Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [classical-electrodynamics]

Classical electrodynamics is the discipline that studies electromagnetic phenomena – such as electric and magnetic fields, radiation, and the dynamics of charged bodies – in classical terms.

0
votes
1answer
58 views

A changing magnetic field passes through a wire loop but the loop itself is not in the field. Is an EMF induced in the loop? [duplicate]

$$\nabla \times \vec{E} =-\frac{\partial{\vec{B}}}{\partial{t}}$$ Applying Stokes' theorem: $$\oint_{loop} \vec{E} \cdot d\vec{l}=\int_S -\frac{\partial{\vec{B}}}{\partial{t}} \cdot d\vec{S}$$ ...
3
votes
1answer
69 views

Does this example contradict Earnshaw's theorem in one dimension?

This is basically a continuation of the post here. Consider electrostatics in $1$-dimension (say, the $x$-axis). Now consider a positive charge $+q$ located at $x=0$, and two equal negative charges $...
5
votes
2answers
1k views

Facing a paradox: Earnshaw's theorem in one dimension

Consider a one-dimensional situation on a straight line (say, $x$-axis). Let a charge of magnitude $q$ be located at $x=x_0$, the potential satisfies the Poisson's equation $$\frac{d^2V}{dx^2}=-\frac{\...
0
votes
0answers
10 views

Are the microwaves in an ECRIS plane polarized?

Or randomly polarized? Are the photons in phase, like in a laser or maser? What is the theory behind how an electron in an ECRIS responds to a microwave photon?
1
vote
0answers
27 views

Space translation of coordinates, classical field theory

Consider the Lagrangian density $L = -\frac{1}{4}F_{\mu\nu}F^{\mu \nu}$ with $F_{\mu \nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu} $. After deriving the Euler-Lagrange equations for this ...
1
vote
1answer
43 views

Divergence of current density and electric field within a wire

In the following exercise: I concern myself with the validity of my interpretations of (b). Here I am more confident slightly. The divergence of the current density is merely $- d \rho / dt$, so as ...
0
votes
0answers
21 views

Deriving magnitudes for $\mathbf J$ and $\mathbf E$ from the shape of a conductor

In the following exercise: I have no idea how to infer the magnitude of $\mathbf J$ nor $\mathbf E$ given the shape of the wire. The only clear thing to me here is that A, B and C all have different ...
1
vote
1answer
48 views

Destructive interference in vacuum (energy conservation) [duplicate]

The following question was proposed by a student durante a lecture for a grad course in EM. What happens to the energy of an EM wave during destructive interference in vacuum, in regards to the ...
2
votes
1answer
83 views

Formula for all the Lorentz boosts that result in $E$ parallel to $B$?

This is a follow-up question to a previous question regarding a minimum-energy invariant of the electromagnetic field. @ChiralAnomaly showed that there is indeed an invariant minimum energy density ...
0
votes
0answers
33 views

About density charged in the Liénard - Wiechert Potential to Point Charge?

I'm reading Griffiths Ch. 10. In the 10.3.1 section, there's a proof of this integral $$ \int \rho(r^\prime, t_r) \mathrm{d} \tau^\prime $$ which is not equal to the charge of the particle, but ...
0
votes
0answers
19 views

Relation between potential and charge of a group of conductors

studying electrodynamics I encountered a few weeks ago this statement regarding a set of conductors in space with no free charges: I could not find an explicit proof of this in any book and I did not ...
2
votes
1answer
37 views

Does the magnetic field move with the current element that induced it?

Initially, a point($P$) is defined in space, with a magnetic field($-B\hat{k}$) produced by a current element ($Idl$). If the wire begins to move in the $-v\hat{i}$ direction, would that produce an ...
1
vote
1answer
81 views

Is this a correct argument why $c$ is the cosmic speed limit, and what does it mean for the speed of massless particles? [closed]

I am now in my second bachelor, taking both an electrodynamics and a quantum mechanics course. This made me think of an argument to explain why particles cannot exceed the speed of light. So far I ...
0
votes
2answers
57 views

Different between spatial change in magnetic field & motional emf?

For the case of a stationary loop, and a changing magnetic field producing a non-conservative electric field $E_{nc}$: If the induced emf (${\Large{\varepsilon}}$) is due to both the change in ...
0
votes
0answers
97 views

Pressure radiation using Lorentz force

We know from theory and experiment that an electromagnetic wave that incides on a surface will generate a radiation pressure normal to that surface as a result of the change in momentum of the wave ...
0
votes
0answers
81 views

How to correctly differentiate the Lienard-Wiechert four-vector potential to get the EM tensor?

The retarded 4-vector potential for a moving charge is given by $$ A^\alpha = \left. \frac {eV^\alpha(\tau)}{V\cdot[x-r(\tau)]} \right|_{\tau = \tau_0} $$ where $e$ is the charge, $V$ the four-...
1
vote
1answer
28 views

How do we find the surface charge density of a charged conductor?

