Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Why the exponential part of Electromagnetic waves on both side of interface must be equal? Why the exponential part of Electromagnetic waves on both side of interface must be equal which implies equality of phases at the boundary at all the time to satisfy the boundary conditions. My text book and all the textbooks say... | For simplicity, assume normal incidence at the point $r=0$, so that all the electric fields are parallel to the interface.
The E-field parallel to the interface is continuous (the same either side of the boundary).
$$ E_i \exp(i\omega_I t) + E_r \exp(i\omega_R t) = E_t\exp(i\omega_T t).$$
Now suppose that $\omega_I$, $... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/597022",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Accessible States in the Ergodic Hypothesis According to Wikipedia, the ergodic hypothesis is the assumption that
all accessible microstates are equiprobable over a long period of time.
My question is about the precise meaning of "accessible" here.
Consider the microcanonical ensemble for a thermally insulated system... | Generically, you could absolutely fix any number of conserved quantities you like. The energy need not even be among them. The energy is just the most common thing to fix because the second most common conserved quantity, momentum, is often not conserved due to boundary conditions in the systems that are common in stat... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Why is it that we ignore height difference when applying Bernoulli in an airfoil I learn physics myself and sorry if this is a very simple question
*
*Why is it that we can apply Bernoulli on above and below the plane even if the are not in the same streamline?
*Why do we ignore height difference when doing so?
Any... | Bernoulli shouldn't be used to try and explain the workings of an airfoil, as the NASA page I linked to clearly explains.
It's basically an 'urban myth' that the Bernoulli principle can explain airfoil lift; it isn't true.
| {
"language": "en",
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What does Art Hobson mean/explain in his article "There are no particles, there are only fields" regarding the double slit experiment? So I have been reading about fields in physics. I am reading Art Hobson's "There are no particles, there are only fields" published in The American Journal of Physics in 2013, and I am ... | The double slit experiment with presumed material particles, such as electrons, demonstrates that the inner reality of nature is represented by waves. In quantum field theory (QFT) the fields are regarded as most fundamental than particles, which are described as oscillations (quanta) of the fields.
Note: The nodes ref... | {
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"question_score": "2",
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Mathematical Definition of Power I am a high school student who was playing around with some equations, and I derived a formula for which cannot physically imagine.
\begin{align}
W & = \vec F \cdot \vec r
\\
\frac{dW}{dt} & = \frac{d}{dt}[\vec F \cdot \vec r] = \frac{d\vec F}{dt} \cdot \vec r + \vec F \cdot \frac{d\vec... | As others have already answered, $W = \mathbf F \cdot \Delta \mathbf r$ is a simplification and works only in a special case of constant $\mathbf F$. And so does your formulae.
One way to look at it physically is to recognize that work is not a function of position. Mathematically we usually describe it using the conce... | {
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Why is are induced electric field's non-conservative while static electric fields are conservative? I have learned that the $E$-field induced by changing magnetic flux, such as in 'motional emf', is non-conservative in nature.
I am also aware that static $E$-fields are conservative in nature.
What is the reason for thi... | A force feild is called conservative when it can be expressed in terms of a potential energy (or potential), as
$$
\mathbf{F}(\mathbf{x}) = -\nabla U(\mathbf{x}).
$$
Helmholtz decomposition means that any field can be expressed as a sum of a potential and a solenoidal components, i.e., as
$$
\mathbf{E}(\mathbf{x}) = -\... | {
"language": "en",
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Is $(L^2, L_z)$ a complete set of commuting observables? According to the main definition we define a (C.S.C.O.) complete set of commuting observables $(A,B,C, \dots)$ if:
*
*Every commutator between the operators of the list is $0$
*If we fix the eigenvalues of the operators there exists a unique eigenvector with t... | What you’re missing is to account for “complete”. In practice this means: do you have enough observables to uniquely label quantum states? In the case of $L^2$ and $L_z$, it will not be enough to uniquely label hydrogen states, or the states of a 3d harmonic oscillator, or for that matter the states in any 3d central... | {
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Dimension of matrix elements in scattering cross section This question is probably going to be somewhat trivial, but I am a little confused about the dimension of the matrix element that appears in the formula for the cross section of a scattering process.
I had always assumed that this kind of matrix elements should b... | We wondered the same thing with a group of friends and came to the conclusion that you are right.
As a confirmation, if you have a copy of Peskin and Schroeder, you can deduce from equation (4.74) that $|\mathbf{p_1} \dots \mathbf{p_n}\rangle$ has mass dimension $-n$, and going back to the definition of the matrix elem... | {
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Why does thermal conductivity of an alloy becomes nearly flat from $0.2I am trying to understand the effect of alloying on the thermal conductivity of an crystalline alloy. I have found a great many papers where I see thermal conductivity sharply decreases from 0 to 0.2 X of the alloy composition, then becomes nearly f... | bufferlab gives a good answer, but I'll dumb it down a bit. Since you tag phonons (and not electrons), I'm assuming that most of your thermal transport is due to phonons.
When $x$ is near 0 or 1, then you basically have a crystal with some impurities. E.g. if you're dealing with $Si_xGe_{1-x}$, then if $x$ is close to ... | {
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Pressure, entropy and enthalpy For an ideal compressible flow, this relation holds:
$$P = P(s, h)$$
where $s$ is the specific entropy and $h$ is the specific enthalpy.
I don't know why: I know that $s = s(e, v)$, but even using Maxwell relations, the density or the temperature is still inside the definition of $P$, so ... | In Thermodynamics, it is quite common to have some perplexity on the possibility of expressing state functions as a function of other state functions. It is a safe attitude since it is not true that everything can be considered a function of everything else. A counterexample is the case of $P=P(T,v)$ (here, I am using ... | {
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Derivation of curl of magnetic field in Griffiths Can someone please derive how $$\frac{d}{dx} f(x-x') = -\frac{d}{dx'} f(x-x')~?$$
In Griffiths electrodynamics, this is directly mentioned. I'm really confused, can someone elaborate!
| I think that this can come from this argument:
You can prove that $$\frac{\vec{x}-\vec{x}'}{|\vec{x}-\vec{x}'|^3}=-\nabla\left(\frac{1}{|\vec{x}-\vec{x}'|}\right)$$
If you take this change $$\vec{x}\rightarrow\vec{x}'$$ and $$\vec{x}'\rightarrow\vec{x}$$ the previous equality become $$\frac{\vec{x}'-\vec{x}}{|\vec{x}'-... | {
"language": "en",
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What is gravity's relationship with atmospheric pressure? I'm asking for clarification here. If Earth had the same atmospheric mass per square unit of ground but the Earth had suddenly gained mass so it had twice the gravity at the surface, would the Earth now have twice the atmospheric pressure just because of the dou... | Air pressure has everything to do with height because it is caused by the force exerted from the volume of air above weighing down.
