Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
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... | Also the earth is rotating and is a non-inertial (accelerating) reference frame. For long-range projectiles the effect of the Coriolis force is important. Some of the earliest numerical simulations using computers were calculations of long-range projectile motion.
| {
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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... | Updated Response per comment by @Chemomechanics.
This response assumes the nitrogen container is well insulated and considers only evaporation (mass transfer) and not heat transfer as the dominant phenomenon. At atmospheric pressure the saturation temperature for liquid nitrogen is -196 C. The air in the room is proba... | {
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Ball in magnetic field, not understanding boundary condition So I am reading solution of following exercise:
A conducting sphere with radius $R$ moves with constant velocity $v=ve_x$ inside a constant magnetic field $B=Be_y$. Find the induced charge distribution on the sphere to 1st order in $v/c$ in the laboratory ine... | Because it implements the correct boundary conditions for the electric field in the primed reference system $E'(r) \to E_0\vec{e}_z$. Check that this is true by taking the gradient.
Conversely you may ask what potential fulfills $\nabla \Phi(r,\theta) = E_0\vec{e}_z$ which will lead to the expression.
Admittedly, it is... | {
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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?
|
I have heard that even atomic clocks lose a second every billion years or so.
That would be a small misunderstanding on your part. The second now is defined by atomic clocks. So, if all atomic clocks were consistently slow, then that would mean that the definition of a second was wrong... by definition.
That doesn't ... | {
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Yukawa matrices It is known that the masses of fermions in the Standard Model are represented in the form of singular values of complex Yukawa matrices (Yukawa couplings). The question is, are the values of the masses/couplings themselves real numbers? If so, what are the values of the imaginary part, what role do they... | Yes, the values of the masses are real positive numbers. Recall how you find them out of the complex matrix Y.
Note first that $Y Y^\dagger$ is hermitian, and has positive-eigenvalues, and so can be written as
$$
Y Y^\dagger = U D ^2 U^\dagger
$$
for some unitary U and diagonal real D with no zero entries, for simplici... | {
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Where is this magnetic rail gun getting its energy? Please watch this youtube video of a magnetic rail gun moving a marble.
So as you already know, Conservation of Energy states that "energy can be neither created nor destroyed, but can only change form". Where is this rail gun getting the energy to move the marble? It... | There is no deviation from the laws of physics. Magnetic potential energy is the another source of energy apart from gravitational potential energy. As long as magnetic property of the metal exists it will last
| {
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Intrinsic carrier concentration in doped semiconductors For an intrinsic semiconductor, due to thermal energy we get some charge carriers whose concentration is known as intrinsic carrier concentration.
Now if we dope the material we'll have carriers both due to donation and due to thermal energy generation.
In this ca... | Intrinsic carrier concentration is the concentration of electrons or holes in a pure, undoped, semiconductor. Doping a semiconductor changes the concentration of electrons and holes but it doesnt change the intrinsic concentration. It just stops being an example of an intrinsic semiconductor. Its the same as heating wa... | {
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How can we say load is static in case of simple tension test if it is changing with time? In simple tension test the load acting is static load(which doesn't change with time), but in this test the load varies with time i.e., the load increases, so how can we say it is a static load test?
| In a static test, the load is nearly constant and acts in only one direction. In a dynamic test, the load varies rapidly and can act in both directions i.e., for example in reversed bending.
Static tests are used to determine yield point, ultimate strength, etc. and dynamic tests are used to determine fatigue resistanc... | {
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Finding power stored/given by each element in a circuit
I will start with my attempt.
I recall my professor saying that $i_1 = 0$ due to the fact that current cannot flow through an open circuit. That makes the current-dependent-voltage-source on the right circuit generate no voltage at all.
Looking at the left circui... | Your approach is correct, for the most part.
*
*Consider the leftmost circuit. Since, as you said, $R_2$ is short-circuited, we can redraw the leftmost circuit:
Thus, applying Kirchhoff's Current Law on the blue node yields $i_1=0$.
*No current flows through $R_2$, since it is short circuited.
*The voltage acro... | {
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Why is Minkowski metric diagonal? Why is the Minkowski metric a diagonal in a 4x4 matrix? What does the diagonal do?
| Being diagonal is a coordinate-dependent concept: the components of the matrix associated to the metric tensor depend on the coordinate system you use. Thus a very simple example of a non-diagonal metric is the standard Euclidean metric $\delta = dx^2 + dy^2$ on $\mathbb R^2$ in the coordinate system $(x,z) = (x, x+y)$... | {
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What does the "true" visible light spectrum look like? When I google "visible light spectrum", I get essentially the same image. However, in each of them the "width" of any given color is different.
What does the "true" visible light spectrum look like, then? It can't be that each and every image search result is corre... | If you're really curious, buy a cheap prism, and take it outside in sunlight.
You'll be dispersing the frequencies present in sunlight, and in addition, your eyes are more or less sensitive depending on the frequency, but that's a good start for being able to see what a "real spectrum" of visible light is.
A monitor do... | {
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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(... | The definition is
$$
\psi(x)=\langle x| \psi\rangle, ~~~\leadsto \\
|\psi\rangle= \int dx ~~\psi(x) | x\rangle , ~~\leadsto \\
|\Psi\rangle= \int dx ~~ A e^{-ikx}| x\rangle .
$$
Wavefunctions are coefficients of coordinate kets.
NB You may also then check
$$\langle p|\Psi\rangle= \int dx ~~A e^{-ikx} \langle p|x\rangl... | {
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Oscillation coil: where is the electric field? Let assume a simple RF coil fed with an alternating current at RF frequencies, say 100MHz.
I believe that no one doubts that the coil will radiate RF energy in the form of radio waves.
A radio wave is classically composed of an electric vector and a magnetic vector orthogo... | One has to distinguish the near field (i.e., the field near the coil) and the far field, i.e., the propagating electromagnetic waves far away from the coil (far away on the scale of the wave length). The major factors that determine whether the oscillating electric current will produce a propagating electromagnetic wav... | {
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Are there fields (of any kind) inside a black hole? It is said that nothing escapes from black holes, not even light. All particles are now thought to be excitation of different fields (electric field, electromagnetic field, photon field, etc).
Does it follow that there are no fields (of whatever kind) inside the event... | Besides the electric field, as Jerry Schirmer mentions in his answer, that is non-zero everywhere one can have a scalar field that is exists inside the horizon. See for example the (2+1) dimensional solution reported here Conformally dressed black hole in 2+1 dimensions.
