Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Does a planar object balance on a unique point? Consider a horizontal planar convex 2D object (say lying on x-y plane) with uniform density. Under constant gravitational force (say in -z direction), does it always balance on a unique point lying on the object (i.e. sum of the torques vanishes with respect to a unique ... | If there is a point in the planar object from which it can be balanced, the net torque is zero from this point. Choosing the point as origin:
$0 = \tau = \int_v \mathbf r \times d\mathbf F$
Considering the object in the $xy$ plane, the weight in the direction $-z$, density and $g$ constants, and its thickness = $t$; th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$P=\epsilon_o \chi E$ or $\epsilon_o \chi E_o$ Suppose the polarisation inside a dielectric is given by $P$, then is it related to the electric field as $\vec{P}=\epsilon_o \chi \vec{E}$ where $E$ is the field inside the dielectric or is is $E$ the original field that would have been present in that region in absence o... | It is a matter of definition. The usual definition of $\chi$ implies the electric field $\bf E$ actually present inside the dielectric.
However, one has to notice that in the special case of a dielectric completely filling a large parallel-plates capacitor at a fixed difference of potential between the plates, the resu... | {
"language": "en",
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How can we discern so many different simultaneous sounds, when we can only hear one frequency at a time? As I understand it, the eardrum works like any other kind of speaker in that it has a diaphragm which vibrates to encode incoming motion into something the inner ear translate to sound. It's just a drum that moves b... |
But that still means the ear is only moving at one frequency at any given time
No, it doesn't mean that at all.
It means the eardrum is moving with a waveform that is a superposition of all the frequencies in the sound-wave it is receiving.
Then, within the inner ear, hair cells detect the different frequencies separ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/606949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Determining the curvature by symmetries of the metric Given the Kahn-Penrose metric:
$$
ds^2=2dudv-(1-u)^2dx^2-(1+u)^2dy^2
$$
I calculated the Riemann Tensor and found that all elements equal 0.
Is there some symmetry principle by which I could have easily deduced zero curvature or other properties directly from the me... | One way to see that the curvature tensor is zero is to start with the fact that there is no dependence of metric components on null coordinate $v$. So performing Kaluza–Klein reduction along the Killing vector $\partial_v$ we would obtain a Newton–Cartan spacetime with two spatial coordinates ($x$ and $y$), while $u$ ... | {
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Do photons "lose energy" when they are absorbed? Recently in my biology class I learned about an experiment in which isolated and illuminated chlorophyll pigments fluoresce in the red part of the spectrum, but also, the solution of the pigments gets hotter. Are the photons that are reflected as the electrons fall back ... | Strictly speaking: when a photon is absorbed, it ceases to exist, so it doesn't make sense to ask whether it loses its energy.
In the experiment that you describe the green light photons are absorbed, and red photons (having lower energy) are emitted as fluorescence. The likely reason for this is that the energy of an ... | {
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Can sound be used for propulsion? I'm no physicist so this might seem absurd.
I Remember watching a cartoon as a kid where the character uses a powerful speaker to propel his cart and I was wondering if this was actually possible.
Being a highschooler I am aware to propel forward you shoot something backward.
So maybe ... | This is a difficult question, even for people who are physicists. I say this after reading:
https://www.physics.princeton.edu/~mcdonald/examples/hidden_sound.pdf
which discusses the concept of hidden momentum, a concept I've never heard of. Moreover, based on that article, it appears experts argue whether it does or do... | {
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Polarity in a magnetized Möbius strip When a flat iron or Alnico washer is magnetized one of the faces develops a north polarity and the other, south. The geometric shape here is simple.
However, when a standard Möbius strip (or one of given thickness, radii of curvature and torsion of edges) is magnetized, which regi... | The answer is easily obtained by remembering what creates the permanent magnetic field. All subatomic particles are magnetic dipoles and in some materials - natural or man-made - subatomic particles are aligned by their magnetic dipoles in such a way that a stable macroscopic magnetic field is present.
The stability of... | {
"language": "en",
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What happens to the rest of the 95 percent in quarks? Quarks are bound by gluons. Gluons have a mass of 0, while mass of quarks is only 5%.
Where is the missing 95%?
| In the level of quarks and gluons one is in the realm of quantum mechanics and special relativity. Special relativity assigns to each particle a four vector , whose "length" is the invariant mass of the particle .
The length of this 4-vector is the rest energy of the particle. The invariance is associated with the fa... | {
"language": "en",
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Volumetric Dilatation Rate, Material derivatives, and Divergence in class we derived the following relationship:
$$\frac{1}{V}\frac{dV}{dt}= \nabla \cdot \vec{v}$$
This was derived though the analysis of linear deformation for a fluid-volume, where:
$$dV = dV_x +dV_y + dV_z$$
I understood the derived relation as:
$$\fr... | The continuity equation reads $$\frac{\partial \rho}{\partial t}+v\centerdot \nabla \rho+\rho \nabla \centerdot v=0$$where $\rho$ is the fluid density. Dividing this by $\rho $ gives $$\frac{1}{\rho}\left(\frac{\partial \rho}{\partial t}+v\centerdot \nabla \rho\right)+\nabla \centerdot v=0$$But, since the density is t... | {
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If a jet engine is bolted to the equator, does the Earth speed up? If a jet engine is bolted to the equator near ground level and run with the exhaust pointing west, does the earth speed up, albeit imperceptibly? Or does the Earth's atmosphere absorb the energy of the exhaust, and transfer it back to the ground, cancel... | Yes, but it is so minuscule that it wouldn't be noticed. It's the same as if your friend was driving a car and you stick your head out of the window and blew in the opposite direction of the car's movement
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/608372",
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If the sea surface were absolutely calm should the Sun reflection be the area of a circle instead a ribbon? Although waves produced on the sea can cause different points of the sea surface to reflect sunlight towards the same observer, how is that kind of ribbon image produced? Why isn't the reflection stretched also p... | Yes, the image of the sun or moon in a perfectly flat lake or ocean would be round. The "ribbon" is due to ripples in the surface of the water. To understand why it's a ribbon instead of a broad expanse of reflected light, all you need to do is shine a flashlight, at the top section of a reflective ball, move the la... | {
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Third-order Feynman diagrams of 2-point function in $\phi^4$-theory $\newcommand{\Braket}[1]{\left<\Omega|#1|\Omega\right>}$
Hello,
I am currently studying QFT and have a problem concerning the 2-point correlation function in $\phi^4$-theory. When I draw all the Feynman diagrams contributing to $\left<\Omega|\phi(x)\ph... | If you denote the external points by $x,y$ and the internal points by $1,2,3$ than the diagrams of the last line in your picture are $ x1, 11,12,23,23,23,3y$ and $x1,12,12,12,23,33,3y$ with integration over the internal points and what is between brackets is contracted. Yes, they are the same.
| {
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What is the propagation direction of plane wave? As far I know, plane wave equation is given by:
$$\vec{E}(\vec{r},t)=\vec{E_0} \cdot e^{i(\vec{k}\cdot \vec{r}-\omega t)} \hspace{2cm} \tag{1}$$
In some textbook propagation direction of $(1)$ is given as $\vec{k}$;
while in some other books, i found propagation directio... | $\vec E(\vec r,t)$ is at a peak when $e^{i(\vec k \cdot \vec r-\omega t)} = 1$, and a minimum when it is $-1$. Maxima happen when $i(\vec k \cdot \vec r-\omega t) = 0$ or $2 \pi i$. It is at a minimum when the exponent is $\pi i$.