It is mentioned in Feynman volume 2 that it is quite algorithmic , where the surface charge density is first guessed then check whether it is equipotential at the metal surface.. My question is for a ...
4
votes
1answer
75 views

Conservation of magnetic charge

It is well known that the electric charge of a system can be thought of as the Noether charge associated with isotropic large gauge transformations. That is, given Einstein-Maxwell theory $$S=\frac{1}...
2
votes
0answers
45 views

Questions about the energy density and momentum density in linear media

Precisely which energies are accounted for in the energy density of a linear medium given by $$u = \frac{1}{2}\left(\epsilon E^2 + \frac{1}{\mu}B^2\right)?$$ For example, is there a kinetic energy ...
1
vote
0answers
26 views

Pulsar distance estimation

This is quite an interesting problem in astrophysics so I thought it would be a good idea to ask here so we can archive the solution for future reference. Consider a pulsar that emits pulses of ...
0
votes
2answers
55 views

Energy-momentum vs charge-current

Is there a simple intuitive way to explain why energy-momentum density requires a tensor, while charge-current density is a vector? $\partial_{\mu} J^{\mu} = 0$ is a statement, in effect, that the ...
-3
votes
2answers
108 views

Which forces violate Newton's third law of motion? [closed]

Forces arising from magnetic fields do violate Newton's third law of motion under certain circumstances. What other forces violate the third law?
3
votes
1answer
402 views

Applying the Euler-Lagrange equations to Maxwell's Theory

In Prof. David Tong's notes, specifically on page 10, he gives the Lagrangian of Maxwell's theory to be $$ \mathcal{L} = -\frac{1}{2}(\partial_\mu A_\nu)(\partial^\mu A^\nu) + \frac{1}{2}(\partial_\...
0
votes
1answer
25 views

Lorentz force within a flux tube

I know the formula for the Lorenz force exerted on a point charge $q$ moving with velocity $\vec{v}$ is given by $\vec{F} = q\vec{v} \times \vec{B}$. Now consider the flux tube with cross section $S$ ...
1
vote
1answer
44 views

Changes in boundaries with the application of Faraday's law

Reviewing Faraday's law of an induced electric field due to a changing magnetic field $$ \nabla \times E = -\frac{\partial B}{\partial t}$$ In integral form via application of Stokes theorem: $$ \...
0
votes
1answer
70 views

Electromagnetic Angular Momentum: Problem with vector integrals

I found in the following reference (p. 10) an interesting decomposition for the electromagnetic angular momentum in terms of an orbital terms $\vec{L_{orb}}$ and an spin term $\vec{L_{spin}}$. However,...
0
votes
0answers
58 views

Polarization ellipse for an EM wave [duplicate]

In Chapter 7 of Jackson's book on Classical Electrodynamics, there's the following statement: Introducing the complex orthogonal unit vectors: $$\epsilon_{\pm}=\frac{1}{\sqrt{2}}(\epsilon_1\pm\...
1
vote
1answer
54 views

Integral of the divergence of a vector field multiplied by the component of another vector field

In Forces in Molecules by Richard Feynman (Phys. Rev. 56, 340 (1939)), eq. (5) implies that $$\int(\nabla\cdot \textbf{F})E_\mu^\alpha dv=-\int F_\mu(\nabla\cdot E_\mu^\alpha)dv,$$ being $\textbf{F}...
0
votes
1answer
225 views

What is longitudinal and transverse component of electric field? [closed]

What is longitudinal and transverse component and how are they interpreted?
1
vote
1answer
74 views

Derivations of Maxwell equations

In my book of electrodynamics, the Maxwell equations are always used for specific conditions (electrostatics, magnetostatics, …). But nowhere I see a complete derivation of the equations. Maybe it ...
1
vote
0answers
52 views

Electromagnetic wave in a prism

Imagine an electromagnetic plane wave entering perpendicular to one of the faces of a prism with the form of a triangle rectangle, which is made of a certain material of refraction index $n$. The wave ...
1
vote
2answers
71 views

Is it possible to create EM radiation moving in an opposite direction to that from an accelerated charge?

The Lienard-Wiechert retarded solution to Maxwell's equations has the radiation fields diverging and propagating away from an accelerating charge; the advanced solution has radiation fields converging ...
1
vote
2answers
84 views

Maxwell's equations, nonlinear media, and dynamic response

Maxwell's equations in the vacuum with electric permittivity $\epsilon_0$ and magnetic permeability $\mu_0$ are given as: $$\nabla \cdot \vec E = \frac{\rho}{ \epsilon_0}$$ $$\nabla \cdot \vec B = 0$...
0
votes
0answers
34 views

Does Jackson's result for the vector potential of current loop correct?