The metric function is found to be:
$$f(r) = \cf... | {
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Are angles ($\theta$ and $\phi$) in spherical coordinates treated as operators in quantum mechanics? Position is specifically considered as an operator in quantum mechanics. I want to know if $\theta$ and $\phi$ are explicitly considered as operators in quantum mechanics for solutions to 3D Schrodinger equation. Also, ... | Yo can define such an operator. Let $\{|\mathbf{r}(r,\theta,\varphi)\rangle\}$ be the eigenstates of the position operator $\mathbf{r}$:
$$
\mathbf{r}|\mathbf{r}(r,\theta,\varphi)\rangle = \mathbf{r}(r,\theta,\varphi) |\mathbf{r}(r,\theta,\varphi)\rangle.
$$
Let us try the following definition of an "angular-operator" ... | {
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What are the necessary and sufficient conditions for a wavefunction to be physically possible? Often times it is stated in books that a quantum state is physically realizable only if it is square integrable. For example, in Griffiths (2018 edition) page 14 he stated
Physically realizable states correspond to the
squar... | In addition to what John Rennie said in the comments, I would like to highlight another thing using the quote you provided:
Physically realizable states correspond to the square-integrable
solutions to Schrödinger’s equation.
the delta function isn't a solution to Schrödinger’s equation.
a physical wavefunction there... | {
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Is entanglement time-symmetric? It is common to describe an experiment as "causing" entanglement. For example, two quantum particles that interact become entangled as a result of their interaction, so we are likely to say that the interaction "caused" the entanglement.
However, quantum mechanics is time-symmetric, so ... | The picture is correct. The key fact is that interaction can both create and destroy entanglement, so there is no conflict with time symmetry.
Suppose for example that two qubits initially in the product state $|\psi_i\rangle = |+\rangle|0\rangle$ are allowed to interact in a way that performs the CNOT gate from the fi... | {
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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... | Keep in mind three facts:
*
*If you look at the Lorentz force, a static magnetic field never imparts kinetic energy onto a charged particle; it only curves its trajectory. You need electric field to speed up or down a charge.
*If you look at Faraday's law, you will see the curl of the electric field is zero when the... | {
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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
$$\omega^{ij} = \epsilon^{ijk}\theta_k$$
$$\omega^{i0} = -... | In the following text, we use $\eta_{\mu\nu} = \text{diag}(-1,+1,+1,+1)$.
For an infinitesimal homogeneous Lorentz transformation, we have
$$ {\omega^\mu}_\nu = \begin{pmatrix} 0 & \zeta_1 & \zeta_2 & \zeta_3 \\ \zeta_1 & 0 & -\theta_3 & \theta_2 \\ \zeta_2 & \theta_3 & 0 & -\theta_1 \\ \zeta_3 & -\theta_2 & \theta_1 &... | {
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Is there a limit to the energy density of a battery? Better battery technology is very important today: improving the energy stored per volume or mass. This led me to wonder whether there is a theoretical limit. (I'm not expecting that we are at all close to it. Real life just inspired the question.)
One extreme bat... | For a battery powered by electrochemistry, there will be a natural limit on its energy density of the following form:
Batteries work by capturing and diverting the electron transfers occurring in chemical reactions that happen in solution (commonly). This means that a chunk of, say, zinc metal in a zinc-copper battery ... | {
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Newton's 3rd Law Coil in Magnetic Field When a current-carrying rectangular loop is placed in a magnetic field, the forces acting on either side of the loop which is perpendicular to the field provides a torque which rotates the loop.
However, according to Newton's 3rd Law, there should be some equal and opposite forc... | Newton's third law does cause magnets to have equal and opposite effects on each other, whether electromagnets or permanent magnets. The equal and opposite forces are transferred by the magnetic field. In an electric motor or generator, just as the field magnets are pushing /pulling on the armature via their magnetic f... | {
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Kinetic Energy of a Block-Bullet System A bullet of mass $m$ is fired towards a wooden block of mass $M$. At a particular instant of time when the bullet is inside the block, the speed of the block is $V$ and the speed of the bullet, relative to the block, is $v$. I would like to find the total kinetic energy of the sy... | The kinetic energy of system depends on the choice of reference frame. To compare the energy of different objects or use conservation of energy, each energy must be defined in the same reference frame.
The most common choice would be to define a "lab frame." This is your reference frame, standing at rest next to the ... | {
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Lagrangian for a fixed number of non-interacting, non-relativistic bosons In my book on QFT, (during an explanation of superfluidity) the author states that the lagrangian for a fixed number of non-interacting, non-relativistic bosons is $$i\Phi^{\dagger}\partial _{0}\Phi-\frac{1}{2m}\nabla\Phi^{\dagger}\cdot \nabla\Ph... | Since $N$ is constant $\mu \int \psi^\dagger \phi= \mu N$ is fixed, so it just sets the zero of energy to be be that of the ground state of the $N$-body system. Note that $\mu$ is usually negative for bosons.
| {
"language": "en",
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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... | Mist is a suspension of tiny water droplets in air.
Light traveling through the mist gets randomly scattered, mainly by bouncing of the droplets. That makes mist far less transparent than bulk water.
I don't think mist is literally gray in colour but the fact that mist is far less transparent than pure air (or bulk wat... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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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?
| A large amount of water (i.e. if the path of light through it is long) will simply start absorbing light, as it's not completely transparent. For smaller amounts, as when pouring it from one container to another, this is mostly negligible. However, there is also surface reflection. A small amount of the incident light ... | {
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What are the "derivations" of the inverse-square law? Besides the derivation mentioned in this Wiki article, I want to know, if there exists any other derivation of the inverse-square law based on some profound physical/philosophical concepts.
| Based on an informal assumption, we could derive the inverse square law for gravitational force and Coulomb force.
Assumption
Suppose everything in the space is scaled up by a factor of $k$, and time stays the same, then we shouldn't expect anything to change.