One peak is found where $\vec k \cdot \vec r =\omega t$. As t gets larger, the peak will ... | {
"language": "en",
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Hypothetically, why can't we wrap copper wires around car axles and turn them into electromagnets to help charge the batteries? We already have a magnetic core, why can't we use it to recharge the batteries? The only problems I see with it are potentially wiping magnetic data, but doesn't the electromagnet have to be r... | This is basically what happens in the alternator. The car's engine, which turns the wheels, also turns the alternator's rotor. The magnetic rotor is surrounded by coils of wire, and induces a current that charges the battery. It is important to note, though, that this does take energy from the engine: nothing is for fr... | {
"language": "en",
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In QFT why does the degree of the interaction terms in Lagrangian start from 3? I'm new to QFT so it's not obvious to me why there is no quadratic interaction terms in Lagrangians.
For example, the Lagrangian for a real scalar field is
$$L=\frac{1}{2}\partial_\mu \phi \partial^\mu \phi-\frac{1}{2}m^2\phi^2-\sum_{n\geq ... | Contrary to what the other answer's claim, we do add these $g\phi^2$ terms as interaction term.
Check any QFT text book, specifically the section on mass renormalization and field renormalization counter terms.
| {
"language": "en",
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Are all atoms spherically symmetric? If so, why are atoms with half-filled/filled sub-shells often quoted as 'especially' spherically symmetric? In my atomic physics notes they say
In general, filled sub-shells are spherically symmetric and set up, to a good approximation, a central field.
However sources such as her... | Your quote references "filled sub-shells". You then write in your question "all atoms are really perfectly spherically symmetric".
Not all atoms have only filled sub-shells. Noble gasses are strong examples of atoms with only filled sub-shells and the small ones behave approximately spherically symmetrically.
Most at... | {
"language": "en",
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Question regarding Lorentz Transformation and Space Contraction- Contradiction I stumbled upon this question regarding Special relativity- and have seemed to reach a contradiction.
I am trying to find the distance that the ball travels
I am obviously not looking for the numerical answer, but I'm trying to understand wh... | You would end up in the same contradiction (wich is not) in Newtonian mechanics. Position is relative in both newtonian mechanics and Special Relativity. If i am moving relative to you you would see me covering a distance, but i would see myself standing still covering no distance.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/610530",
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Luttinger Liquid Parameter Physical Meaning - Attractive or Repulsive For 1+1D systems at long wavelength, it is known that the (Tomonaga)-Luttinger Liquid can be rewritten in terms of bosonic parameters, where the Hamiltonian densities can be written
\begin{align*}
H =& \left[\psi_L i\partial_x \psi_L - \psi_R i \part... | Unfortunately the literature on Luttinger liquid theory and bosonization suffers from a lack of consistent notation. This is just one of many inconsistencies you will find between different authors. Generally any single author will (hopefully) be self consistent, but comparing between different authors can require quit... | {
"language": "en",
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How warm are radioactive metals? I read that radium is warm to the touch -- is that because of actual heat or is that because, for example, the radiation it emits creates the sensation of warmth? How high of a temperature can a radioactive element or isotope actually have?
| Yes radioactive materials are warmer than they would otherwise be. There are two reasons for this.
One is that the decayed atom will have picked up some kinetic energy in recoil and that is heat.
Another is that some of the radiation does not make it outside but hits another atom and transfers its energy to it as more ... | {
"language": "en",
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Aharanov-Bohm Effect Gradient of Line Integral In Griffiths' Quantum Mechanics 2nd edition section 10.2.3 the phase
$$g(\mathbf{r}) = \frac{q}{\hbar}\int_{O}^{\mathbf{r}}\mathbf{A}(\mathbf{r}')\cdot d\mathbf{r}'$$
is defined. It is noted that this integral is only defined if $\nabla\times\mathbf{A} = 0$ throughout the ... | I think $\mathbf{r}$ is just an arbitrary position in space where the particle could be for the integral in the definition of $g$. So it doesn't depend on $\mathbf{R}$ in $g$. It is the potential $V$ used in other parts of the setup which confines the particle's position $\mathbf{r}$.
We can treat the space outside the... | {
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Meaning of adding a term to the Hamiltonian in a quantum harmonic oscillator Let $H$ be the Hamiltonian in a harmonic oscillator,
$$
H = \sum_{n=0}^{\infty} \hbar \omega \left (n+\frac{1}{2} \right ) |n\rangle \langle n|.
$$
Suppose we introduce the interaction
$$V = \sqrt{2} \hbar \omega
(|0\rangle \langle 1|+|1\r... | You can think of the initial Harmonic oscillator with the Hamiltonian $H=\sum_{n}\hbar\omega(n+1/2)|n\rangle\langle n|$ as a free system, that is under time evolution a state prepared in any one of the $|n\rangle$ remains in that state. However, the effect of adding an interaction term of the form that you proposed now... | {
"language": "en",
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Why is energy lost here? Let's say a $1 \ \text{kg}$ block is moving.
With a speed of $1 \ \text{m/s}$ so its kinetic energy is $\frac{1}{2} \ \text{J}$. Now let's gently place a block of mass $3 \ \text{kg}$. Now as linear momentum is conserved due to lack of external forces on the system the blocks move together with... | Energy is lost due to work done by friction . Try to analyze each block individually.
| {
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Non-Analytic Equations and Chaos Could anyone please tell me an example of an equation with no analytic solution(s) that is not a chaotic one? And what is the physical meaning of having analytic solution? For instance, the three body problem does not have in general analytic solution and it leads to chaos. But I don't ... |
Could anyone please tell me an example of an equation with no analytic solution(s) that is not a chaotic one ??
A simple fifth order polynomial ($k_5 x^5 + k_4 x^4 + k_3 x^3 + k_2 x^2 + k_1 x + k_0 = 0$) has no analytic solution, but is not chaotic.
And what is the physicall meaning of having analytic solution ??
T... | {
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Noether's Theorem and Liouville's Theorem Liouville's theorem states that for Hamiltonian systems the phase space volume $V(t)$ is a conserved quantity, i.e., $\frac{d}{dt}V(t)=0$. This is related to the fact that trajectories in phase space do not cross and a point in phase space has a unique time evolution.
Noether's... | There are several versions of Liouville's theorem. One version states that a Hamiltonian vector field (HVF) $X_H=\{H,\cdot\}_{PB}$ on a symplectic manifold $(M,\omega)$ is divergence-free
$$ {\rm div} X_{H}~=~0.$$
One may view the above HVF as the underlying symmetry.
| {
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Is projectile motion an approximation? Doesn't the acceleration vector points towards the center of the Earth and not just downwards along an axis vector. I know that the acceleration vector's essentially acting downwards for small vertical and horizontal displacements but if the parametrization of projectile motion do... | It is an approximation, as everything in physics is an approximation based on mathematical/statistical modeling. And by definition, a model is an imperfect representation of reality. Models employ simplifying assumptions in order to make problems tractable so we can explain what is happening and hopefully make predic... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/611916",
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Will liquid nitrogen evaporate if left in an unopened container? SOS! I left work today and got a horrible feeling that I forgot to put the lid back on a large container of liquid nitrogen which contains many racks of frozen cells in it. If this did happen, how long would it take liquid nitrogen to evaporate? Does it s... | This is because the temperature at which liquid nitrogen changes phase to gas is below room temperature. I suppose you could apply Newton's law of cooling differentially to the surfaces of the canister. The open end is exposed to the atmosphere, and you can imagine volume elements that cool before the liquid nitrogen a... | {
"language": "en",
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Is it even theoretically possible for a perfect clock to exist? I have heard that even atomic clocks lose a second every billion years or so. That raises the question, is it even theoretically possible for a perfect clock to exist, one that never gains or loses time?
| Well, there is no concept of absolute time or a perfect ' tick tock ' in the universe. Phenomena happen at their own rate.