That said, if Earth had twice the gravitational field strength at the surface, by the inverse square law this would mean that the Earth doubled in mass. This would double the gravitational... | {
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Why every system tends to be more stable? An entire topic of inorganic chemistry, i.e. chemical bonding, which is also one of the most important topics, is based on the idea of stability. But whenever I ask why every system tends to get stable, I never get a satisfactory answer. People say that's how nature "wants" to ... | Consider how many somethings there are in a mole. A whole lot, right?
Imagine you have a mole of something. And imagine that something has 3 states it can get into easily, and a lot of rare states. After awhile, statisticly it will be mostly in the common states. Because it's easy to get into them, and hard to get into... | {
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Why is stress defined in the way as it is? Stress is like pressure and it doesn't matter in which direction the force acts (given it is perpendicular to the surface).
I read in my book that if we have a rope which is being pulled on both sides by a force $F$ , then the stress at any cross section of the rope is define... | The standard definition is the correct one. You can see that by investigating some different configurations where stress is not constant.
First, consider the rope being pulled with only one force, let’s say only the force to the right. In that case the rope will accelerate to the right. The stress on the left side of t... | {
"language": "en",
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Nature of Work done by friction I have always been told that work done by friction can, at most, be zero, but never positive.
But consider two blocks placed one on top of the other, such that their surfaces in contact are rough. If we give the block on the top a certain horizontal velocity, then in crude words, we can ... | The energy conservation principle tells us that energy is never consumed or produced, just converted. And the very nature of friction is that it only converts from kinetic energy to heat, never the other way round.
If we ignore the heating part (as the sentence "work done by friction can, at most, be zero, but never po... | {
"language": "en",
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Formula for centripetal acceleration: simple proof that does not use calculus? I teach physics to 16-year-old students who do not know calculus and the derivates. When I explain the formula for centripetal acceleration in circular uniform motion, I use this picture:
Here,
$$\vec{a}_{\text{av}}=\frac{\Delta \vec{v}}{\D... | Your students might like this approach. It uses Newton's technique of modelling a smooth change as a series of sharp, sudden changes.
Consider a particle bouncing around the inside of a cylinder, following a path that is a regular polygon.
The change of velocity at each bounce, for example at C, is clearly
$$\Delta v=... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is it easier to handle a cup upside down on the finger tip? If I try to handle a tumbler or cup on my fingertip (as shown in fig), it is quite hard to do so (and the cup falls most often).
And when I did the same experiment but this time the cup is upside down (as shown in fig), it was quite stable and I could han... | Think of it like this: The center of gravity of the cup wants to be as low as possible, right? Consider your fingertip as a fulcrum point. So when the cup is above your fingertip, so is the center of mass, so the slightest tilt allows the cup to rotate so that the center of mass drops. When the cup is the other way, th... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/600066",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "31",
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On Poincare group’s Casimir operators We’ve defined Casimir operator for a group as an operator which commutes with all generators of that group. For the Poincare group we’ve found two Casimir operators: $p_\mu p^\mu$ and $W_\mu W^\mu$ where $W_\mu$ is the Pauli-Lubanski vector.
In checking that they are indeed Casimir... | Unfortunately, Lorentz invariant operators are not automatically Casimir operators - you can see this since there are essentially infinite independent Lorentz scalars you can construct from $M_{\mu\nu}$ and $P_\mu$, whereas the dimension of the Cartan subalgebra of the Poincaré group can be shown to be finite. An examp... | {
"language": "en",
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Galaxy Superclusters Okay quick question... why is it that when galaxies group together in superclusters they form a sort of branch-like shape? I personally would think they’d all just group together in some sphere or disk just like regular galaxies.. So what’s the physics behind the branching?
| galaxy clusters are formed where there was higher density of dark matter after the inflation epoch. The dark matter creates some gravitational wells in where the galaxy clusters are formed. These dark matter clusters are connected by lower density dark mater regions which may look like a branch. Some galaxies may have ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/600271",
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Engine rotating a generator at its rated speed, but engine power exceeds power required? What would occur if a gasoline engine would be driving a generator (specifically, a permanent-magnet synchronous AC 3-phase sinusoidal generator) as its only load at the output shaft and if that engine is rated at i.e. 35kW @ 8000 ... | Most gasoline engine powered small AC electric generators have a governor on the throttle to keep them running at a fairly consistant rate of revolutions per minute. This keeps the AC phase steady. When the generator has a load increase or decrease the governor will open or close the engine throttle accordingly to keep... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/600470",
"timestamp": "2023-03-29T00:00:00",
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Hamiltonian formalism of General Relativity Textbook I've been reading Wald's book on General Relativity and in appendix $E_{2}$ it discusses the Hamiltonian formalism of General Relativity.I would like to understand it more, can you recommend me a textbook about ADM formalism?
P.S.
I have already read Eric Poisson boo... | I can only point you to the summary by R. Arnowitt, S. Deser, C.W. Misner "The Dynamics of General Relativity", p. 227 from "Gravitation: An Introduction to Current Research", Editor Louis Witten, John Wiley & Sons, 1962.
from which you can see the original list of articles by the trio:
Arnowitt, R., and S. Deser, 1959... | {
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What quantum gates are needed to get the state $|01\rangle+|10\rangle$ from $|00\rangle$? I was wondering if I start with two qubits in the state
$$|00\rangle$$
If it's possible to apply gates to get it to the state
$$\frac{|01\rangle + |10\rangle}{\sqrt{2}}$$
I have tried applying the Hadamard Gate, Controlled X etc, ... | Applying Hadamard + CNOT takes you from $|00\rangle$ to $\dfrac{|00\rangle + |11\rangle}{\sqrt{2}}$. Now, just apply the single-qubit $X$ Pauli operator (which swaps $|0\rangle$ with $|1\rangle$ and vice-versa) to either one of the two qubits, and you get the target state $\dfrac{|01\rangle + |10\rangle}{\sqrt{2}}$.
| {
"language": "en",
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How does light look like when it is 180° out of phase? When two lights are 180° out of phase, would it look like this?
In the photo below, the left side is flipped and it is a mirror image.
| When people talk about phases of a light field, they usually refer to the electric field $E$. To understand why this is of interest, consider two light sources in 1D. Their (skalar) electric fields at the position $z_0$ are given by $E_1(z_0)=|E_1| e^{i\phi_1(z_0)}$ and $E_2(z_0)=|E_2| e^{i\phi_2(z_0)}$. The relative ... | {
"language": "en",
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Will the velocity be constant in a vertically oriented tube?