Derivation
Since we are in a 3D space, any volume would be ... | {
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Problems deriving the Quantum Hamilton-Jacobi equation This is my first question at Physics SE so please be kind. I am well versed in the etiquette over at Math SE, but not so much here. Anyway, I thought this question was better suited to this site because it is less a problem of understanding mathematical computation... | I have solved this problem myself. If we write
$$\Psi=e^Z$$
Where $Z$ is now allowed to be complex, we get
$$i \hbar \partial _{t} Z=\frac{-\hbar ^{2}}{2m}\left( \nabla ^{2} Z+\Vert \nabla Z\Vert ^{2}\right) +U$$
Letting $Z=\frac{iS}{\hbar}$ will yield the desired result.
| {
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Can the Auger effect cause a second electron to be just excited instead of ionised and emitted from the atom? From what I understand, the Auger effect is usually defined as when an electron deexcites but instead of releasing its change in binding energy as a photon, it transfers it as kinetic energy to another electron... | Let me first note that Auger process is due to Coulomb interaction between electrons, so it may be beneficial to think of it in terms of the Fermi golden rule:
$$
w_{i_1 i_2\rightarrow f_1 f_2}=\frac{2\pi}{\hbar}|\langle i_1, i_2 | V|f_1, f_2\rangle|^2\delta(\epsilon_{i_1} + \epsilon_{i_2} - \epsilon_{f_1} - \epsilon_{... | {
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If radio signals attenuate when travelling through space, then what kinds of emissions are we looking for when searching for extraterrestreal life? My understanding is that radio waves travel forever, like ripples in a pond, but attenuate with distance. They get mixed with other signals and become cosmic noise.
I'm loo... | You are right, an omnidirectional radio broadcast would be very faint by the time it reached earth, and very difficult to distinguish from background noise. Projects such as SETI use large amounts of computing power and sophisticated signal analysis algorithms to try to detect a faint signal with a pattern indicating a... | {
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Are there any books/portions of books/particular topics of study that go into great detail to describe the physics of a rope (or any ropelike body)? I don't know why, but lately, I can't get the physics of a rope of my mind, specifically what happens to a coiled, or otherwise not taught/pulled straight rope when it is ... | Continuum mechanics, but it has too many topics except for the rope. The principals are the same.
Besides, there does exist some books about rope only. Such as Theory of wire rope . Just try googling with book about rope/wire/string mechanics and you will find more.
I did some research about this before and if you are ... | {
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Is perfectly monochromatic light always polarized and vice-versa? Is perfectly monochromatic light always polarized and polarized light always monochromatic? I am not totally sure but I think that the answer to the first is 'YES'. Because if a radiation is unpolarized, its polarization changes randomly with time so tha... | You are correct, but the light does have to be really really monochromatic and the polarization has to be strictly defined and exactly fixed.
As you point out, if you examine the light through a fixed polarizing filter and if its amplitude changes over some period of time, then there must be more than one frequency p... | {
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Is it impossible to use a 1m by 1m suction cup? I was learning about fluids and I randomly thought of suction cups. I think they're a really cool application of air pressure. When you squeeze the air out, the outside air pressure exerts a huge force on the cup, meaning it will hold in place.
However, I was thinking abo... | I don't think the air pressure matters - that's always present. By your logic as stated, you couldn't push all the air out of an air mattress either. All that matters is the resistance of the suction cup material itself, as that is what you're deforming.
| {
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How are propagator and two-point function related? Assume that we have a QFT with one scalar field $\phi$ with mass $m$ and the Lagrangian
$$\begin{aligned}
\mathcal{L}_{\mathrm{EFT}, \mathrm{off}}=& \frac{1}{2}\left(\partial_{\mu} \phi\right)^{2}-\frac{1}{2} m^{2} \phi^{2} \\
&-\frac{C_{4}}{4 !} \phi^{4}-\frac{C_{6}}{... | The main point is that the 2-pt functions for the generator $W_c[J]$ of connected diagrams and the generator $\Gamma[\phi_{\rm cl}]$ of 1PI diagrams are each other's inverse (up to factors of $i$), cf. e.g. this & this related Phys.SE posts.
In particular note that for a 1PI diagram the external legs are stripped/amput... | {
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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... | It is a good question. The single standing waves are the modes that get emphasized in teaching, but the real motion of strings is due to sums of such modes.
Consider a Slinky. One can make it swing in the fundamental mode or in modes with one or two nodes by adding energy from your hands in just the right intervals. Yo... | {
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How do position operators ($\hat{x}$, $\hat{y}$, $\hat{z}$, $\hat{r}$) act on orbital angular momentum states? Consider an orbital angular momentum state $\vert l,m\rangle$, I am pretty sure when $\hat{r} = \sqrt{\hat{x}^2 + \hat{y}^2 + \hat{z}^2}$ and $\hat{z}$ act on it, the resulting states will still be $\vert l, m... | To find the action of $x,y,z$ on $|l,m\rangle$ you need to use the Clebsh-Gordon procedure for combining the $l=1$ vector defined by $x-iy \sim |l=1,m=-1\rangle$, $z\sim |l=1,m=0\rangle$, $x+iy\sim |l=1,m=+1\rangle$ to the $|l,m\rangle$ state. You will get a linear combination of $l-1$, $l$, $l+1$
| {
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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 $$
we can use the conservation of energy and momentum in order to determine the minimum photon energy in order to make this reaction possible, which happens ... | If you analyze the threshold reaction in the center-of-momentum frame, rather than the lab frame where the deuteron is initially at rest, the energy-conservation equation becomes
$$
pc + m_d c^2 + \frac{p^2}{2m_d} = m_p c^2 + m_n c^2
\tag{1*}
$$
Here $p$ is the magnitude of the initial momentum for both the photon and ... | {
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What's the difference between "optical amplification" and "magnifying"? Optical amplification used widely in astronomical observatories and "magnifying" used in microscopes.
What's the difference between "optical amplification" and "magnifying"?
| Optical amplification is a process where one captured photon triggers the release of
an electron inside the apparatus which then releases more electrons, etc. in a process called photomultiplication.
This can also be done when the incident photon causes the injection of an electron into a semiconductor junction, which ... | {
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Can buoyancy be explained in terms of kinetic and potential energy exchanges between the buoyant object submerged in a fluid?
As observed in the above diagram, a wooden block is held submerged in water within a container by an (external, such as a string attaching the block to the bottom of the container) force which... | When the wood block is released, the upward force $F$ is the pressure difference between its top and bottom surfaces minus the weight. That force is constant until the block reaches the surface of the water, so we can say that the potential energy is $Fh$, where $h$ is the depth of the block from the water surface.
How... | {
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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... | If the wire is straight, then no, due to axial symmetry magnetic field is just compressing the wire a little but no net force is present.
However, if the wire isn't straight, then net magnetic force due to wire on itself may be non-zero. For example, consider wire in shape of upside-down letter J, in which current flow... | {
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If the world had four spatial dimensions, then area would be a tensor? In three dimensions area is a vector because two dimensions have a direction relative to the third. If the world had four spatial dimensions then area would be a tensor?