You can't quantify their 'perfection'. You can quantify the errors you made while measuring their physical aspects.
| {
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How long does it take an electron to emit (or absorb) a photon? A photon is emitted (or absorbed) by a transitioning electron. How fast is this process?
| The emission process always takes some time. How much depends on the kind of transition.
If the transition is spontaneous dipole transition, like when excited electronic state of an atom decays, the rate of spontaneous transition as found using quantum electrodynamics is
$$
\Gamma = \frac{\omega_{12}^3|\mu_{12}|^2}{3\p... | {
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Problem deriving entropic uncertainty relation In this paper the authors state that the inequality near the bottom of page 2 reduces to inequality (1) when $N=1$. However, I am struggling to get that result, as I have an extra minus sign in front of the integrals. Can anyone try this for themselves and see if they get ... | Due diligence first. The first term of the first equation you wrote is missing a sign, as you should have easily checked yourself, from the coefficient of $\ln \pi$.
That is, write down the parent $W(q)$ right above the equation you start with, around your minimum q =2, i.e. $q\equiv 2(1+\epsilon)$, so that $p= 2(1-\... | {
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Explosion of an asteroid in orbit I've learned from an answer here on this site that if a body were to split apart in orbit the center of mass will continue to be on the same orbit. (Couldn't find the post)
But let's say an asteroid is to blow up in two pieces such that the smaller piece reverses its direction and velo... |
if a body were to split apart in orbit the center of mass will continue to be on the same orbit.
This is not always true. It would be true if we were talking about a projectile in a uniform gravitational field, but for an orbiting satellite the gravitational field is non-uniform. That means that for large displacemen... | {
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Why the formula of kinetic energy assumes that the object has started from an initial velocity of zero? According to my physics textbook, the formula of kinetic energy is:
$$
W = \frac{1}{2}mv^2
$$
Where $m$ is is mass of the object and $v$ is the velocity of the object. The equation is calculated from this (according ... | The $s$ in equation $W=mas$ can be replaced from 3rd equation of motion.
$$W=\frac{ma(v^2-u^2)}{2a}$$
$$W=\frac{1}{2}m(v^2-u^2) \rightarrow K.E.=\frac{1}{2}m(v^2-u^2)$$
Or else, if you still want to substitute $s$ from 2nd equation of motion:
$$W=ma(ut+\frac{1}{2}at^2)$$
$$W=ma(ut)+\frac{1}{2}m(a^2t^2)$$
From 1st equat... | {
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Is a wave function a ket? I just started with Dirac notation, and I am a bit clueless to say the least. I can see Schrödinger's equation is given in terms of kets. Would I be correct to assume if I were given a wavefunction, say $\Psi(x)=A\exp(-ikx)$, would I be able to just use the notation $\lvert \Psi\rangle =A\exp(... | Yeah, as a beginner you can basically do this with no problems, but a physicist would tend not to do that because it's mixing notations. $|Y\rangle$ is a state, i.e. a vector. $Y(x)$ is a function that eats a value of $x$ and spits out the value of the wave function $Y(x)$. So whereas $Y(x)$ "depends" on an input, $|Y\... | {
"language": "en",
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Oscillation coil: where is the electric field? Let assume a simple RF coil fed with an alternating current at RF frequencies, say 100MHz.
I believe that no one doubts that the coil will radiate RF energy in the form of radio waves.
A radio wave is classically composed of an electric vector and a magnetic vector orthogo... | Since the magnetic field is varying with time, by Maxwell-Faraday's induction law there is a rotation of the electric field. One can then distinguish the propagating and the non-propagating part but that is outside the scope of the question.
| {
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Definition of Ensemble Im studying statistical mechanics and came across the ensembles.
*
*Now system of large number of particles can be defined by an ensemble which contains elements (infinite of them) where each element is the mental copy of system at a particular time and time average of any quantity of system ca... | They are just the same. This ensemble definition is a bit archaic and was useful back in the days in order to visualize the Ergodic principle ($\lim_{T\rightarrow \infty}\frac{1}{T}\int^{t=T}_{t=0} dt \ ...=\frac{1}{Z}\int_{\Gamma} \ d\alpha \ ...$ ).
| {
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Doppler Effect and Radar Sensors Radar sensors make use of the Doppler effect to measure the radial velocity of an object.
The radar's Tx antennas emit an electromagnetic wave which travels to the moving objects, is reflected and the radar's Rx antennas detect the incident wave.
Due to the movement of the measured obje... | In the case of radar velocity measurement by doppler shifts, note that the velocity of the object being measured is of order ~10 to 1000 m/s. This is so slow compared to the speed of the radar pulse itself that relativistic corrections do not have to be made in order to get a velocity measurement accurate enough to jus... | {
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Circuit (Pseudo-Homeworks) I'm studying Kirchoff's Laws and how to find $\Delta V$ between two points in a circuit (CC). I already know how to use it but I look on internet for more exercise to learn more and find this circuit
I'm a little confuse about it, because there is a wire in between that don't understand exac... | First, notice that the supplied current of the current source, must be equal to the current that flows through the 'back' of it, and therefore must be equal to the current that flows 'upward' from node $C$:
Then, applying KCL on node $C$:
Yields:
$$I_1 = I_1 +I_2$$
$$I_2=0A$$
| {
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Is $\phi^4$ theory an attractive or repulsive force?
*
*Does the well known $\phi^4$ theory, with Lagrangian $$\mathcal{L}=\frac{1}{2}
\partial_\mu \phi\partial^\mu \phi-\frac{m^2}{2}\phi^2 -\frac{\lambda}{4!}\phi^4,$$
yield an attractive or repulsive force for the $\phi$ particles? (Here we use the $(+,-,-,-)$ sign ... | If $\lambda>0$, the force is repulsive. If it $\lambda<0$, the force is attractive, but the system is unstable to vacuum decay.
To be more specific $\lambda \phi^4$ interaction is the relativistic version of the Schrodinger delta-function potential.
| {
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Will a changing $E$ field induce a current in a loop similar to a changing $B$ field? An induced current in a wire loop that is caused by a changing B field is a common EM question. However, I couldn't find examples online where the B field was substituted for a changing E Field.
The following question was given on a t... | $\nabla\times E = -\frac{\partial B}{\partial t}$ says there is "circulation" of E around the loop. E will push charges along the loop.
This follows from Stoke's Theorem: $\int_{loop} E \cdot \space dl= \int_{surface} \nabla \times E \space\text{ds}$
$\nabla \times B = \frac{\partial E}{\partial t}$ says there is "circ... | {
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How do you make more precise instruments while only using less precise instruments? I'm not sure where this question should go, but I think this site is as good as any.