General form of Maxwell equation is given by $$ \nabla_\mu F^{\mu\nu} = 4\pi J^\nu $$ where $F_{\mu\nu}=\nabla_\mu A_\nu-\nabla_\nu A_\mu$ is the tensor of EM field. Then Maxwell equations can be ...
1
vote
1answer
62 views

Charged particle moving in magnetic field using cylindrical coordinate [closed]

It's knowen that a charged particle take a helix trajectory in a uniform magnetic field $B =B e_z$ I tried to study this problem using cylindrical coordinate and i get that $$F=q v ×B = m a$$ in ...
2
votes
0answers
145 views

Maxwell stress tensor for electromagnetic wave

Sorry if this is a naive question but I've been struggling in trying to proof this for a week. Consider an electromagnetic wave with wave vector $\vec{k}=k\hat{n}$, the Maxwell stress tensor can be ...
1
vote
0answers
101 views

Magnetic field $\vec{A}$ as momentum potential

I was reviewing some topics on electromagnetic field theory and I came across the following interesting assertion: the electromagnetic moment $P_{EM}$, which is defined in vacuum as: $$P_{EM}=\frac{1}...
2
votes
2answers
65 views

Physics solely in terms of local observables

Practically all of the physics equations I've encountered are written in terms of what might be called "remote observables", such as the distances between objects in Euclidean space or between events ...
1
vote
5answers
296 views

Concerning the energy stored in electromagnetic fields

How do we know that $$u = \frac{1}{2}\left(\epsilon_0E^2 + \frac{1}{\mu_0}B^2\right)$$ gives the energy density of electromagnetic fields? Is it a postulate of classical electrodynamics? Griffith ...
1
vote
1answer
43 views

Question about the Lorenz gauge in classical electrodynamics

The Lorenz gauge is the gauge such that $$\nabla \cdot \mathbf{A} = -\mu_0\epsilon_0\frac{\partial\Phi}{\partial t}.$$ This condition dictates what $\lambda$ is in $$\mathbf{A}' = \mathbf{A} + \nabla \...
0
votes
1answer
31 views

Symmetry operations on an infinite uniform sheet of charge

My book has a section on symmetry operations. It says, (if the plane of charge is the yz plane) translation symmetry along the y-axis and z-axis implies that the electric field is constant if one ...
4
votes
1answer
111 views

Why can we pick the divergence of the vector potential? [duplicate]

I'm aware that the vector and scalar potential in E&M can be modified using a function $\lambda(t)$ in the following way: $$\mathbf{A}' = \mathbf{A} + \nabla\lambda,\;\; \textrm{ and } \;\;\Phi' =...
3
votes
2answers
68 views

Could magnetic fields really be completely substituted by relativity and electric fields?

In many textbooks (especially those for undergraduate level), magnetic fields are described merely as a relativistic side product of electric fields when considering frames in motion relative to ...
6
votes
5answers
365 views

Does classical electromagnetism really predict the instability of atoms?

I will try to give a concise summary of what I wrote below. I understand that it is very long and apologize if I am wasting your time. I used the Liénard-Wiechert potential and the Lorentz force ...
1
vote
0answers
65 views

Green's function in a region between a conductor sphere and two charged spheres inside, with point charges inside of each

Please, help me. I have to find the Green's function in the following region, but I don't have any idea how to find it: I have a conducting spherical shell of radius a; in the center there are 2 ...
2
votes
1answer
98 views

Electromagnetic field tensor and antisymmetry

Why does the inner product between the four force (caused by the electromagnetic field tensor) and the four velocity equaling zero imply that the electromagnetic field tensor is antisymmetric? This ...
1
vote
2answers
93 views

inverse square law and far-field radiation

This question is related to Feynman on inverse square law of EM radiation. It is basicly the same question except that I don't see that the question was ever answered, and I hope someone will answer ...
7
votes
1answer
161 views

On the applicability of Coulomb's law and the Biot-Savart law

Jefimenko's equations are $$\textbf{E}(\textbf{r}, t_r) = \frac{1}{4\pi\epsilon_0}\int \left[\rho\left(\textbf{r}', t_r\right)\frac{\textbf{r} - \textbf{r}'}{\left|\textbf{r} - \textbf{r}'\right|^3} + ...
0
votes
0answers
30 views

Ampere's Law, Interface conditions for magnetic field

I'm failing to understand the derivation of the interface conditions for the tangential components of the magnetic field given her (based on d.j,griffiths) Ampere's law in integral form is given as $$...
8
votes
4answers
277 views

Mathematics supporting the classical explanation of why the phase speed of light slows down in a medium

Consider the answer here by Chad Orzel which explains how a monochromatic light can slow down in a medium. He explains, You can think each of the atoms (of the medium) as being like a little dipole,...