And what form then the laws of physics which imply the concept of area, as elec... | Roughly speaking, Yes!
In 3 spatial dimensions a 2D thing (an area) uses 2 of the 3 available dimensions. So even though it isn't really a vector (an area should have twice the units of a vector) an area can be described by picking out the single direction that is orthogonal to it.
In 4D an area has a 2D space orthogon... | {
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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 green line is the line that the "zig-zag" runner is running.
Thus:
$$s\cos(\alpha)=a\tag 1$$
So if he is doing this n times during the half-marathon you obtain
that:
$$n\,(s-a)=\Delta L\tag 2$$
With Eq. (1) and (2) you obtain that:
$$\cos(\alpha)=\frac{a\,n}{\Delta L+a\,n}$$
Edit
with the remarks from @AgniusVasil... | {
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Can the Hamiltonian be interpreted as the "speed" of unitary evolution? The Schrodinger equation
$$i\hslash \frac{d}{dt} \psi = H \psi$$
means a quantum state $\psi(t)$ evolves unitarily, that is,
$$\psi(t) = \exp(-\frac{i}{\hslash} H t) \psi(0)$$
where $\psi(0)$ is the initial state at time $t = 0$.
Suppose if we sca... | The Hamiltonian itself is not a speed, but you're right that evolution speed is proportional to the energy scale that defines the dynamics.
To define quantum evolution speed, consider first a classical signal with a finite range of frequencies in its Fourier spectrum. The width of the range tightly bounds the number o... | {
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Could gravity be a weak force because gravitons are absorbing gravitons before they reach a target rest mass? Could gravitons be similar to the gluons in the colour force? Can gravitons absorb other gravitons before they reach their target rest mass?
| Classical general relativity has many well-known phenomena involving gravitational fields interacting with themselves, most dramatically the geon solutions which describe gravitational radiation collapsing to a black hole.
Since classical general relativity has self-interacting gravity, it would make sense for there to... | {
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Cosmology - an expansion of all length scales From the link Is non-mainstream physics appropriate for this site?
"a question that proposes a new concept or paradigm, but asks for evaluation of that concept within the framework of current (mainstream) physics is OK."
Here is a concept, evaluation within the framework of... |
the frequency of the received photon would be lower
Why would it ? Since $c=\lambda f$ and $c$ and $\lambda$ change in the same proportion then $f$ is constant. All you are doing is changing the units in which length is measured. You get exactly the same effect if you measure the wavelength in furlongs instead of met... | {
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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... | A person would not be able to localize a sound using the Doppler effect created by shaking their head.
Say a person shakes their head at 20 cm/s. The speed of sound is about 330 m/s. This gives a frequency change of 0.06%.
The "just noticeable difference" to discern two frequencies played in succession is about 0.6% ... | {
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Singularity in Robertson Walker metric with flat spatial slices In Sean Carroll's GR book, pg. 76, a special case of the Robertson-Walker metric, where the spatial slices are flat is given by
$$ds^2=-dt^2+a^2(t)[dx^2+dy^2+dz^2].$$
It was said that $t=0 $ represents a true singularity of the geometry (the 'Big Bang') an... | Remeber that the solution to the Friedman equations for the scale factor $$a(t) = a_0 t^{\lambda}$$ where $\lambda$ is a constant. This is obviously zero at $t=0$. At this point the spatial part of the metric $$ds^2=-dt^2+a^2(t)[dx^2+dy^2+dz^2]$$ vanishes.
| {
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Spherical Lens Instead of Parabolic Lens I know that using the paraxial approximation, spherical lenses behave like parabolic lenses.
It seems that there is no reason to use spherical lenses instead of parabolic (because they are used in the same way, and parabolic lenses do not required paraxial approximation) apart f... | Once you deal with practical lens systems that operate at higher numerical apertures, for example a microscope objective, the paraxial conditions no longer hold. Spherical surface can be made highly accurately, and then combined in order to control aberrations precisely in a way that would be much more expensive with a... | {
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Is there a way to prove that different angular momentum components anticommute without using a specific matrix representation? I know spin-1/2 Pauli matrices satisfy the anticommutation relationship $\{\sigma_i, \sigma_j\}=2\delta_{ij} \mathbb{I}$. I wonder how this can be proved without writing down the matrix represe... | It will not be possible to derive the anti-commutators from the commutation relations alone, because not every representation of the commutation relations (i.e., of the algebra $\mathfrak{su}(2)$) satisfies $\{L_i, L_j\} = 2\delta_{ij}$. For example, $\sigma_x$ and $\sigma_y$ anti-commute, but the spin-1 matrices
$$ L_... | {
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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... | An accelerated frame is only locally equivalent to a gravitational field. Globally, you will not be able to "fake" the gravitation of a planet by just accelerating. Only if the passenger ignores tidal forces will he be unable to distinguish an accelerated frame from a static gravitional field.
| {
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Newton's Laws of Motion, Pulleys, Rope and tension I was solving some questions to apply my concepts, and I came across the atwood machine and pulley block problems.
Consider the following for example:
The pulley is massless and frictionless, string, too, is ideal.
Why does the book say that the tension in the green s... | Imagine removing the black string and masses and instead just grabbing the pulley with your arms. If you pull with a force of $T$ with each arm, shouldn't the green string in the top then hold back against both? The green string tension should be $2T$.
Back to your scenario, the situation is the same. The green string ... | {
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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... | A loudspeaker can be modeled as a linear AC motor driving a flexible membrane. The motor components (in this case, the voice coil and the cone) have a certain amount of mass, and the membrane's clamped circumference possesses compliance, and when taken together they result in a fundamental resonant frequency. When the ... | {
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High school experiments recommendations I am from India and you know teachers don't show the experiments and they only give the theory but I want to see experiments video at least so I'm satisfied that physics is correct , can you tell me some websites / channels which show high school experiments .
| Well I think as you are asking for High school course, I believe that you should prefer YT videos by Walter Lewin here.
He was a professor at MIT and has an unique way of teaching the subject with the demonstrations wherever it is required. The videos cover Newtonian Mechanics, Electromagnetism, Vibrations and Waves,Bo... | {
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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 = ... | It is true that $V/I$ is a constant for resistors, and also that $I/V$ is a constant. But, of course, they are not the same constant.
$R=V/I$ gives the resistance of a resistor, while $G=I/V$ gives the less commonly-used conductance of a resistor.