When humankind started out, all we had was sticks and stones. Today we have electron microscopes, gigapixel cameras and atomic clocks. These instrument... | Greeks used a compass and straight edge. A compass is as precise as an A-frame anchored with a needle, and a straight edge is some wood that is sanded flat by a carpenter. One wooden straight edge can be used to create a straighter straight edge because the sanding grit is small and it evens out the bumps.
Distances we... | {
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Determining the partial derivative of a metric tensor Im new to the Tensor Calculus and General Theory of Relativity, and I have one question. I want to determine the Christoffel symbols in FRW metric.
This is the general equation of Christoffel symbols:
$$\Gamma^{\mu}_{\hphantom{\mu}\alpha\beta}=\frac{1}{2}g^{\mu\nu}\... | FRW metric is written as
$$-c^2 d\tau ^2 = -c^2 dt^2 +a(t)^2 d\textstyle{\sum}^2$$
where for simplicity take $k=0$ in Cartesian coordinates which refers to zero curvature then $d\textstyle{\sum}^2 = dx^2 +dy^2 + dz^2$. Where the metric tensor $g_{\mu \nu}$ is
$$g_{\mu\nu}=\begin{bmatrix}
-1 & 0 & 0 & 0\\
0 & a^2(t) & 0... | {
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What does it mean that a free particle has no definite energy in quantum mechanics? In the quantum mechanics case of the infinite square well, the general solution to the Schrodinger equation is a linear combination of solutions with definite energy states. When you measure the particle, it will take one of these energ... | Short answer: physically no plane waves really exist. Mathematically, we use them all the time. If we are desperate for normalization, we normalize them by particle flux, rather than the total probability.
Let me first deviate into discussing photons, snce this situation has been extensively coveregd on this site: in o... | {
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Matrix Representation of Lorentz Group Generators Let $\Lambda^{\alpha}{}_{\beta}$ denote a generic Lorentz transformation.
Then, an infinitesimal transformation can be written like
$$\Lambda^{\mu}{}_{\nu} = \delta^{\mu}{}_{\nu} + \omega^{\mu}{}_{\nu} $$
where
$$\omega^{ij} = \epsilon^{ijk}\theta_k$$
$$\omega^{i0} = -... | Thanks to @Charlie and @Cosmas Zachos I was able to find the correct answer.
It simply suffices to develop the sum
$$\frac{\omega^{\alpha \beta}}{2}\left(J_{\alpha \beta} \right)^{\mu}{}_{\nu} = -\delta_1 \left(J_{01} \right)^{\mu}{}_{\nu} - \delta_2 \left(J_{02} \right)^{\mu}{}_{\nu} - \delta_3 \left(J_{03} \right)^... | {
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Free body force diagram with 3 pulleys I am certain this has already been asked and answered, but I have not been able to find it. It has been about 25 years since I last sat in a physics class, and some parts are a bit rusty! I am planning to hang some heavy gear in my garage and am trying to determine the weight at... | Let T be the tension in the rope. The force acting on weight is 2T upwards. So T is half the weight of the box. Well I can't say T is 100lb as pound is unit of mass not force but anyways I hope you understand what I am trying to convey.
The hinge force at A is T, at B is 2T and at C is T; hence hinge forces are calcula... | {
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What does "commuting with the Hamiltonian" mean? In quantum mechanics an observable or an attribute to a particle (like spin) is conserved if and only if it commutes with the Hamiltonian. What does this mean? What observables do not commute with the Hamiltonian?
| Two operators $A$ and $B$ commute if (and only if) their commutator $[A,B]$ vanishes
\begin{equation}
[A,B] \equiv AB - BA = 0 \implies A,B\ {\rm commute}
\end{equation}
Consider a Hamiltonian operator for a single particle in 1 dimension
\begin{equation}
H = \frac{p^2}{2m} + V(x)
\end{equation}
where $x$ the position ... | {
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Why is mist gray but water clear? I was walking outside one cold afternoon with my mask on and my glasses began fogging up. The mist was initially gray.
I kept walking without cleaning my glasses and eventually enough mist collected that that it transformed into clear water droplets.
This got me thinking: why is mist g... | Regarding water droplets collecting on the surfaces of your glasses:
Those water droplets backscatter the incoming light in random directions, including ones away from your eyes. This means that any glass lens surface populated with water droplets will appear less bright than it would without the droplets, and the rand... | {
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Why does water cast a shadow even though it is considered 'transparent'? If you pour water from a container, the flowing water stream seems to cast a shadow. I am not sure you can call it a shadow, but it definitely is not letting all light through it. How is this possible and what uses can it have?
| This is just to add an illustration to noah's and Ralf Kleberhoff's answers which correctly point out that refraction is the main reason.
Note that although most of the light rays do make it through the water drop, most of them do not continue on the path with the rest of the light bundle, but end up somewhere else. A... | {
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What do the different "levels" in a Kac-Moody algebra tell us physically? I'm reading a book on conformal field theory and I am on a section which derives the current algebra:
$$[j_m^i,j_n^j]=\sum_l f^{ijl}j_{m+n}^l+k \;m \delta^{ij}\delta_{m,-n} $$
We can prove this result assuming we are working with chiral fields of... | The scale of the algebra is always chosen so that the length of the longest root in the underlying finite Lie algebra is 2. This choice simplifies a number of formulae, and in particular ensures that the level $k$ is an integer.
| {
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Can the Auger effect cause a second electron to be just excited instead of ionised and emitted from the atom? From what I understand, the Auger effect is usually defined as when an electron deexcites but instead of releasing its change in binding energy as a photon, it transfers it as kinetic energy to another electron... |
My question is why is this process defined with the second electron being emitted from the atom instead of just excited to a higher energy state sometimes.
The atom is a unit tied up quantum mechanically . To observe transformations of an atom, there must be an interaction that can be measured. An emitted photon can ... | {
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Interdependence of $P,V$ and $T$ Can we prove that the thermodynamic state of a system is completely determined by any two out of the three factors $P,V$ and $T$ ? (Without using statistical mechanics. Only using Thermodynamics)
NB: I have not learnt the axiomatic formulation of Thermodynamics.
| There exist systems with other degrees of freedom other than pressure temperature and volume, for example in magnetic systems you can also talk about the degree of magnetization and the applied field. So the completely general answer is no, because it is not always true.
Once you have limited yourself to systems that o... | {
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Quantum Field theory, solving delta / green function I have read an equation as follow
$$[-(k^2-m^2)g^{\mu\nu}+k^{\mu}k^{\nu}]D_{\nu\lambda}(k)=\delta^{\mu}_{\lambda}$$
The solution is given as:
$$D_{\nu\lambda}(k)=\large{\frac{-g_{\nu\lambda}+k_{\nu}k_{\lambda}/m^2}{k^2-m^2}}$$
I can only verify the answer but do not ... | We can make use of the projection operators
\begin{align}
& P^{T}_{\mu \nu} = g_{\mu \nu} - \frac{k_{\mu}k_{\nu}}{k^{2}}, \\
& P^{L}_{\mu \nu} = \frac{k_{\mu}k_{\nu}}{k^{2}},
\end{align}
where $P^{T}_{\mu \nu}$ and $P^{L}_{\mu \nu}$ are the transverse and longitudinal projectors. $P^{T}_{\mu \nu}$ and $P^{L}_{\mu \nu}$... | {
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Why do bullets shoot through water but not through sand? There are a few questions only on this site about this but none of them answer my question.
