, AC can flow without a "complete circuit" - that's what happens in LC circuits all the time. An LC circuit is technically not complete - the capacitor of LC circuit contains an insulator between its plates and so electrons are unable to flow through the capacitor (unless it fails). Still the oscillations in the ... | {
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Do higher-order mass moments have any physical meaning?
*
*The zeroth moment of mass of an object is simply its total mass.
*The first moment of mass yields an object's center of gravity (after normalization).
*The second moment of mass yields an object's moment of inertia.
Is there an analogous physical interpre... | In statistics the third moment is used to calculate skewness. I would guess this has a physical analogy. Although I haven't thought this through, I'd guess it would be possible to take a disk and deform it asymmetrically so that the centre of mass and moment of inertia remained the same, but the third moment changed. I... | {
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accelerated charged particles and interaction with magnetic field In high school we are taught that magnetic field perpendicular to velocity of an charged particle experience perpendicular force that causes it to move in circular path by relation
$$qvB=\frac{mv^2}{r}$$ but in drawbacks of Bohr's theory it was proposed ... | Yes, as the charged particle is accelerated, it will emit radiation. But it never stops at a point because of the uncertainty principle. Refer Energy of electron spinning in a magnetic field
| {
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Probabilities in statistical mechanics I am reviewing some concepts in statistical mechanics and am becoming confused with how to calculate probabilities when a system has $N$ non-interacting particles.
For instance, let's say we have $N$ electrons with magnetic moment $\vec{\mu} = (g e/2 m)\vec{S}$. If we apply a str... | It seems like you want to consider $N$, distinguishable, non-interacting spins with spin $1/2$, so let's assume we don't have to consider subtleties arising because of identical particles. When there are $N$ distinguishable, the Hilbert space $\mathcal H_N$ of the system has dimension $2^N$, and when they don't intera... | {
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Calculate exact time of Solar midnight I want to calculate real time of midnight (Solar midnight), but I am unable to find any formula or algorithm for that. I have times of sunrises and sunsets for every day of year. How can I get Solar midnight times from them?
Thank you all.
| The formula for calculating the Local Time equivalent for Local Solar Time is $LST=LT+\frac{TC}{60}$, where $TC$ is the Time Correction factor $TC=4(Longitude-LSTM)+EoT$. $LSTM$ is the Local Solar Time Meridian, calculated by $15º-ΔT_{GMT}$, where $ΔT_{GMT}$ is the difference, in hours, from GMT. (For instance, here on... | {
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Do bifacial solar panels operate at lower temperature than normal solar panels? Bifacial solar panels have a glass backplane instead of the usual aluminum and are designed to absorb and convert to electricity light incident on either face. Many manufacturers claim that their bifacial panels operate at lower temperatur... | For what it's worth, lg310's seem to run a little cooler than gxb300's, at least on my roof.
See also http://dx.doi.org/10.1016/j.egypro.2014.02.148
but read Table 4 carefully. The differences seem to be small.
| {
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Why do chimneys have these spiral "wings"? While walking around I noticed something very peculiar. Many chimneys had spiral "wings", while others didn't. I came up with two possibilities:
*
*The wind circles around the chimney upwards which pushes whatever gases being released even higher into the sky.
*The wind ci... | The spirals are used to prevent the formation of Kármán vortex sheets downwind of the chimney. They work by diverting the wind upwards on one side of the chimney and downwards on the other, creating a three-dimensional airflow pattern that disrupts the vortex sheet. Without them, the vortex shedding could cause vorte... | {
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Projectile motion of a charged particle in a homogeneous electric field I'm reading an article about projectile motion, but I'm having some trouble with how the author found the equations of motion when a homogeneous electric field is considered.
To allow an immediate comparison as an example of non-gravitational
acce... | I think:
1) The first equation is just a rearrangement of the standard expression for the norm of the four-velocity $u=\frac{dx}{d\tau}$ in special relativity: $u_\alpha u^\alpha=1$ (with $c$=1).