Neither of these is a "derivation" of the resistance- $R=V/I$ is a defin... | {
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Excess Pressure on a curved surface with two radius of curvature While studying surface tension, I noticed the following formula to calculate excess pressure on a curved liquid film made use of two radii of curvature:
$$2T\left(\frac{1}{R_{1}} + \frac{1}{R_{2}}\right)$$
I have not been able to understand the significan... | The best way to visualize two different radii for any given surface. The best way is to cut the surface by a pair of perpendicular planes. Since, now you are viewing the section of surface cut by a plane you will have a planar surface which has a radius of curvature.
Try this out:::: Use a torus to apply the above argu... | {
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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... | Feynman once asked more or less the same question (page 129 of "Quantum Field Theory" by Lewis H. Ryder):
I remember that when someone had started to teach me about creation and annihilation operators, that this operator creates an electron, I said 'How do you create an electron? It disagrees with conservation of char... | {
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Mesoscopic Bose-Einstein Condensate Bose-Einstein condensates of molecules of a few daltons have been already created, so I was wondering: would making a Bose-Einstein condensate on a system of Quantum Dots, due to their properties, cause the system to display any different effects?
| There is no boson condensate. It was imagined to fulfill the idea of big bang inertial gravity, energy without matter and all that resulted in dark energy, dark matter. Below is image to show how finiteness of particles gives boson distribution and there is no infinite or high energy accumulate in given region.
This ... | {
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How do I find the approximate surface area of a chicken? I'm working on building a chicken army and I'm trying to find out how much metal or kevlar (still deciding) I need to make armor for the chickens. this measurement does not need to be exact I'm just trying to get an estimate for how much I will need. You will be ... | The astrophysics answer.
Take a representative chicken, put it in a cold room that is lined with infrared detectors measuring the flux in several wavelength bands.
Assume the chicken is a blackbody and fit a Planck function to estimate both temperature and emitting surface area.
| {
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Understanding a Poynting vector equation I'm reading this section in the Griffiths Introduction to Electrodynamics book.
I trying to understand where equation 9.57 comes from (the middle part of the equation at least; I see where the $cu\;\hat{\mathbf{z}}$ part on the right comes from).
Does it come directly from calcu... | For a electromagnetic wave, $\mathbf E$ and $\mathbf B$ are orthogonal to each other and both are orthogonal to the direction of wave propagation. For the monochromatic plane wave propagating in the $z$ direction, if $\mathbf E$ is in $x$ direction, we deduce that $\mathbf B$ is in $y$ direction. We also know that the ... | {
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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 ... | The latter. I think it is most intuitive to think about the F=Ma equation as a statement about cause and effect. The force on an object arises due to something physical (ie. a stretched spring connected to your object), and it's magnitude depends on the configuration of your system. The acceleration is a consequence of... | {
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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 ... | If it can be determined what the stopping distance was for the ship, such as by a measurement of the depth of penetration of the ship into the canal wall, and can ignoring the resistance of the water to the ship movement, one can estimate the average impact force using the work energy theorem, which states that the net... | {
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Does physics explain why the laws and behaviors observed in biology are as they are? Does physics explain why the laws and behaviors observed in biology are as they are? I feel like biology and physics are completely separate and although physics determine what's possible in biology, we have no idea how physics determi... | We are very good at describing small quantum mechanical systems, because today we have QM, and we do know that the world is ultimately quantum mechanical in nature. That being said, when it comes to predicting bigger biological systems (just like our own human nature), our capabilities are very limited. We are all made... | {
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How does Stefan Boltzmann law work for absorption? I have come across the Stefan Boltzmann law and I have a couple of doubts based on its use in the net heat flow expression
$$dq/dt = e A\sigma T^4$$
Now according to my textbook
If the temperature of the surrounding is $T_1$ and the temperature body is $T_2$ and the bo... | I believe it was Gustav Kirchhoff who found out that the emissivity $\epsilon(\lambda)$ of a body at wavelength $\lambda$
equals its absorptivity $A(\lambda)$ at the same wavelength,
$$\epsilon(\lambda) = A(\lambda)$$
In physics, we are not really able to answer why this is the case. However, let's consider the theore... | {
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How would velocity of sound, the fundamental frequency and wavelength of sound vary when the temperature of an organ pipe is increased? Here is my approach to this:
neglecting any thermal expansion of the pipe:
By the Laplace formula for the speed of sound,
$V=\sqrt{\frac{\gamma P}{\rho}}$ where P is the pressure, $... | The wavelength is independent of temperature.
You can see here that the wavelength depends only on the length of the organ pipe and the harmonic of the resonance, rather than the temperature of the gas itself.
The only way of changing the wavelength is by increasing the harmonic or the length of the pipe itself.
Hope ... | {
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Force on the bottom of a tank full of liquid - Hydrostatic Pressure or Gravity
Imagine a tank filled with water that has some height $h$ and at the bottom area $A$ but as it goes up, for example at height $h/2$, it's area is now $A/2 $. What's the correct way to calculate the force at the bottom of the tank? (Let's ig... | Imagine what will be the case if you consider a closed tank and put the pressure inside under high pressure. While the hydrostatic pressure is high, the weight of the water on the bottom of the tank will always be equal to the weight of the water. so when you calculate pressure is a different thing from calculating wei... | {
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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... |
But how do we know that the errors are in the measuring or in the law about which we make measurements?
Laws in physics theories are extra axioms to pick up from mathematical solutions those solutions that are descriptive and predictive of data. Whenever data do not fit predictions , one finds the dimensions of valid... | {
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What is the physical importance of topological quantum field theory? Apart from the fascinating mathematics of TQFTs, is there any reason that can convince a theoretical physicist to invest time and energy in it?
What are/would be the implications of TQFTs?
I mean is there at least any philosophical attitude behind it?... | TQFTs were not discovered by mathematicians - they were actually discovered by physicists, so one should expect there to be physical motivation for the theory. One reason why that this is difficult to discover is that mathematicians have taken over the theory so it is hard to recognise the physical motivation.
One reas... | {
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"url": "https://physics.stackexchange.com/questions/626151",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
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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:
$$
\frac{1}{V} \ln Q(z, V, T) \underset{V \rightarrow \infty}{\longrightarrow} \text { Finit... | There is no mathematical proof just because, in general, it is not true that the limit exists or it is finite. Of course, we would expect a finite limit as a precondition for a thermodynamic interpretation of the statistical mechanics formula.