Can cannonballs go through water?
Why does a bullet bounce off water?
I find it hard to understand why bullets shoot through water at longer distances but stop in sand alm... | The shape
The mechanical displacement of water in water requires less energy than the displacement of sand in sand. This is because the movement of a water molecule through the conglomeration of water molecules requires less energy for the displacement and rotation of each molecule than the displacement of grains of s... | {
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Gauss' law from Hamiltonian density of electromagnetic field I am going through David Tong's QFT course, for which lecture notes and exercises are available online at http://www.damtp.cam.ac.uk/user/tong/qft.html.
In Question 1.8 we have the Lagrangian (density)
$$L = -\frac{1}{4} F^{\mu \nu} F_{\mu \nu} + \frac{1}{2} ... | Main points:
*
*A total spacetime divergence in the Lagrangian (or Hamiltonian) does not change the EOMs, cf. e.g. this Phys.SE post.
*If we know that the fields vanishes on the boundary, e.g. by imposing pertinent boundary conditions, we can use the divergence theorem to argue that a divergence term cannot contrib... | {
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Why is the electric flux through a surface between a dipole zero
In surface $S_3$, as per the Gauss Law, the net electric flux is zero as there is no charge enclosed in the surface. Also the other reason the book mentions for this that since the amount of flux entering is equal to the amount leaving, the net flux is z... |
Even if the Gaussian surface near a charge that itself doesn't contain
the charge has the electric field lines passing through it. So
shouldn't it have some flux rather than it being zero?
Yes there is electric flux. But there's a difference between electric flux and net electric flux. Gauss' law says that the net el... | {
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Meaning of 1+1 dimensions I came across the notion of 1+1 dimensions in a condensed matter context and, in particular, while studying bosonization, which relates to 1D quantum systems. Indeed, the Wikipedia article about it makes reference to 1+1 dimensions in the very first sentence, though the hyperlink redirects to ... | To simplify a problem and perhaps gain some insight in how to solve it in its full form, physicists sometimes create a "toy universe" which is missing one or two spatial dimensions, and recast their problem in that toy universe. Sometimes the problem can be solved in that universe and sometimes that solution is of use.... | {
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Why is the radial velocity considered zero? I had recently come across a question which is stated as below:
A disc placed on a large horizontal floor is connected from a vertical cylinder of radius $r$ fixed on the floor with the help of a light inextensible cord of length $l$ as shown in the figure. Coefficient of fr... | Because the cord is inextensible, its tension does no work on the disc (in other words, the disc always moves perpendicular to the cord). Therefore we can model the situation as a linear deceleration under friction. It would be the same if the cylinder's radius were zero and the motion were circular.
| {
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Standing waves - why do wavelengths fit perfectly? When reading about standing waves it is always said that only certain wavelengths are "allowed". I understand that these wavelengths are a requirement for there to be a standing wave due to the boundary conditions, but what does "allowed" mean in this context?
When cre... | The reason they are "standing" is precisely because the wavelength fits perfectly. If the wavelength didn't fit, it wouldn't be standing but moving along.
So your thinking is good enough, just the wrong way round.
Why a given piece of string (or, for that matter, a bridge) has a tendency to create standing waves is ano... | {
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How does Inertial force arise? Consider the following scenario$-$
A ball sits on the floor of a bus, which was originally at rest w.r.t the ground. Suddenly it accelerates forward, and we observe the ball moving backwards. Well, originally the ball does not move, it is only the bus that moves forward. Why doesn't the b... | Because the rolling resistance of the ball is less than the force of acceleration transferred to the ball from the floor of the bus
| {
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Kepler's laws for circular orbits Kepler's first law states that planets revolve around the sun in an ellipse with the sun at one focus of this ellipse. (a special case would be a circular orbit with the sun at the center).
The second law states that the areal velocity is a constant. Thus we can write ($dA = c dt$). If... |
My question is, should we swap out "semi-major axis" and replace it with "radius"...
We can always do that. As you noted, circle is a special case of ellipse.
The second law states that the areal velocity is a constant. Thus we can write (dA=cdt).
Correct. Here c is areal speed.
If we integrate over one complete c... | {
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Application of the principle of conservation of angular momentum and the principle of conservation of energy I came across this question and was left confused.
A satellite is launched in a direction parallel to the surface of the earth with a velocity of $36900 \; \mathrm{\frac{km}{hr}}$ from an altitude of $500 \; \ma... | Starting with the given velocity and altitude, you can use conservation of energy to find the velocity at the point of nearest approach. (This will be parallel to the surface.) Then you can use conservation of angular momentum to find the (error) angle (measured from the parallel to the surface) at the launch point.
| {
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Disintegration of the deuteron Considering the scattering of gamma rays on a deuteron, which leads to its break up acording to:
$$ \gamma+ d \longrightarrow p +n $$
we can use the conservation of energy and momentum in order to determine the minimum photon energy in order to make this reaction possible, which happens ... | The mass of a system is constant in reactions and is a Lorentz invariant quantity, where the mass is defined to be $$E^2-p^2c^2.$$
$E$ is the total energy of the system, and $p$ is the magnitude of the net momentum of the system. For your initial system, in the lab, you have
$$E=p_{\gamma }c+m_d c^2\text{ and } p=p_{\g... | {
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Interesting inertia problem Consider the following.
A car is accelerating with acceleration $a$. A string is attached to the roof of the car and to the bottom of the string, an object of mass $m$ is attached. Given $\theta$, the angle between the vertical and the string (which is not $90^\circ$ due to inertia of the o... | So if we consider the x-component of the tension $Tsin(\theta)$ and given the car moves with acceleration $a$ then
$$ma - Tsin(\theta) =0$$
and so
$$a=\frac{Tsin(\theta )}{m}$$
for a mass $m$. Remember that the mass experiences an inertial force and so $a$ is the acceleration of the car and mass for an observer inside ... | {
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Is there a material which opposes electric fields? I was wondering if a material existed which opposed applied electric fields, analogously to a diamagnetic material, which opposes magnetic fields. So the flow of charge would be against the field, rather than toward the field. In other words, instead of electrons flo... | If charge flows opposite the direction of the applied field, you have a material with negative resistivity. No such material exists.
However, it is possible to build a circuit that, within a limited range of applied voltage, delivers an opposing current (i.e. a negative resistance circuit). Such a circuit must necessar... | {
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Does the magnetic field produced by a current carrying wire, exert a magnetic force on the wire itself? I have to calculate the pressure on a current carrying wire. Since there is a pressure on the wire, there must be a force on it, which is a magnetic force. Does the magnetic field produced by the wire, exert a magnet... |
I have to calculate the pressure on a current carrying wire. Since there is a pressure on the wire
This is not relevant to the main question, but what do you mean by “pressure”? Pressure normally means the ratio of force to area. What area would you be using here?
Does the magnetic field produced by the wire, exert ... | {
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Convert newton formula to acceleration formulas I am trying to understand a problem from the book Solving Problems in Scientific Computing Using Maple and MATLAB. The problem is about trajectories of tennis balls.