The tube here is placed into a water stream moving with a velocity $v$
My query is will the velocity be constant throughout the pipe or only in some places?
I have conflicting opinions on this, the equation of continuity would say the velocity is constant th... |
My query is will the velocity be constant throughout the pipe
No. Flow will slowdown going up due to the stopping gravity force of water column. Check this scheme :
Water will raise-up to the height $h$ only due to dynamic pressure. Neglecting atmosphere pressure, maximum height until water can go-up can be calculat... | {
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Why don't opaque objects reflect light? My sister was doing a quiz and I tried to point her in the right direction by giving her scenarios to imagine. One of the questions in the quiz was:
Which of the following objects do not reflect light:
*
*Polished metal
*Mirror
*Undisturbed water
*Book
She suggested that ... | In my opinion, one more thing which is possible here is that may be the question is
Which among the following causes diffuse reflection ?
Undisturbed water can also give specular reflections but the surface of a book have many microscopic as well as macroscopic irregularities which causes diffusion of the reflected ray... | {
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General plane motion and freely floating rigid body
Consider a rigid rectangular plate of length $l$, width $w$ and thickness $t$ which is at rest and is floating freely in space (no gravity). The center of the plate is at $O_L$ with respect to global coordinate frame $O_G$. The initial pose (position and orientation)... | The torque calculated from a point of an inertial frame (for example the origin $O_G$) is the time derivative of the total angular momentum: $$\tau = \frac{d\mathbf L}{dt}$$
And the angular momentum of the plate at a given time is:
$$\mathbf L = \int_v \mathbf r_G \times d\mathbf p = \int_v \mathbf r_G \times \frac{d\... | {
"language": "en",
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Is (net) force invariant in special relativity? I am aware that acceleration is not invariant under lorenrz transformations, but I was sure that the first postulate of special relativity implied that newton’s second law in its original form, F=dp/dt, where p is the relativistic momentum, was invariant. However, the fo... | The fact that Newton’s Second Law in the form
$$\mathbf F=\frac{d\mathbf p}{dt}$$
is relativistically form-invariant, meaning that in another inertial frame
$$\mathbf F’=\frac{d\mathbf p’}{dt’},$$
does not mean that force doesn’t transform under a Lorentz boost. It does transform, in the same way as the time derivative... | {
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Generalization of first law of thermodynamics What is the mathematical statement for the first law of thermodynamics, accounting for kinetic energy, potential energy, internal energy, work, heat and most importantly taking into consideration the work-energy theorem?
Also, is $∆U=∆Q-∆W$ only valid for systems whose cent... | The relationship you state is for a closed system with no change in center of mass kinetic or potential energy, viewed from an inertial frame. You also have to consider whether the system is open or closed. An open system can have mass flow in/out but a closed system cannot.
For the detailed relationships, see any go... | {
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Why can't dark matter lose energy by gravitational waves and collapse into itself? Because of lack of electromagnetic induction, dark matter can't lose its gravitational potential energy. That is preventing it from collapsing like an ordinary matter cloud in space.
But why can't dark matter lose energy by gravitational... | Dark matter was originally postulated to explain galaxy rotation curves, which needed an extended in space matter envelope to explain them. So it does collapse gravitationaly, but by the fact that its postulated extent is so large its effective density has to be low.
Evidently, if it is composed of particles , in order... | {
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Why does air pressure decrease with altitude? I am looking to find the reason: why air pressure decreases with altitude? Has it to do with the fact that gravitational force is less at higher altitude due to the greater distance between the masses? Does earth’s spin cause a centrifugal force? Are the molecules at higher... | As you go higher, there are less air molecules (less weight) on a given area this is basically one reason why it decreases.
From the barometric formula, one can get the relation between the pressure and altitude. It's defined as
$$P = P_{0}e^{-\frac{mgh}{kT}}$$
so the relation between pressure and altitude is $P\propto... | {
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Strong and Weak Interactions How do I determine whether an interactions is strong or weak if what i've been given is just the particles and the type of reaction:
Determine whether the following are strong or weak interactions?
*
*a muon decays
*a kaon decays
*pions are produced
| There are different arguments to use depending on the reaction. For instance :
*
*muon decay is due to weak interaction, leptons are not concerned by strong
*kaon decay, is also due to weak interaction, because kaons have a strange number equal to $+1$ while its decay products have no strangeness at all
*if pions a... | {
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Direction of dipole moment? Why is the direction of dipole moment taken from negative to positive charge when the field is directed in the opposite direction ?
| It actually does not matter as long as you keep track of the signs. It was probably defined this way in order to give a similar formulation for torque
$$\vec{\tau} = \vec{r}\times\vec{F}$$
dipole moment is defined as $\vec{p} = q\vec{d}$ and torque on a dipole
$$\vec{\tau} = \vec{p}\times\vec{E}$$
If it was defined oth... | {
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Geometrical linearization in continuum mechanics In continuum mechanics, we often make use of "physical and geometrical linearization", e. g. during derivation the Navier-Cauchy equations (c. f. https://en.wikipedia.org/wiki/Linear_elasticity). I understand that physical linearization refers to the assumption that the ... | Imagine two infinitesimally close points $a$ and $b$, the difference between their squared distances before and after the deformation can be measured by $((I+H)(b-a))^T (I+H)(b-a)-(b-a)^T(b-a)$, where $I$ is the identity matrix. So it is possible to think of the nonlinear strain tensor as $(I+H)^T(I+H)-I$. The accurate... | {
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Is it the gravitational field created by an object which exerts force on another object or the interactions between their fields that exerts force? We are taught that Gravitational force exerted by an object is a two-step process:
*
*The object creates a field around it.
*The field exerts a force on bodies present i... | Here is the wikipedia article on classical fields that will help clarify:
A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations. The term 'classical field theory' is commonly reserved for describing those physical theories that describe... | {
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Why is the net force acting on a massless body zero? I know that massless bodies can accelerate (in theory) even with the net force equaling to zero. But, why cannot there be a net force on a massless object? Why does it always have to be zero as a resultant in the end?
I'm talking about object whose mass is assumed to... | In truth, the answer to your question is that no massless objects exist.
Sometimes an object's mass is negligible compared to something else - such as strings carrying weights in a pully system, or such as a human being on the surface of the earth - and then we can safely ignore it in our calculation with no significan... | {
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Did I make an electric circuit with my cat? This is definitely the dumbest question I've asked, but I'm curious. My cat was resting her head on my left hand while I pet her with my right. Suddenly, my left hand began receiving regular static electric shocks. Was I pulling electrons from her fur with my right hand, thus... | Yes, you made a circuit, although, due to the bad conductivity of the contact between your left hand and the fur of your cat, the current was not continuous but concentrated in the sparks.