The right question is not about the reason for a finite limit, but to ask th... | {
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In thermodynamic limit, how is $\frac{V}{(2 \pi)^{3}} \int \mathrm{d}^{3} k= \int \frac{\mathrm{d}^{3} r \mathrm{~d}^{3} \boldsymbol{P}}{h^{3}}$? In the book Intro. Statistical Physics by K.Huang, on page 106, it is given that
Because of indistinguishability; the $N$ -body wave function is labelled by the set $\left\{... | It is simply a change of variables. With the definition $$\vec{p}=\hbar \vec{k} = \frac{h}{2\pi} \vec{k},$$
we can conclude that for every direction, $i\in\{x,y,z\}$
$$\frac{dk_i}{2\pi} = \frac{dp_i}{h}$$
therefore
$$\int \frac{d^3\vec{k}}{(2\pi)^3} \rightarrow \int\frac{d^3\vec{p}}{h^3}$$
the other factor is just volu... | {
"language": "en",
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Series combination of springs When a spring mass system is connected vertically with two massless springs in series whose spring constants are $k_1$ and $k_2$ to a block of mass $m$ we know that equal forces act on both the springs. Let that force during oscillations be $F$.
When we calculate effective spring constant ... | When the springs (assumed to be massless) are hung upside down, they will have a zero extension. So in fact $$mg = k_1x+k_2x$$ when the mass is added to the system.
You wrote that the resultant force $$F_r = F_1+F_2$$ and concluded that this should be $2F$ as if the force in both springs were equal.
So what you should ... | {
"language": "en",
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Angular momentum commutation relations The operator $L^2$ commutes with each of the operators $L_x$, $L_y$ and $L_z$, yet $L_x$, $L_y$ and $L_z$ do not commute with each other.
From linear algebra, we know that if two hermitian operators commute, they admit complete sets of common/simultaneous eigenfunctions. The way I... | If two observables $A$ and $B$ commute, i.e. if $[A,B]=0$, then there exists a common eigenbasis. In other words, there is a basis $\{|\phi_n\rangle\}_n$ for which
$$A|\phi_n\rangle= a_n\, |\phi_n\rangle \quad\text{and}\quad B|\phi_n\rangle= b_n\, |\phi_n\rangle \quad.$$
Now consider the case where $A$ also commutes w... | {
"language": "en",
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Why are electronic transitions in atoms modelled as oscillating electric dipole radiation? Sources such as Eugene Hecht and Griffiths claim that oscillating electric dipole radiation is a great approximation for radiation generated from atoms and molecules during electronic transitions. I don't really understand why th... | It seems that they refer to the dipole approximation, which is not the same as describing the radiation as that of an oscillating dipole (the description that work well for antennas, but not for atoms). The essence of the approximation is that the interaction between the atom and the electric field can be represented a... | {
"language": "en",
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Hyperbolic isometries in the context of General Relativity In the context of hyperbolic geometry, it is possible to create a classification for isometries.
I would like to know if these isometries have any particular meaning in the context of general relativity.
Is it possible to understand these isometries from the po... | There is a Bianchi classification of 3d Lie algebras upto isomorphism. Hence this also classifies all 3d Lie groups.
Now, in the ADM formalism of General Relativity, we consider the evolution of a 3d spatial slice. The symmetry group of this slice is a Lie group. When this symmetry group is 3d, we can use the Bianchi c... | {
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What exactly does an exchange of particle labels for identical particle wave functions mean *physically*? We know that the wavefunction of identical particles behaves as follows:
$$\Psi(1,2)=\begin{cases}-\Psi(2,1) & \text{for fermions} \\ +\Psi(2,1) & \text{for bosons} \end{cases}$$
Now, what exactly does exchanging p... | The exchange is an artifact of the formalism. When we write it down, we tend to label the particles as 1 and 2 (or something else), but physically there is no difference between the particles. They are not "labeled" in any way in the physical world. Therefore, we need rules for our formalism when we exchange the partic... | {
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How does one integrate the Fermi-Dirac distribution using the zeta function? I've seen in my physics book that:
$$n=\frac{g}{2\pi}\int_0^\infty\frac{E^2dE}{e^{E/T}\pm1}$$
Regarding the number density of a relativistic gas of either bosons ($-1$) or fermions ($+1$). The solution of both of the integrals is given to be:
... | As a hint, try to show that:
$$\frac{1}{e^{E/T}+1} = \frac{1}{e^{E/T}-1} - \frac{2}{e^{2E/T}-1}.$$
| {
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Solving for transmission coefficient in the finite square well Consider a finite square well of depth $V_0$ and which extends from $-a$ to $a$. For $|x|>a$, $V=0$. The wavefunction ansatz one can propose for an incoming wave from the left $Ae^{ikx}$ is:
$$ \psi = Ae^{ikx} + Be^{-ikx}, x<a $$
$$ \psi = Ce^{ik_2 x} + Be^... | You have mistakes in the equations for boundary conditions, they would be:
$$\psi\text{ continuous at }x=-a\longrightarrow Ae^{-ika}+B e^{ika}=C e^{-ik_2a}+De^{ik_2 a}$$
$$\psi'\text{ continuous at }x=-a\longrightarrow ik(Ae^{-ika}-B e^{ika})=ik_2(C e^{-ik_2a}-De^{ik_2 a})$$
$$\psi\text{ continuous at }x=a\longrightarr... | {
"language": "en",
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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... | Recently gravity was measured between two 1 mm gold spheres. (Measurement of Gravitational Coupling between Millimeter-Sized Masses by Westphal et al)
Gravity cannot be separated from "bending spacetime". Any force that affects everything equally in a place can alternatively be described as a bent spacetime. So those 9... | {
"language": "en",
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"source": "stackexchange",
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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... | In the standard model the $α=(g-2)/2$ of an elementary particle should be calculable , the calculations as accurate as the higher orders are computed.
For the electron the calculations coincide with the experimental value to great accuracy
The muon $α=(g-2)/2$ has different diagrams dominant so the theoretical value ... | {
"language": "en",
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What is momentum? Momentum tells you the mass of the object and how fast it is going right? So if I have a 2 kg ball moving at 2 m/s, then the ball has 4 kg⋅m/s of momentum. My question is why do we multiply mass and velocity to get momentum. (From the example above) Why cant we just say the ball is 2 kg moving at a sp... | The deep reason for introducing the momentum is that momentum is a quantity that in some circumstances can be conserved, while this is not the case for the velocity. The case of one particle is not really enlightening, but as soon as we move to systems of more than one particle the advantage of introducing momentum is ... | {
"language": "en",
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Is there a fundamental reason why in the Standard Model, there is no Feynman rule for a vertex with more than 4 legs? In Standard Model, there are vertices with 3 legs (example: $W l\nu$) and with 4 legs (example: $WWWW$).