The author gives two formulas $D_L(v)$ for drag force and $M_L(v)$ for magnus force:
$$D_L(v)=C_D\frac{1}{... | From $v=\sqrt{\dot{x}^2+\dot{z}^2}$ we get that $\dot{y}=0\Rightarrow y=constant$. With that on mind, the cross product is
$$\vec{\omega}\times\vec{v}=\omega_y \dot{z}\hat{i}+(\omega_z \dot{x}-\omega_x \dot{z})\hat{j}-\omega_y\dot{x}\hat{k},$$
where $\hat{i} , \hat{j} , \hat{k}$ are unitary vectors in $x , y , z$ direc... | {
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Approximating the angle between the trajectory I started to learn physics this semester and I found the following task:
A contestant is participating in a half-maraton tournament(straight line length $L =21095$ meter) running in a zig-zag manner (constantly surpassing other contestants), holding a stable angle $\alpha$... | The angle is small but it is not zero. The problem relies on the fact that the angle is small when it asks to find an approximate value without using a calculator.
You find $cos(\alpha)=\frac{L}{L+\Delta L} \approx 0.977 $.
But, for small angles, you can approximate $cos(\alpha) \approx 1-\frac{\alpha^2}{2} $
So, by co... | {
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What keeps dry flat pieces of sand crumbles together (even if you microwave them)? Originally these flat pieces get created as the top layer on the beach where water pulls back (low tide), leaving the sand to dry on the Sun. The top layer dries and creates a separate layer, making these flat solid pieces.
Sometimes th... | The water which was mixed with the sand contained not only salt but lots of microorganisms called plankton. Taken together, they make a simple sort of glue that tends to absorb moisture and retain it even when heated, and keep the grains stuck together (weakly).
This is testable, by taking a sand sample and scrubbing i... | {
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Propagator in string theory and boundary conditions I would like to understand how to compute propagators for open and closed strings.
My references are Tong's lecture notes, https://arxiv.org/abs/0908.0333 and Blumenhagen, Lust, Theisen book "Basic Concepts of String Theory".
Tong uses a path integral approach: he st... | There is no an algorithmic way to solve the Poisson equation in two dimensions (except in the case of a disk). The problem you want to address must be done in a case by case manner.
The whole chapter six in Polchinski string theory texbook (page 170,Vol. 1) is dedicated to find scalar Green functions for wolrdsheets of... | {
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Why voltage is same across parallel circuit if it is work done per unit charge? Suppose we have the following circuit:
If voltage is work done per unit charge, why voltage is same across each resistor if the charge has to do more work in resistor R2 than in resistor R1?
| I will explain this in terms of the water analogy. Consider a river bed that follows the same setup as the circuit. The water level is the voltage. Higher voltage = higher water level = more energy per unit of water/charge. The resistors are dams that restrict the water flow. The battery is like a pump that works to ke... | {
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Do resistance have to heat? Not a physicist.
Is there a way to build resistances that do not heat when opposing current?
More generally, is it necessary to waste energy to resistance?
If I cut a wire, there will be an almost infinite resistance (unless the tension is large enough to break the air), but no energy cost.
... | This is possible for alternating current (AC). For AC voltage sources the definition of resistance is extended to what is called impedance. If you supply a voltage of the form
$$V(t)=V_0\sin(2\pi f t)$$
then for simple elements (resistor, inductor or capacitor) the current will be of the form
$$I(t)=I_0\sin(2\pi ft+\ph... | {
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Could the observable universe be bigger than the universe? First of all, I'm a layman to cosmology. So please excuse the possibly oversimplified picture I have in mind.
I was wondering how we could know that the observable universe is only a fraction of the overall universe. If we imagine the universe like the surface ... | We could discover that the actual universe is larger than the observable universe when there is a non-zero constant curvature everwhere. Then, this would be like standing upon the earth and seeing as far as the horizon. In the picture drawn above what we can observe would be limited by a kind of cosmological horizon.
| {
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Do humans use the doppler effect to localize sources of sound? Consider a source of sound such as a person speaking or a party of people which makes a continual drone sound of the the same frequency. If a human shakes their head side-to-side with sufficient angular speed, they are in effect obtaining different frequenc... | Humans DO use dopler effect to estimate a sound source position, they just dont use it exactly the way you imagine.
The simplest example is the distance to a passing by object (a car, an airplane, a mosqito or even a talking human). A near flyby makes a rapidly lowering tone. An object passing away from you will change... | {
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Larger aperture for objective lens in a compound microscope A telescope has a large aperture to collect more light and hence improve the visibility (brightness) of the image. It also helps in improving the resolution as $\theta_{\text{min}}=\frac{1.22\lambda}{a}$ where $a$ is the radius of the aperture (assuming it to ... | Even under laboratory conditions, the light intensity collected by a microscope can be low, so a larger aperture is in general desirable. However, as you correctly point out, what matters is not the physical aperture (i.e. the radius of the objective lens) but the angle $\beta$. The quantity
$$ NA = n\sin \beta$$
is ca... | {
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Energy of a $n$-particle system in special relativity Consider an inertial frame $S'$. With respect to this frame of reference consider a system of $n$ particles. The $k$-th particle has rest mass $m_{0,k}$ and it moves with speed $u_k$. Can we say that the mass of the system is $$\mathbf{m=\sum_{k=1}^{n}m_{0,k}\gamma_... | $\sum\gamma_k m_k c^2$ is called the "total relativistic energy" $E_{rel,tot}$.
Although some have called $\frac{1}{c^2}E_{rel,tot}=\sum\gamma_k m_k$ the "relativistic mass",
because of misconceptions by novices, its use is discouraged.
(Further comments at Invariant rest mass vs Proper velocity )
The "invariant mass ... | {
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Is there a way to prove that different angular momentum components anticommute without using a specific matrix representation? I know spin-1/2 Pauli matrices satisfy the anticommutation relationship $\{\sigma_i, \sigma_j\}=2\delta_{ij} \mathbb{I}$. I wonder how this can be proved without writing down the matrix represe... | The pauli matrices are simply a matrix basis which (up to a factor of i) represents a basis for Cartesian bivectors. In this sense they are isomorphic to the quaternions, meaning they form a representation of the Clifford algebra of $\mathbb{R}^{3}$ .
As another answerer has pointed out, the dimension of one's represen... | {
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Newtonian physics and equivalence principle: a doubt on acceleration and gravity First of all, the famous Einstein's elevator experiment is quite clear in my head, both of versions.
But now, consider the following:
Suppose then you wake up inside a car that is traveling in perfect straight path in a autoban (but you d... | If the acceleration of the frame is due to rotation, there is, in addition to the centrifugal force which locally looks like gravity, also Coriolis force, which acts perpendicular to the motion of the test particle, and makes it possible to distinguish the frame from one which doesn’t rotate.
| {
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Christoffel Symbols of a parallel shift Regarding the transformation $p=3x+5y;q=x+y$ (where $x,y$ are cartesian coordinates) I want to look at the christoffel symbols for the covariant coordinate system for a parallel shift (using ruler and triangle).
Therefore I calculated the covariant basis vectors and then looked a... | Location $(x,y,z)$ isn't a vector in most spaces. $(dx/dt,dy/dt,dz/dt)$ is a contravariant vector.
$dp = (dp/dx) dx + (dp/dy) dy$
I believe the basis vectors transform "opposite" to that.