It was a circuit because your right hand was acting as a generator, where the electromotive force was due to the triboelectric effe... | {
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Parity transformations and Dirac Spinor I'm reading "No-Nonsense quantum field theory" and I have some doubts about the transformation law for Dirac Spinors as explained by the author. In the book the left chiral spinors $\chi$ and right chiral spinors $\xi$ are introduced as objects that have two components and behave... | Under parity in spherical coordinate we have,
$$\textbf{P} \theta = \pi - \theta \\ \textbf{P} \phi = \pi + \phi $$
This explains why ,
$$ B^{(\xi z)} (\phi) \rightarrow B^{(\chi z)} (\phi)$$
Now we need to know what do you mean by a left-chiral spinor. Left-chiral spinor is an object which transforms like this boost,
... | {
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If i wrote something in the sky what would the radius of visibility on the ground? If i wrote something in the sky 8500-10,000 ft high 40 ft tall letters what would be the visibility radius on the ground? Or what formula would i use to come up with the answer?
| The size of the letters and their distance will be proportional to size and distance where they are to be calculated. This is arrived by calculating angle those letters subtend at eye.
Let say if their visibility radius is to be calculated at 1 ft distance then:
$$\frac{sz}{1 ft} = \frac{40 ft}{10000 ft}$$
=> $$sz = .... | {
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Does the speed of light in our definitions take vacuum energy into consideration? We know that the speed of light decreases as it goes through a medium, and we also know that there is a certain vacuum energy that creates a sea of particles coming in and out of existence, which, naively, one could assume has an effect o... | The constant $c$ is commonly referred to as the speed of light, but really it's better interpreted as a conversion factor between distance and time units, or a maximum speed of cause and effect. The speed $c$ is the only one that observers in all frames agree on. Therefore if we found that the speed of light was not $c... | {
"language": "en",
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Can the gravity of one entire galaxy slow down an astrophysical jet emitted from its central black hole? Let say we are talking about our Milky way and Sagittarius A* emitted a astrophysical jet. If the galaxy was 150 000 ly wide would its gravity with dark matter included eventually slow down that jet keeping in mind ... | No.
These jets are jets because they are faster than the escape speed of the various relevant objects (central black hole / Active Galactic Nucleus / host galaxy). Thus, gravity is not relevant in slowing it down. For order of magnitude, the escape speed of the Milky Way galaxy is about 650km/s, but these jets are rela... | {
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Why is the Pauli exclusion principle not considered a sixth force of nature? Why is the Pauli exclusion principle not considered a sixth force of nature, given it produces such things as repelling of atoms and molecules in solids?
| The Pauli exclusion principle as understood nowadays, is a consequence of the spin-statistics behavior of fermions. We know that Spin representations are related to wave-functions with specific commutation relations which enforce anti-symmetry (all of this encoded in the Dirac-equation, which already accounts for spin)... | {
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What would happen if a 10-kg cube of iron, at a temperature close to 0 kelvin, suddenly appeared in your living room? What would be the effect of placing an object that cold in an environment that warm? Would the room just get a little colder? Would it kill everyone in the room like some kind of cold bomb? What would h... | The heat capacity of iron at room temperature is 0.444 J/K per gram (it changes with temperature, but let us leave that aside). That means it will want to absorb about 1,332,000 J of heat. That is a lot, but a bathtub with 300 litres of 40 °C water (about 10 degrees above a 300 K) will have about 12,900,000 J of intern... | {
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Understanding dependent/independent variables in physics How does one determine the independent and dependent variables?
What do the terms mean?
Can they be derived from a formula?
For example I saw in a textbook $F = k\Delta l$, Hooke's Law, that $F$ is the independent variable. Is this because $\mathbf {F} $ is the s... | In an equation there is no inherent distinction between dependent and independent variables. There are to my knowledge only two contexts where the distinction makes sense.
Experimental: In an experimental context the independent variable is the one that the experimenter is controlling in the experiment. It is the treat... | {
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Will this spaceship collide with the star? (time dilation) I thought of the above thought experiment and arrive on 2 conflicting conclusions. I can't seem to identify the flaw in my reasoning.
Suppose there is a star 4 light years from earth that has will explode and turn into a white dwarf in 3 years (as measured in t... | It is a synchronization problem. When the two observers synchronize their clocks at earth (Let us assume the traveler accelerates almost instantaneously so it reaches full speed still at earth), the star that is current for the traveler is in the future for the person on earth. So for the traveler, the time left to exp... | {
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Does Work become state function in an Isothermal Process and what are other processes in which it happens? In a reversible isothermal process and for an ideal gas we know from the definition of Helmholtz free Energy
$dF= -SdT -PdV$.
And as temperature doesn't change for an isothermal process, $dT$ must be zero. So dF c... | The equation dF=TdS-PdV applies only to two closely neighboring (i.e., differentially separated) thermodynamic equilibrium states, where P is the pressure calculated from the (equilibrium) equation of state (e.g., the ideal gas law) for the fluid. In an irreversible process, even if the boundary of the system is held ... | {
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Fugacity of Bose-Einstein Condensation I'm studying Bose Einstein Condensation.
In the book "Huang K Statistical Mechanics 2 edition", page 288, the author gets the following result for the fugacity ($z$) as a function of temperature and specific volume (lambda is the thermal wavelength and small $v$ the specific volum... | I am assuming you don't know how to get from 12.41 to the second equation of 12.52?
If you start from 12.41:
$$ \frac{1}{v} = \frac{1}{\lambda^3}g_{3/2}(z) + \frac{1}{V}\frac{z}{1-z}, $$
and take the infinite volume limit $V\rightarrow \infty$, so that the $1/V$ term above goes to $0$.
Then, you are left with:
$$ \frac... | {
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What does Gibbs Energy represent physically? I was reading about Thermodynamics in Chemistry.
I know that Enthalpy change at a constant pressure by a system is actually released as heat by the system.
My question:
What exactly does Gibbs Free Energy and its change represent physically (on a similar note to what Enthal... | The Gibbs Free Energy represents the energy that is free to do useful work for a spontaneous process. In other words, it is the max work done by a process (at constant $T$ and $P$). The Gibbs free energy can tell us whether a process will proceed spontaneously or not.