Is there a fundamental reason why in the Standard Model, there is no Feynman rule where a vertex ... | Fundamental? Power counting. The SM is renormalizable, that is, without dimensionful couplings. The lagrangian must have dimension 4, and any vertex with 5 legs or more would dictate a coupling of dimension -1 or less. So the gauge couplings g,g', the Yukawa y, etc, are all dimensionless.
(You do have Feynman diagram... | {
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Hybridisation of orbitals When we talk about the electronic configuration of boron , sulphur , nitrogen.
What I got to learn new was about their hybridisation. For example , boron has electronic configuration as $1s^2 2s^2 2p^1$.now , there is one pared electron at 2s^2 and one unpaired electron at 2p1.
Now , during a ... | This is called Atomic excitation and there can be multiple sources of energy that can lead to this effect, f.e. atom can absorp energy of an photon which will then lead to excitation (but any energy source can lead into it). You can also take a look on wikipedia Excited state. Hope this helps.
| {
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Double slit interference question a level physics The question attached asks you to calculate the angle between the central fringe and the second bright fringe. The mark scheme says that you should use tan to work out this angle using the distance between the slits and the plane and the fringe separation. But I'm wonde... | It's actually correct (and, IMHO, better) to use the equation you used. The problem solution uses a couple of approximations that are basically correct but lead to small enough errors that the final answer differs at 2 significant figures.
Using $s \sin \theta = n \lambda$ as you did, you would obtain $\theta = 0.1865... | {
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Hamiltonian classical electrodynamics After coming across the Lagrangian density of the Maxwell equations
$$
\mathcal{L} = -\frac{1}{4\mu_0} F_{\mu\nu}F^{\mu\nu}-J_\mu A^\mu = \frac{\varepsilon_0}{2}||\mathbf{E}||^2-\frac{1}{2\mu_0}||\mathbf{B}||^2 -j_\mu A^\mu
$$
I was wondering whether there is a corresponding Hamilt... | According to these lecture notes, the combination of the following Hamiltonian density and constraint gives rise to the Maxwell equations:
$$\mathcal{H} = \frac{\varepsilon_0}{2}\mathbf{E}^2 + \frac{1}{2\mu_0}\mathbf{B}^2 - j_\mu A^\mu$$ and $$\nabla\cdot\mathbf{E}=\frac{\rho}{\varepsilon_0}$$
| {
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Different versions of Schwinger parameterization One common used trick when calculating loop integral is Schwinger parameterization. And I have seen two versions among wiki, arxiv and lecture notes.
$$\frac{1}{A}=\int_0^{\infty} \mathrm{d}t \ e^{-tA}$$
or,
$$\frac{-i}{(-i)A}=-i\int_0^{\infty} \mathrm{d}t \ e^{itA}$$
wh... | The Schwinger parameter itself is manifestly positive. In particular, it is not Wick-rotated, so there are not different versions of it. Rather it is OP's $A$ operator that is Wick-rotated.
OP lists a few references in above comments.
*
*Ref. 1 works in Euclidean signature, so it's well-defined.
*Ref. 2 & 3 only us... | {
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Why is the half-wave dipole the most used antenna design? When producing em waves using a dipole antenna (of length L), you could theoretically use any L and adjust the frequency of the oscillating voltage to get the desired wavelength. Then why are most antennas half a wavelength long? I'd also like to know why it's u... | An antenna is a resonator. If you are not feeding it with its natural frequency, it is not going to oscillate with sufficient amplitude. Of course you can always think about increasing voltage, but usually everything in technology is about efficiency. Imagine you would have to carry a heavy car battery and a high volta... | {
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Perception of simultaneous events I have a two-fold question about the light-cone structure of spacetime, specifically about space-like separated events.
As far as I understand it, any two events that happen at the same time in a given reference frame are space-like separated. If so, any two simultaneous events occurri... | The notion of 'perceiving an event' is misleading.
Perception happens here and now, so all perception at a given time for a given observer takes place within its own single event (the observer here-and-now as a point in spacetime).
What is perceived is causally determined by whatever happened in other events within tha... | {
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Commutation relations inconsistent with constraints In section $9.5$ of Weinberg's Lectures on Quantum Mechanics, he uses an example to explain the clasification of constraints. The Lagrangian for a non-relativistic particle that is constrained to remain on a surface described by
$$f(\vec x)=0\tag{1}$$
can be taken as
... | On one hand,
$$ 0~=~[0,0]~=~[f(x),\vec{p}\cdot\vec{\nabla}f]~=~i\hbar (\vec{\nabla}f)^2.$$
On the other hand, a constraint function $f$ typically satisfies a regularity condition
$$ \left .\vec{\nabla}f \right|_{f=0}~\neq~\vec{0}.$$
| {
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Anticommutation of variation $\delta$ and differential $d$ In Quantum Fields and Strings: A Course for Mathematicians, it is said that variation $\delta$ and differential $d$ anticommute (this is only classical mechanics), which is very strange to me. This is in page 143-144:
If we deform $x$ we have
$$\delta L = m \l... | To elaborate on Qmechanic's answer to show why anticommutation in a bigraded differential algebra is natural, consider a manifold $X$ and its exterior algebra $\Omega(X)$. Suppose that there is a bigrading on $\Omega(X)$ such that $$ \Omega(X)=\bigoplus_{(r,s)\in\mathbb Z^2}\Omega^{r,s}(X), $$ where the sum is a direct... | {
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Magnetic flux through circular loop due to infinite wire I’m trying to calculate magnetic flux that’s going through circular loop with radius $R$, due to magnetic field of a infinite wire that is in distance $d$ from the center of the loop. $\vec{B}$ vector is parallel to $\vec{dS}$ vector. I know that magnetic field o... |
The convenient infinitesimal surface $\rm dS$ is shown in the Figure-01 :
\begin{equation}
\mathrm{dS} \boldsymbol{=}\mathrm{hdw}\boldsymbol{=} (2R\sin\theta)( \mathrm d\ell\sin\theta)\boldsymbol{=} (2R\sin\theta)( R\mathrm d\theta\sin\theta)
\tag{01}\label{01}
\end{equation}
so
\begin{equation}
\mathrm{dS} \boldsym... | {
"language": "en",
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Can a body float in the middle of a fluid? Let's say we have a cubic body of side $a$ and made of a material with density $\rho$ and we measure its immersed height in a fluid of density $\rho_f$ by the variable $y$. Then, its potential energy (and considering a gain of potential due to buoyancy) can be written as:
$V =... | The buoyancy on an object in a liquid is constant as long as the object and the liquid are constant. Whether the object is completely or only partially submerged is irrelevant. So is whether it is floating on top or in the middle somewhere or laying on the bottom. What makes it tricky in practice, is the fact that it i... | {
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Why are constant volume and constant pressure heat capacities basically the same for solids? Are degrees of freedom involved? I knowv that $C_V=\frac{\frac{f}{2} Nk_B}{m}$ and $C_P=\frac{(\frac{f}{2} +1)Nk_B}{m}$. Since for solids their values are very close to each other, I would assume $\frac{f}{2} +1$ is very close ... | It is because solids and liquids are very close to being incompressible. So it doesn't matter whether the pressure is changing or not.
| {
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Definition of electric polarisation and the potential due to a polarised body I've two questions, the second one depends on the first.