2) Since $qE/m=g$, the acceleration in the z-axis $\frac{du^3}{d\tau}$ due to the electric field is obtained from the standa... | {
"language": "en",
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What is phenomenological equation and phenomenological model? I come across these terms in some papers. My understanding is that it is an equation or model describing a phenomenon. Usually, the equations are given and claimed to be true with only some explanations and justification, but not derived from the first princ... | You could read the definition in dozens of dictionaries and encyclopedias like this
http://en.wikipedia.org/wiki/Phenomenology_%28science%29
My feeling is that the term is related to the word "Phenomena" that is something that is observed. Thus, the phenomenological models and equations describe rather instrument readi... | {
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"source": "stackexchange",
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Minimal Extension of Wave Equation to Include Dispersion Let's say you are modeling some process with the wave equation $\frac{1}{c^{2}}\frac{\partial^{2}\psi}{\partial t^{2}} = \nabla^{2}\psi$. You wish to improve your model by including dispersive effects, but you want your model to be as simple as possible for compu... | What you want to do is change the wave equation into a Klein-Gordon equation:
$$\frac {1}{c^2} \frac{\partial^2 \psi}{\partial t^2} - \nabla^2 \psi + \alpha^2 \psi = 0,$$
where $\alpha$ is a constant of appropriate dimension and usually (in quantum theory) given by
$$\alpha=\frac {m c}{\hbar}.$$
Inserting an ansatz o... | {
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What is the difference between phase difference and path difference? The path difference is the difference between the distances travelled by two waves meeting at a point. Given the path difference, how does one calculate the phase difference?
| Let's assume that, two stones are thrown at two points which are very near, then you will see the following pattern as shown in the figure below:
let's mark the first point of disturbance as $S_1$ and the other as $S_2$, then waves will be emanated as shown above. By having a cross-sectional view, you will see the sam... | {
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Why do meteors explode? A report on the Chelyabinsk meteor event earlier this year states
Russian meteor blast injures at least 1,000 people, authorities say
My question is
*
*Why do meteors explode?
*Do all meteors explode?
| Airburst, I think, it is called.
An article on wired.com covers exactly this question
the rock was already going very fast when it entered the Earth’s atmosphere. There is no way the air could get it down to terminal velocity – there just wasn’t enough
distance for a rock this large. But this air resistance is es... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 0
} |
Does light loses its energy when it passes through denser medium? I know it does not because it emerges out of denser medium at 300,000 KM per second, but according to $E=mc^2$ and given that speed of light decreases inside denser medium with refractive index greater than 1, does not it suggest that energy of light ins... | Speed of light is a function. When light traverses through a non-vacuum medium, its speed decreases compare to its speed in vacuum. (That is why, index of refraction, n, as a ratio of speed of light in vacuum per speed of light in non-vacuum, increases).
It means, perhaps, the decreasing of speed of light when light t... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/76240",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 4
} |
Controlling Neutrinos for Communication Neutrinos travel straight through earth at the speed of light. Therefore, it seems that they could be great for intercontinental communication. Of course, I assume a lot still needs to be learned about detecting, producing and controlling neutrinos before they can be used for the... | We already know a lot about detecting, producing and controlling neutrinos.
Production in a controllable (switchable) way requires a particle accelerator. Enormous energy is not required but high current is. These are complex and expensive facilities. There are not something you can buy off-the-shelf.
Likewise detecti... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/76531",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
What does imaginary number maps to physically? I am taking undergraduate quantum mechanics currently, and the concept of an imaginary number had always troubled me. I always feel that complex numbers are more of a mathematical convenience, but apparently this is not true, it has occurred in way too many of my classes, ... | If you really wanted to, you could formulate the laws of physics without using imaginary numbers - see, for example, Can one do the maths of physics without using $\sqrt{-1}$?
Let's say you need to do a fourier decomposition of a function $f$ in order to find how some responds to being driven at $f$.
If you think back,... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/76595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 7,
"answer_id": 3
} |
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