The Gibbs free energy is not magical. Instead it is... | {
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Are we seen or not by an observer from a galaxy farther away than the age of our solar system? I looked at this question. There's one answer that alludes to what I'm asking, but I don't find it satisfactory, because to me the following is still a paradox:
The most distant galaxies we have ever observed appear to be ar... | Because, not accounting for the expansion of space, we are seeing what was in their galaxy 10 billion years ago if they are 10 billion light years away, we do not know what it looks like right now. If observers there are looking towards our part of space right now they will see what was here 10 billion light years ago,... | {
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Interpretation of Hooke's Law I often see people interpreting Hooke's Law $σ=Eε$ as,
"The deformation $ε$ that occurs when you subject a material to a stress $σ$."
This makes it sound like stress is an external stimulus that causes the material to deform. But from what I know, stress is an internal phenomenon, not an... | Hooke's Law—in the standard form as you've written it—says that the normal stress $\sigma$ and normal strain $\varepsilon$ are linearly coupled by a constant of proportionality $E$ (termed Young's modulus). This is a good approximation for a long rod of a stable, constant-temperature solid for small axial deformations ... | {
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Why do we need instantaneous speed? I am new to this topic and was just wondering about the use of instantaneous speed. I mean, we use to calculate the speed of car let us say at 5 sec. So we take the distance travelled in 4.9 to 5.0 seconds and divide it by time. We get instantaneous speed. We could simply as well hav... | One way of looking at it is that instantaneous speed gives you more details about your journey, especially when your journey consists of variable speeds. The smaller the time interval in which you measure, the more information you have about your journey. But if your speed were constant, instantaneous speed would make ... | {
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What causes water droplets to drop in periodic, but not uniform time intervals? A little while ago I noticed water droplets forming from a slightly overflowing reserve in my sink. They dropped in a special periodic time pattern, which was not uniform. Instead two droplets would be created and fall right after each othe... | The droplet formation process depends on gravity, surface tension, the nozzle diameter, and any velocity that the water has before it begins forming up into a droplet at the nozzle tip. When one droplet breaks off the nozzle tip, it tends to leave behind a velocity field in the water right next to the nozzle tip which ... | {
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Trapped Radiation inside a Faraday Cage Say you have a Faraday cage mesh with holes small enough to reflect the radio waves. What happens to radiation when it is emitted from inside the Faraday cage?
Does it keep reflecting off of the inside of the cage forever?
| No. Unless the cage is a perfect conductor (which it is impossible in real life to make or find), a portion of the energy of the waves will get absorbed by the walls of the cage on each reflection, eventually causing the wave to attenuate or "die out". Given the incredible speed of light, these reflections happen close... | {
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Proper terminology for comparing decibels Decibels are a log-scale rather than linear unit, so for example 10 dB + 10 dB is about 13 dB, not 20dB. What then would be the proper terminology for comparing decibels?
For example, how would we describe the relationship between 13 dB and 10 dB? Would we say that "13 dB is 3 ... | As mentioned by The Photon in the comments, you add decibels according to normal arithmetic, so 10 dB + 10 dB = 20 dB. However, you need to be careful with what that means. Saying that 13 dB is 3 dB louder than 10 dB signifies that the pressure amplitude of the 13 dB signal is roughly twice that of the 10 dB signal.
... | {
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Help proving bound on POVM measurement probabilities I am trying to follow Nielsen and Chuang's 1 proof that the difference in measurement probabilities is bounded by the difference between two unitary operators applied to a given state.
Can someone show me how to get from Equation 4.66 to 4.67 in the proof below (see ... | $$|\langle \psi| AB \phi\rangle | \leq ||\psi|| \: ||AB \phi|| \leq ||\psi||\: ||AB|| \: ||\phi|| \leq ||\psi|| \: ||A||\:||B||\: ||\phi||$$
In our case $||\psi||=1$ and $||A||, ||B|| \leq 1$.
because one of $A$ and $B$ is unitary and the other is part of a POVM.
| {
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Where does the law of conservation of momentum apply? Take the scenario of a snowball hitting a tree and stopping. Initially, the snowball had momentum but now neither the snowball nor tree have momentum, so momentum is lost (thus the law of conservation of momentum is violated?). Or since the tree has such a large mas... | The ground applies a frictional/constraint force , so there is a net external force on the system and hence, Conservation of momentum can't be used on the snowball and tree as a system.
| {
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Can thermodynamics be considered logical? One of the laws says that heat won't flow from cool to warm and at the same time this same theory claims that there is a finite (albeit tiny) chance that it will, because there is always such a microstate.
We can also have a situation where all air molecules in the room can be ... | You need to distinguish between the net transfer of energy in the form of heat between two bodies and the energy transfer that can occur between the individual molecules of the two bodies.
Take the simple example of heat transfer by conduction. Body A is placed in contact with body B where $T_{A}>T_{B}$ prior to contac... | {
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How does a deadcat work? This is a deadcat. It's a fluffy cover you put on top of a microphone to prevent wind noise on the recording. Lavalier and head-mounted microphones often use pieces of foam over the mic for the same reason. But how? I can't come up with any reasonable physics justification for why it would elim... | Wind noise is generated when wind hits a surface. The rigid structure of the surface resonates at particular frequencies, generating the noise. A fluffy cover or foam has no resonant frequencies, so does not generate noise, and at the same time shields the microphone itself from the wind. Likewise the fluffy cover mini... | {
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What experiment confirms $\mathbf{J}^2 = \hbar^2 j(j+1)$? I learned that if we measure the spin angular momentum of an electron in
one direction $J_z$, we get $\pm \frac{1}{2} \hbar$. But if we measure
the magnitude of the angular momentum $\mathbf{J}^2$, we should get
$\frac{3}{4} \hbar^2$. What experiment gives the... | It is true that $J_x$, $J_y$ and $J_z$ does not commute.
But $J_z$ does commute with $\textbf{J}^2=J_x^2+J_y^2+J_z^2$. So there can be an eigenstate of both $J_z$ and $\textbf{J}^2$, where you can simultaneously measure both the z component of the angular momentum and the magnitude of the angular momentum.
If the possi... | {
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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 ... | Saying that the object balances on a point, suggests to me that the point is on the perimeter of the object (lke a corner). Given that, then the object can be in an unstable equilibrium on any extruding point as long as the point is below the center of gravity. If you are going to drill a hole and insert a axle, then ... | {
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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... | Good question! $\vec{E}$ is the total field. i.e. the external field in the absence of the dielectric plus the field due to polarisation of charges in the dieletric.
Source: Griffiths, Introduction to Electrodynamics 4Ed., p181. (Griffiths actually says the field due to anything except the polarisation, which could ... | {
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What does it mean that a neutron has a 'negative' magnetic moment? Most questions about this ask why, or how, a neutron has a magnetic moment at all, or why it is negative....