$e_p = (dx/dp) e_x + (dy/dp) e_y =(dx/dp) \hat{x} + (dy/dp) \hat{y}$
As they said, these are all constants so the derivatives are go... | {
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How do speakers vibrate for a complex music? I understand how a speaker could produce simple sound and constant frequency. How does it produce more complex sounds like music? How can you calculate what frequency to oscillate at when there are multiple instruments and voices in a song? There must be a limit to the compl... | Unlike an acoustic instrument like a guitar string or a triangle that emits mostly a single frequency (and a set of its harmonics) defined by its physical characteristics (shape, tension etc.), a speaker is driven by electric signal, and its motion is controlled by this signal, rather than by speaker's shape.
At each p... | {
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If $I \propto V$, then why is $R = V/I$ and not $I/V$? I know that the current flowing through a conductor is directly proportional to the potential difference across its ends (by Ohm's Law).
Hence,
*
*I ∝ V
*V ∝ I
*R = V/I, where R is a constant (Resistance)
But why can't it be derived this way?
*
*I ∝ V
*I = ... | Chris' answer is perfectly right, but I think it's missing a key point: try it out!
Suppose you're the one working with Ohm for his formula (but you have today's equipment for simplicity's sake). You do your reasoning, and you arrive at the two conclusions you pointed out. What to do? You try them out. Build a circuit,... | {
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Is the coefficient of restitution of a bouncing ball constant with respect to height? Is the rebound rate (or ratio) of a bouncing ball actually constant?
Edit: To clarify, I am considering a given ball and whether it has a different coefficient of restitution depending on the height dropped from (or velocity at impact... | No, it's not constant. It decreases with the drop height.
https://www.researchgate.net/profile/Khairul-Ismail-6/publication/260991737_Coefficient_of_restitution_of_sports_balls_A_normal_drop_test/links/57a021d808aec29aed214c06/Coefficient-of-restitution-of-sports-balls-A-normal-drop-test.pdf?origin=publication_detail
... | {
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Is radial time a solution of Einstein's equations? Imagine a (flat) 4D space where we measure time outwards in a radial direction from the origin.
So that 3D space at a given time would consist of a spherical shell. (As such this would be a closed Universe.)
In a far distant time the spherical shells at any given posit... | I believe that this is equivalent to a Universe which is closed spatially but not in the time dimension since the point at $r=0$ will transform to $t=-\infty$, the transform will be $r=e^t$. The singularity at $r=0$ merely represents the infinite past. We have merely 'squished' the time dimension up to make it appear i... | {
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If electrons can be created and destroyed, then why can't charges be created or destroyed? I read on Wikipedia that electrons can be created through beta decay of radioactive isotopes and in high-energy collisions, for instance when cosmic rays enter the atmosphere. Also, that they can be destroyed using pair annihilat... | Charges can be created and destroyed. Total charge cannot.
Whenever you create an electron, charge $-1$, you must also create a positron, charge $+1$. That gives total charge $0$.
Whenever you create a proton, charge $+1$, you have to create an anti-proton, charge $-1$. That gives total charge $0$.
As far as we're awar... | {
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Normalization factor for symmetric state As per Wikipedia,
The above discussion generalizes readily to the case of $N$ particles. Suppose there are $N$ particles with quantum numbers $n_1, n_2, ..., n_N$. If the particles are bosons, they occupy a totally symmetric state, which is symmetric under the exchange of any tw... | I am also looking for an answer to this, but I believe it is the latter. I think about it like this:
There are definitely $\frac{N!}{\prod_{n}(m_n!)}$ distinct permutations that change $|n_1 \rangle \cdots |n_N \rangle$ which are orthonormal and for each of those distinct permutations, there are $\prod_{n}(m_n!)$ permu... | {
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Is no acceleration a cause or consequence of no net force? If a body is moving with constant velocity, or is at rest, then the net force on it must be $0$. If the net force on a body is $0$, then it must be moving with constant velocity or must be at rest.
Is $0$ net force a consequence of being at rest or moving with ... |
Is 0 net force a consequence of being at rest or moving with constant velocity
Yes. It's not a physical consequence, though. Constant velocity (which is also the case for a body at rest) implies zero force (which is connected to acceleration by a factor m), but it doesn't cause it. Force on a body is caused by some a... | {
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How much force applied to canal wall from that cargo ship given 220,000 tons and 12.8 knots? In case you've been hiding under a rock, or are reading this in the future: "that cargo ship" is a huge story right now (3/26/2021). A brief summary: well basically a few days ago one of the world's largest cargo ships somehow ... | For the volume of water displaced... Archimedes principle says that "a floating body displaces it's own weight of the liquid in which it floats".
So the cargo ship displaces 220,000 tons or 200,000,000 kg of water. For a density of water of 1000kg per cubic metre that's 200,000 cubic metres of water displaced.
Appare... | {
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Why phonons are Goldstone modes? I read this in the lecture notes by David Tong:
"Gapless excitations often dominate the low-temperature behaviour of a system, where they are the only excitations that are not Boltzmann suppressed. In many systems, these gapless modes arise from the breaking of some symmetry. A particu... | This question is answered in detail in the paper Phonons as Goldstone Bosons. The question What is the difference between a photon and a phonon? is also closely related. Here, I'll just give some basic intuition.
Consider two solids that are identical except that one of them is shifted slightly in space compared to the... | {
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Is the work done on a system always equal to the negative of that work done by the surroundings? I'm trying to conceptualize some aspects of thermodynamics for myself. Many textbooks often define the work done as the external force acting on the system, resulting in the formula that looks like:
$$W = -\int P_{ext}\, \m... | Your interpretation in terms of Newton's 3rd law is absolutely correct. In an irreversible process, the gas does not pass through a sequence of thermal equilibrium states like it does for a reversible process. The ideal gas law is only applicable to thermodynamic equilibrium states, and gives incorrect values for oth... | {
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Question on Faraday's Induction Experiment We all know that moving a magnet through a loop of wire induces a current, like in this youtube experiment here. Similarly, we know that moving one solenoid (with a battery hooked up) through a larger solenoid (with no battery) will induce a current in the larger solenoid, jus... | Faradays law is that the induced voltage in a coil is
$\epsilon = -N\frac{\Delta \phi}{\Delta t}$
Where $N$ is the number of turns in the coil, $\Delta \phi$ is the change of magnetic flux through it and $\Delta t$ is the change in time.