$\mathbf{1}$
How exactly is polarisation defined? Griffiths says
$\mathbf{P} \equiv$ dipole moment per unit volume
How exactly do we go about calculating it?
For example if I need to ... | A general remark, which is too long for a comment:
Landau's book on the subject is appropriately called Electrodynamics of continuous media - important thing here is that we are dealing with macroscopic quantities, i.e., the quantities averaged over a "macroscopically small volume", so that they vary smoothly in space ... | {
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In special relativity, how do we know that distance doesn't change in the direction perpendicular to velocity? In the theory of special relativity it is said that the distance in the direction of the speed changes by a factor of
$$\gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$$
How do we know that the distance perpendi... | I think the best way to understand this is to figure out why we needed special relativity in the first place.
One of the fundamental principles of Physics is that no matter which frame of reference you go to, the laws of Physics should remain same (Also known as lorentz invariance). Relativity was introduced by Galileo... | {
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How to find the falling time of an object when acceleration is not a constant? Let's say we are throwing an object from the surface of the earth, this object reaches 70,000km with initial velocity of $10713 \mathrm{m}/\mathrm{s}$ until it reaches the peak high , the g value at 70,000km is $0.068 \mathrm{m}/\mathrm{s}^2... | Integrating twice based on acceleration due to gravity. (Keplar could also be used, but the result is the same).
m1 = mass of earth
m2 = mass of object, so much smaller than mass of earth it can be ignored
gravitational constant G = 6.6743 x 10^-11
m1+m2 = 5.9722 x 10^24
r is distance between the two objects
v is combi... | {
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Probability current is zero for normalizable stationary state I'm asked to show that the probability current is zero for a normalizable stationary state of the Shrodinger Equation. So we have that $\Psi(x,t)=\psi(x)e^{-iEt/\hbar}$. Now using the conservation of probability we have
$$0=\frac{\partial}{\partial t}|\psi|^... |
I want to say that for the state to be normalizable we must have
$j\rightarrow 0$ as $|x|\rightarrow\infty$, and so $j=0$ everywhere.
I don't know why should be the case.
Normalisable just means that $|\Psi|^2 \rightarrow 0$ "fast enough". The $j$ takes out the phase, which in $|\Psi|^2$ does not matter, so I don't t... | {
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How can parallel rays meet at infinity? I found that in every book (till my 12th) it is written that, in concave mirror, when object is at focus, then reflected rays will be parallel and they meet at infinity to form a real image.
But, as we know, parallel rays never meet. Then, does this mean that all books are wrong ... | It means that they don't meet, because as you correctly pointed out parallel lines never meet.
Then what's the point in saying "they meet at infinity" if they never meet? Because you can obtain a parabola by an ellipse with focal distance $d$ in the limit where $d\rightarrow\infty$. In the ellipse rays from one focus g... | {
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Does Einstein's Equivalence Principle ignore time dilation? It seems Einstein's equivalence principle is neglecting time dilation. If an observer is at rest in an inertial reference frame, free of any gravitation, she will experience time flow at the "native" rate of a universe empty of mass and energy. However, an obs... | There is no way for the observer knows about time dilation. The clock in the frame shows the local time, and there is no comparison with another clocks in distant places. Otherwise it is not a local frame.
| {
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Mathematical proof of charging by induction If we bring a positive charge +Q near a neutral conductor , we know that the surface near the source gets-Q and opposite to it gets +Q, but why do these induced charges have to be equal in magnitude to source charge, why isn't a charge distribution such as -7Q on surface ne... | Charges are free to flow inside a conductor. If there is an electric field inside a conductor, positive charges will flow with the field and negative charges against it. These displaced charges contribute their own electric field that quickly cancels out the original field. This is why we say there the electric field i... | {
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"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Residual symmetry group of a scalar field theory Given a Lagrangian
$$\frac{1}{2} (\partial_\mu \phi)^2 - \frac{\lambda}{4!}(\phi^2 - v^2)^2$$
for a real scalar field theory with $\vec{\phi} = (\phi_1,\phi_2,...,\phi_n)^T$ and $O(n)$ symmetry. Why is the residual symmetry group (or little group) given by $O(n\!-\!1)$ w... | O(n) means you may rotate any n-vector to any other of the same length, or a suitably normalized combination of others. So you make a choice to rotate your reference vector to say, $\phi_1=v(1,0,0,..,0)^T$.
Its little group rotating the n-1 components indexed by 2,3,...,n among themselves is thus O(n-1), and it has the... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/631562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Why force between charges increases when it moves, instead of decreasing? Imagine two positive charges in a space ship moving with a velocity,v with respect to an observer on earth.
according to the person in the spaceship,the electrostatic force between the charges is $F'=(\frac{1}{4π\epsilon_0})\times \frac{q_1q_2}{r... | Since the vector F is perpendicular to v it is not affected by the Lorentz transformation and it is the same in both reference frames.
You can yourself a big favour by using the [covariant formulation of electromagnetism][1]. The force is given by $$f^\mu = F^{\mu\nu} j_\nu$$ where $F^{\mu\nu}$ is the Lorentz covariant... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/631902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Is self-propulsion in Leidenfrost drops placed carefully on a highly heated solid surface possible? As is familiar to anyone who has inadvertently spilt a few drops of water onto a highly heated pan, rather than boiling away in a flash the water gathers itself up into globules and begins to "dance" over the surface as ... | A good guess might be symmetry breaking through randomly bursting vapor bubbles or vapor emitted at the sides of the drop, which would produce small fluctuating forces on the drop. The consequence might be Brownian motion, with smaller drops being more strongly influenced due to their lower inertia.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/632049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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