But I am curious as to what it means, physically or experimentally, for a magnetic moment to be 'negative'....
I am reading that a neutron's 'an... | It means that a neutron’s magnetic moment vector is in the opposite direction from its spin angular momentum vector.
| {
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Will speed of light in any denser medium will be same for all the inertial frame in that medium? As we know the speed of light in vacuum is constant for everyone (at least for all inertial frames), i.e., if we run away or toward a beam of light (in vacuum) the speed would be $c$. It doesn't change.
But imagine a beam o... | If we assume linear media, then it is easy enough to transform the E- and D-fields and show that permittivity is not a relativistic invariant.
Thus the refractive index of a medium is not relativistically invariant and observers in different inertial frames measure a different speed of light in a medium.
For example, i... | {
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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... |
So too much frequency and you do lose the ability to decipher it and it starts to just sound like nosie?
If that happens, it's because of how your brain interprets the signals that it receives from your ears, and not because of the physics of how your ears work. If the total sound pressure level is not so great as to... | {
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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 ... | You use "absorbed" in the title, and "reflected" in the text, so I'm not clear on what your conception of the process is. In fluorescence the incoming photon is completely destroyed, and a new photon of lower energy is generated. This is evident because the incoming light is typically blue or ultraviolet, and the emi... | {
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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 ... | No, you can't. Here is why:
When a loudspeaker is producing sound, it is pushing forward to produce the compression part of the wave in air, then it pulls backwards to produce the rarefaction part of the wave, then forward, then back, etc. This means there is no net force applied to the air from the cone and no net rea... | {
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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... |
When a flat iron or Alnico washer is magnetized one of the faces develops a north polarity and the other, south.
That's one way that a flat washer could be magnetized. But, it also could be magnetized in other directions (E.g., with one edge north and the opposite edge south. Or, it could be magnetized with several a... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/607796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How can magnet field go through non-transparent medium? We know that the change in the magnetic field is carried on electromagnetic waves so when magnets move relative to each other they radiate. But if the medium is not letting em cross in some length then the change in the magnetic field is blocked which means they w... | Normally, electromagnetic response of media is frequency-dependent, e.g. gamma-particles and x-rays can go through most things.
You are interested in magnetic field due to moving magnets - that would be very low frequencies: hertz, milli-hertz etc. I guess a lot of media are transparent at those frequencies.
One needs ... | {
"language": "en",
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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... | Total angular momentum with respect to the center of mass, in practice the center of Earth, is conserved. When the airplane takes off, it acquires an angular momentum antiparallel to the Earths rotation axis. The total angular momentum of Earth plus atmosphere increases by the same amount. This angular momentum will ev... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/608372",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "35",
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"answer_id": 11
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When we use Lorenz gauge or Coulomb gauge, the result formula for electric $E$ and magnetic field $B$ is same or different? Gauge condition can be chosen as you like or not?
is the Lorenz gauge is the only one correct? If Coulomb gauge can obtained exactly same results as Lorenz gauge for the electromagnetic fields E ... | Gauge condition is any human imposed restriction on the functions $\varphi(\mathbf x,t), \mathbf A(\mathbf x,t)$ that does not change electric and magnetic field implied by those potential functions.
For any single physical situation, one can use either potential functions obeying the Coulomb gauge condition or those o... | {
"language": "en",
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Principal quantum number of the classical particle The example 7.9 in this page shows the principal quantum number of the classical particle.
A small 0.40-kg cart is moving back and forth along an air track
between two bumpers located 2.0 m apart. We assume no friction;
collisions with the bumpers are perfectly elasti... | In the link you give it says "as though the cart were a quantum particle", so to ask:
If the cart moved very very slow, can we find the cart at other place
is to ask if the kinetic energy is very very small: "can it behave as a true quantum particle".
In the link they answer using the "bohr correspondence principle".... | {
"language": "en",
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Finding an exact value for energy in perturbation theory
Supose a particle of mass $m$ and electric charge $q$, subject to harmonic potential in 1D, is placed in an area with electric field $\vec E = E \hat u_x$. Determine the exact change in its energy spectrum caused by interacting with this field.
I started by wri... | This isn't intended as a perturbation theory problem. (It actually can be solved, to all orders in perturbation theory, but that would be unthinkably arduous.) The actual point is the notice that the potential, including the linear potential due to the electric field, is still a quadratic function of the position,
$$... | {
"language": "en",
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Finding contradiction in equations Question:- In the figure shown, Coefficient of friction between the blocks C and B is $0.4$. There is no friction between block C and ground.The system of blocks is released from rest in the shown situtation.Find the accelerations of masses.(Given $m_{B}=5kg , m_{C}=10kg , m_{A}=3kg$ ... | There should be different between static frition coefficient $\mu_s$ and the kinetic friction ceofficient $\mu_k$, and $\mu_s$ must be greater than $\mu_k$. If $\mu_s \lt \mu_k$, there will induce non-physical phenomena.
Another concept is that the kenetic friction (as long as two badies have relative motion) is a cons... | {
"language": "en",
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Newton's 3rd Law and a bouncing ball just want to clear something up.
Take a ball that's been dropped to the ground. Gravity acts and this ball as it has mass and then the ball now moves to the ground with a constant force of say ($X$).
Now when the ball makes contact with the ground, Newtons 3'rd Law takes effect (no ... |
Now here's where I get confused.
If the ball (which has a constant force when it hits the ground ($X$))
experiences the same constant force in the opposite direction ($-X$,
minus indicating opposite direction), then the total force acting on
the ball should be net ZERO ( $X + (-X) = 0$).
Yes the ball and the floor (E... | {
"language": "en",
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What is the meaning of $F=ma$? Does it mean that an object with mass $m$ will have an acceleration $a$ if an external force $F$ is applied to it? I know this is a very simple question, but I am just learning physics. I am seeing the basics of how a block on a horizontal frictionless surface gets accelerated by a force ... |
However, I ask myself if F=ma is only used on the object the force is
being applied to, not the source of the force (which could be a
finger, another block, or anything that pushes)
$F_{net}=ma$ is Newton's second law. Newton's third law says that the object the force $F$ is applied to exerts an equal and opposite fo... | {
"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... | Yes. This is how regenerative braking works.
There is a style of brakes that is used in many electric or hybrid cars that, rather than converting the motion of the wheels into heat energy the way conventional brakes do, instead convert the motion of the wheels into electrical energy that recharges the car's battery. By... | {
"language": "en",
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"source": "stackexchange",
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Centrifugal Force Dilemma While learning Classical Mechanics, I am confused on nature and application of centrifugal force.