For infinite coils the $\Delta \phi$ part would be zero, so you are right that the... | {
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What is astrophysical fluid? I am reading a book about astrophysical fluid dynamics, some basic fluid equations are mentioned, but is there any difference between astrophysics case and general case? Is there any definition about astrophysical fluid?
| The fluid dynamics equations are the same (continuity, conservation of momentum, and energy). What makes peculiar the astrophysical fluids are the huge variety of dynamical conditions (from almost stationary conditions to the need of relativistic fluid dynamics), the possible presence of important magnetic fields (magn... | {
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How to know if the error is in a law or in uncertainty of the measurement? I read these words in a (great) answer to this question:
There are errors that come from measuring the quantities and errors that come from the inaccuracy of the laws themselves
But how do we know that the errors are in the measuring or in the... | Every measured value has errors. This is the principle stand of the Guide to the Expression of Uncertainty in Measurements. The magnitudes of the errors can be determined (quantified). They consist of offset (calibration) errors, measurement (device) errors, and random errors. A term with less negative context is uncer... | {
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Why should $\lim_{V\to\infty} \frac{1}{V} \ln Q(z, V, T)$ have a finite limit? In the book Intro. Statistical Physics by K.Huang, on page 174, it is given that
In the thermodynamic limit $V \rightarrow \infty,$ we expect that:
$$
\frac{1}{V} \ln Q(z, V, T) \underset{V \rightarrow \infty}{\longrightarrow} \text { Finit... | In grand canonical ensemble, the partition function $Q$ and the grand potential has relation
$$
\Phi = -K T \ln Q.
$$
The grand potential (also known as Landau free energy) in thermodynamics can be derived using Legendre transformation from internal energy $U(N,V, S)$, where $S$ is the entropy:
The Helmholtz free energ... | {
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Is Rutherford scattering formula inconsistent with reality? On our way to deriving the famous Rutherford scattering formula, we get a formula for the fraction ($f$) of incident alpha particles scattered by $\theta$ or more and this formula has the form
$$f=\pi n t\left(\frac{Ze^2}{4\pi \epsilon_0 K_E}\right)^2\cot^2(\t... | Let's take a step back.
We begin with the impact parameter $b(\theta)$
and the total cross section $\sigma(\theta)$
for an alpha particle being scattered by an angle of $\theta$ or more
by a single atomic nucleus.
$$b(\theta) = \frac{Ze^2}{4\pi \epsilon_0 K_E}\cot(\theta/2)$$
$$\sigma(\theta) = \pi b^2(\theta) = \pi\le... | {
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Series combination of springs When a spring mass system is connected vertically with two massless springs in series whose spring constants are $k_1$ and $k_2$ to a block of mass $m$ we know that equal forces act on both the springs. Let that force during oscillations be $F$.
When we calculate effective spring constant ... | The springs are massless, so the tension at any point in either spring is the same value, $F$. For the two individual springs, with extensions $x_1$ and $x_2$, we have#
$F = k_1x_1=k_2x_2$
and when considered as a single spring they have an effective spring constant $k$ such that
$F = k(x_1+x_2)
\\\displaystyle \Righta... | {
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Angular momentum commutation relations The operator $L^2$ commutes with each of the operators $L_x$, $L_y$ and $L_z$, yet $L_x$, $L_y$ and $L_z$ do not commute with each other.
From linear algebra, we know that if two hermitian operators commute, they admit complete sets of common/simultaneous eigenfunctions. The way I... | Since there's probably a mathematical development of this property in your textbook/notes, I'm guessing you want an intuitive approach on it. If this doesn't help you I'll develop the mathematics:
Imagine $L^2$ has having three properties, let's call them colors. $L^2$ is green, red, and blue. $L_x , L_y , L_z$ are res... | {
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What is light, a wave or a particle or A wave-particle? What is light?
And how do we know that light is an electromagnetic wave?
I asked my teacher and he said that when you place a compass in light's path, the needle of the compass rotates. Which I think is not a valid answer and thats not what actually happens when w... | Electromagnetic waves are solutions of the wave equation that we derive from Maxwell's equations. When we use Maxwell's equation to compute predictions of what we expect to see when making experimental observations with light, we see that the predictions agree with the observation. Therefore, we conclude that light is ... | {
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Translation of Fock's original paper? Has the original paper on the Fock space [1] ever been translated to English? I'm not looking for things like Cook's paper [2], what I want is a faithful traslation from German (if it exists).
[1] V. Fock, Konfigurationsraum und zweite Quantelung, Z. Phys. 75, 622-647 (1932). https... | Yes, this famous article by V.A. Fock has been translated from German to English, and quite recently. It is published in English in:
Faddeev, L.D. et al. - „V.A. Fock. Selected works: quantum mechanics
and quantum field theory”, CRC (Chapman & Hall), 2004, page 191.
Library of Congress Cataloging-in-Publication Data
... | {
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Proof of $\mathrm{tr}(\gamma^{5}\gamma^\mu\gamma^\nu)=0$ Using $\gamma^{5}\gamma^\mu=-\gamma^\mu\gamma^{5}$ and $\mathrm{tr}(AB)=\mathrm{tr}(BA)$ I obtain
\begin{equation}\tag{1}
T_{\mu\nu}:=\mathrm{tr}(\gamma^{5}\gamma^\mu\gamma^\nu)=-\mathrm{tr}(\gamma^\mu\gamma^{5}\gamma^\nu)=-\mathrm{tr}(\gamma^{5}\gamma^\nu\gamma^... | Following the given hint, we note that $\gamma^{\alpha}\,\gamma^{\alpha}\,\gamma^{\alpha}\,\gamma^{\alpha} = \mathbb{I}$, for $\alpha=0,1,2,3$.
Thus, we can write
$$\mathrm{Tr}(\gamma^{\mu}\,\gamma^{\nu}\,\gamma^5) =\mathrm{Tr} (\gamma^{\alpha}\,\gamma^{\alpha}\,\gamma^{\mu}\,\gamma^{\nu}\,\gamma^5\,\gamma^{\alpha}\,\g... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/627420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What is the exponential (or geometric) rule (or law) for uranium enrichment? Uranium ore starts at about .72% U-235...
At ~20% U-235, it is considered to be about '90% of the way' to weapons-grade uranium, which is about ~90% U-235...
Because uranium enrichment in centrifuges follows a geometric (or exponential) law...... | You are basically trying to sort the atoms, reducing entropy. The minimum work per mole of extracting an element that has mole fraction $x$ is
$$W_{extract}=-TR\ln(x).$$
So, the cost difference as you go from $x$ to $2x$ is (assuming $x<1/2$) $$W_{extract}(2x)-W_{extract}(x)=-TR\ln(2).$$ Which is constant, as the expon... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/627642",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Why is the identity not considered when expanding a $2 \times 2$ matrix in the Pauli basis? I am aware of the expansion of a two dimensional matrix $M$ in Pauli basis given by
$$ M = \sum_{\mu=0,1,2,3} c_\mu \sigma_\mu$$
with $\sigma_0 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$, the Identity matrix and $\sigma_{... | The set of $2\times 2$ complex matrices can be understood as a real vector space of dimension $2\times 2^2$, or equivalently, as a complex vector space of dimension $2^2$.
You can easily verify that a basis for the vector space is given by the Pauli matrices and the identity matrix. More precisely, we can write
$$\math... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/627719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
According to general relativity planets and Sun bend the spacetime (explaining gravity), but does this hold true for smaller objects? According to general relativity planets and the sun bend spacetime, and that is the explanation of gravity. However, does this hold true for smaller objects, like toys, pens, etc.? Do t... | It seems to be the case that all matter and all forms energy will bend spacetime. Whether or not there is some kind of matter we're not aware of that doesn't is another question. But, everything you'd encounter in your day to day life will bend spacetime, and produce gravitational waves.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/628115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 9,
"answer_id": 6
} |
Muon $g-2$ experiment: is there any theory to explain the results? The nature of the experiment has been discussed here, but my main question is this: is there any theory that has predicted the results of this experiment or are we completely clueless about what's happening? In other words, have we come up with a new hy... | A Nature paper was published on the same day, which seems to have attracted a lot less press. This presents a recalculation of the muon $g-2$ value, using standard model physics and their value is consistent with the new experimental value (Borsanyi et al. 2021). So there's one theoretical explanation of the result!
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/628266",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
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