In my textbook, it is written that centrifugal force is a pseudo force that, depends on reference frame, but I can't understand that if it is pseudo force then why we feel something pushing us out... | It is not you who is moving outwards and squeezing into the car door. It is the car door which is moving inwards into you.
Your inertia makes your body tend to keep moving straight.
Just like when standing in a bus that brakes - you feel pushed forwards, but in fact it is the bus that is being pulled backwards undernea... | {
"language": "en",
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Prove $\exp (-i \phi(\hat{n} \cdot \vec{\sigma}))=\cos \phi-i(\hat{n} \cdot \vec{\sigma}) \sin \phi$ using certain property I'd like to prove $$e^{-i \phi(\hat{n} \cdot \vec{\sigma})}=\cos \phi-i(\hat{n} \cdot \vec{\sigma}) \sin \phi$$ using $$\sigma_{i} \sigma_{j}=\delta_{i j} I+i \varepsilon_{i j k} \sigma_{k},$$ whe... | Your relation dictates $(\hat{n}\cdot \vec{\sigma})^2=I$, hence, for integer m,
$$
(-\hat{n}\cdot \vec{\sigma})^{2m}=I, \qquad (-\hat{n}\cdot \vec{\sigma})^{2m+1}= -\hat{n}\cdot \vec{\sigma}.
$$
So, what is
$$
\exp \Bigl ( i\phi (-\hat{n}\cdot \vec{\sigma}) \Bigr )= \cos \Bigl ( i\phi (-\hat{n}\cdot \vec{\sigma}) \Big... | {
"language": "en",
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Why do we need to introduce thermodynamic potentials? Each thermodynamic potential (Enthalpy, Helmholtz free energy, Gibbs free energy) is the same as the 1st law of thermodynamics. Then, why do we need them? Why did people introduce them in the first place?
| Each potential describes a different physical situation, i.e. they differ by the controlled variable of the experiment being described by them.
Take for example the Helmholtz free energy. Note that
$$
dF=-PdV-SdT
$$
so F is a function of volume and temperature: $F(T,V)$, since it follows from the equation that small va... | {
"language": "en",
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Quantum Harmonic Oscillator Virial theorem is not holding I'm asked to calculate the average Kinetic and Potential Energies for a given state of a quantum harmonic oscillator. The state is:
$$
\psi(x,0) = \left(\dfrac{4m\omega}{\pi\hbar}\right)^\frac{1}{4}e^{\frac{-2m\omega}{\hbar}x^2}
$$
The thing is, calculating
$\la... | The ground state of the harmonic oscillator is (see Wikipedia for example): $$\psi_0(x) = \left(\frac{\alpha}{\pi}\right)^{1/4} e^{-\alpha x^2/2},\quad \quad \text{where }\quad \alpha =\frac{ m \omega}{\hbar}$$
Your math is correct, it's just that the state you have is not a bound state of the harmonic oscillator, the ... | {
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Why does a wooden spoon creates bubbles when put in hot oil? This question might be a bit weird, but I just asked myself why a wooden spoon creates bubbles when put in oil at about 170°C. My idea is, that the water in the spoon reacts with the Oil, but why does this just starts to happen when the Oil reaches approx 170... | Wooden spoons are porous, and have oil, water, and air embedded in their surfaces after being used for a while.
Putting a spoon like this into hot oil will boil the water in the surface, and expand any air there, thus creating bubbles of water vapor mixed with air.
If you then allow the oil and the spoon to cool off to... | {
"language": "en",
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How warm are radioactive metals? I read that radium is warm to the touch -- is that because of actual heat or is that because, for example, the radiation it emits creates the sensation of warmth? How high of a temperature can a radioactive element or isotope actually have?
| The interesting number for application in radioisotope thermoelectric generators is the power by weight. It is not very high for radium, as the half-life is 1600 years. The Pioneers use generators containing 13 kg of $^{238}$Pu with a halflife of 88 years. That has a power density of 0.54 W/g.
The linked wikipedia arti... | {
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Meaning of adding a term to the Hamiltonian in a quantum harmonic oscillator Let $H$ be the Hamiltonian in a harmonic oscillator,
$$
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... | Well the eigenstates would change, but not all of them; since if $H'=H+V$, $H|n \rangle= H'|n \rangle$ for all $n>1$, where $|n \rangle$ is eigenstate of the original QHO with energy: $E_n = \hbar \omega \big(n+\frac{1}{2} \big)$. Now in the case of $n\leq 1$:
$$H'|0 \rangle=E_0|0 \rangle+ \sqrt2 \hbar \omega |1 \rangl... | {
"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... | "let's gently place a block of mass 3 kg"
Maybe you should rethink the system. You cannot "add stuff" just like this. Energy is conserved in an isolated system. Adding a block just like this is not what I would consider as isolated at all.
Put the two blocks (or one) and then tell us what you want to do.
| {
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"source": "stackexchange",
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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 ... | As a real everyday physical model, consider a non-damping pendulum: ($\ddot{\theta} + \frac{g}{l}\sin\theta = 0$). It does not have a general analytical solution (Wikipedia) and yet it is not chaotic.
| {
"language": "en",
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"source": "stackexchange",
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How to calculate the heat that leaves the furnace through one opening? I recently read in the newspapers that one man died through the flame that left the iron furnace. I wasn't sure about this. Iron is melting at 1500 °C. The furnace they used is 8 cubic meters, he was standing one meter from the furnace. The opening ... | $Q=m×c×ΔT$ is not useful here.
What you can do is apply Stefan - Boltzmann (as an approximation), which tells us that:
$$P=A\varepsilon \sigma T^4$$
where:
*
*$P$ is the power emitted by the grey body radiator,
*$\varepsilon$ is the emissivity, which we can approximate as $1$,
*$\sigma$ equals $5.670373\times 10^{-... | {
"language": "en",
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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... | Just as the motion of body around the earth is ellipse (1st Kepler law replacing sun by earth), so is the motion of a projectile. Notice that almost everything we deal is an approximation, the earth is not a massive perfectly rounded ball, and we are neglecting the air, so it is not a sin to consider the motion of the ... | {
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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... | Depending on the type of container, there’s another risk: condensation and ice build-up. All containers need to be vented, but there’s a reason why the vents are designed to be one-way (nitrogen gas goes out, air can’t freely get in). If left open to air, the air will condense on surfaces near the liquid, and ice will ... | {
"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?
| Imagine a totally perfect clock. It measures the time in the place where it is situated. Its own proper time. But due to general relativity the speed of time is affected by nearby objects, gravitational fields, spacetime curvature, speed of motion and gravitational waves.
The ideally perfect clock would look not that p... | {
"language": "en",
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Momentum of a relativistic atom I have been solving a problem where you should derive the formula for doppler effect when a source of light is an atom moving at a relativistic velocity v. I understood everything in the solution except for why was the formula for impulse of the atom used as if it's not relativistic i.e.... | There is nothing wrong on writing the relativistic momentum as $p = mv$ as long as you are saying that $m = \gamma m_0$, where $m_0$ is the rest mass (inertial mass). In other words, the equation $p = mv$ is relativistic, the Lorentz factor is simply compacted onto the mass, meaning that the mass you are using to calcu... | {
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Is conformal transformation a coordinate transformation or not? I have been looking through several questions and answers on conformal transformation in the stack exchange community. While they have helped me gain a better understanding of the basic issues, I am also somewhat confused. Here are couple of them:
*
*Whi... | Conformal transformation/mapping is a term from the complex analysis. In its original sense it is indeed a coordinate transformation:
The conformal property may be described in terms of the Jacobian derivative matrix of
a coordinate transformation. The transformation is conformal whenever the Jacobian at
each point i... | {
"language": "en",
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Least count of cesium clock and maximum possible significant figures for time We know that a second is defined as being equal to the time duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the fundamental unperturbed ground-state of the caesium-133 ato... | Your conclusion is not correct. Current atomic clocks based on optical transitions are much more accurate and precise than that. So time durations less than 1/9192631770 s can indeed be accurately measured.
The kernel of truth to your conclusion is that such time durations cannot be accurately known in SI units. Howeve... | {
"language": "en",
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Is there a difference between instantaneous speed and the magnitude of instantaneous velocity? Consider a particle that moves around the coordinate grid. After $t$ seconds, it has the position
$$
S(t)=(\cos t, \sin t) \quad 0 \leq t \leq \pi/2 \, .
$$
The particle traces a quarter arc of length $\pi/2$ around the unit... | By definition, $$\left|\text{instantaneous velocity}\right| = \text{instantaneous speed}.$$
However,
\begin{aligned}
\left|\text{average velocity}\right| &= \left|\frac{\text{displacement (i.e., change in position)}}{\text{time elapsed}}\right|\\
&= \frac{\left|\text{displacement (i.e., change in position)}\right|}{\te... | {
"language": "en",
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Explosion of an asteroid in orbit I've learned from an answer here on this site that if a body were to split apart in orbit the center of mass will continue to be on the same orbit. (Couldn't find the post)
But let's say an asteroid is to blow up in two pieces such that the smaller piece reverses its direction and velo... | If a body splits in orbit, the center of mass of the body won't necessarily be in the same orbit, before and after, even though the sum of angular momentum of the pieces will be conserved. As an example, consider an asteroid broken into 2 pieces which both have escape velocity in opposite directions. The center of ma... | {
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Why the formula of kinetic energy assumes that the object has started from an initial velocity of zero? According to my physics textbook, the formula of kinetic energy is:
$$
W = \frac{1}{2}mv^2
$$
Where $m$ is is mass of the object and $v$ is the velocity of the object. The equation is calculated from this (according ... | Formula $W=Fs$ is the formula for work, that is the energy transferred to the body by the force. It is the difference of kinetic energies between the final and initial state. If you're starting from $v=0$ and assume $E_{kin}(0)=0$, you have
$$
E_{kin}(v)
= E_{kin}(0) + ma \cdot \frac12at^2
= 0 + \frac12 mv^2
$$
If you ... | {
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Superposition of eigenstates in statistical mechanics Consider the simplest case in quantum statistical mechanics, where we find the density of states in the case of a cuboidal 3 dimensional box. In the derivation we take only those states which are product seperable into wavefunctions along the three directions i.e. c... | The point is not that there's a priori something special about separable states compared to non-separable states, the point is that you'd like to sum over a complete set of states -- that is, a basis for the Hilbert space. The separable states you describe happen to be particularly convenient states, because they are e... | {
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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(... | You are indeed confusing eigenket and eigenfunction. The eigenkets are the possible forms your system takes after a measurement, they are the eigen vectors (eigenkets) of the measurement operators (ex $\hat{H}$) you choose to apply on your eigenfunction (your system). The eigenfunction of a system can be expressed as ... | {
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"source": "stackexchange",
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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... | Ensemble is many systems, evolving from different initial conditions. Ergodicity, an important assumption in statistical physics reasoning, is that time averaging (for a single system) is equivalent to the ensemble averaging (over many systems.)
Note that the same definition of ensemble is used in quantum mechanics, wh... | {
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"answer_id": 1
} |
Infinite acceleration without loss of energy in a vacuum, gravity-less void? Suppose we have a void that is free of any other objects or forces, even gravity (impossible I know but hypothetically). If one were to propel an object forward, would that object be able to accelerate infinitely without requiring further addi... | A fixed amount of energy will not lead to an object moving faster and faster. A fixed amount of energy would give the object a fixed velocity and that's that.
A fixed force that kept being applied would make the object move faster and faster. But the longer the time interval, the greater the total energy needed to appl... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/614306",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Why is the magnetic field $B$ a pseudo-vector? Physically speaking, "pseudo-vectors" are vectors $v\in \mathbb{R}^3$ which transform as $ v'= (\det {R})v$ if the "system were to transform as $R\in O(3)$". However, what does this mean mathematically? And in particular, why is the magnetic field $B$ a pseudo-vector?
I wo... | I don’t know the math jargon, but the magnetic field is a pseudo vector because it is determined for a given configuration of moving charges by the right hand rule. This means that if you look at a mirror reflection of a physical setup with the B vector drawn in, it will be reversed. Angular momentum is another example... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/614650",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
D'Alembertian of a delta-function of a space-time interval (i.e. on the light-cone) How one differentiates a delta-function of a space-time interval? Namely,
$$[\partial_t^2 - \partial_x^2 - \partial_y^2 - \partial_z^2] \, \delta(t^2-x^2-y^2-z^2) \, .$$
Somewhere I saw that the result was $$4\pi\delta^{4}(x).$$ However... | Using the theory of generalized functions, let us regularize the Dirac delta distribution$^1$
$$ \delta(x^2)~=~\lim_{\varepsilon\searrow 0^+}\delta_{\varepsilon}(x^2), \qquad x^2~:=~x^{\mu}\eta_{\mu\nu}x^{\nu}, \tag{1}$$
via a smooth $C^{\infty}$-function$^2$
$$ \delta_{\varepsilon}(x^2)
~=~ \frac{1}{\pi}\frac{\varepsi... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/614754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
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