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.

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51 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 ...
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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 ...
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Is it true that any system of accelerating charges will radiate?

I was recently told by a physics teacher that "any system of charges in which at least some of the charges are executing some sort of accelerated motion, will radiate and lose energy". This refers to ...
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If you run an electric current through a wire loop, do the accelerated charges radiate?

Does an accelerated charge always radiate? For example the current electrons in an electric circuit when moving through a turn they are accelerated, do they radiate because of that acceleration? If ...
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Semi-classical derivation of maximal magnetic field in the Universe

I'm looking for all (or most) theoretical semi-classical derivations of the maximal magnetic field intensity that there may be in the Universe. As an example, this paper evaluate a maximal field of ...
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surface and volume current density; definition and logic seems contradictory to me

Okay, so in Griffiths' Introduction to Electrodynamics, Griffiths clearly defines surface current density as follows: when charge flows over a surface, we describe it by the surface current density,...
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49 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 ...
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34 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 ...
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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 ...
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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 ...
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82 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 ...
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Relationship between electronic current and electron momentum?

I remembered when reading Laughlin's famous argument for quantum Hall, it implied the actual current should be proportional to the electron's mechanical momentum ($p-eA/c$) instead of $p$ itself. Why ...
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58 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 ...
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98 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 ...
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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}...
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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 ...
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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 ...
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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 ...
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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 ...
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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?
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Classical analog of Bell QM experiment correlation coefficient calculation

This question is motivated by recent experiments in QM entanglement.[1][2] consider the following "simple/ simplified" classical analog of Bells experiment. it has a laser, a standard beamsplitter ...
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410 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_\...
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In Maxwell equations, why time derivatives only appear together with Curl?

In the four maxwell's equations, the time dependence only appear in curl of $E$ and $B$ but not divergence. My question was that: Why time dependence only appear in curl? what's the implication? (I ...
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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 $S$ ...
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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: $$ \...
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Magnetic Multipole Tensor

When the electric scalar potential is expanded into spherical coordinates, one gets \begin{align} \phi (\vec r) = \frac{1}{4\pi\varepsilon_0} \sum_{l=0}^{\infty} \sum_{m=-l}^l \sqrt{\frac{4\pi}{2l+1}...
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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,...
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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 ...
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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\...
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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}...
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272 views

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

What is longitudinal and transverse component and how are they interpreted?
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72 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 ...
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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 ...
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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$...
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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 ...
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1answer
63 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 ...
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152 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 ...
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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}...
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What is the answer to Feynman's Disc Paradox?

[This question is Certified Higgs Free!] Richard Feynman in Lectures on Physics Vol. II Sec. 17-4, "A paradox," describes a problem in electromagnetic induction that did not originate with him, but ...
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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 \...
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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 ...
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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' =...
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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 ...
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Deriving Heaviside-Feynman formula for the electric field of an arbitrarily moving charge from Lienard-Wiechert potential

I've been trying to derive this (which Feynman warns takes a lot of work) for a couple of days now, without success. My current best derivation which however doesn't give the right answer is: First, ...
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163 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} + ...
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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 ...
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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 ...
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110 views

Is Poynting vector conserved or Intensity conserved for reflection and transmission of electromagnetic wave?

When an electromagnetic wave meets a boundary, does energy conservation mean Poynting vector of reflected + Poynting vector of the transmitted wave is equal to Poynting vector of incident wave or just ...
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529 views

Why can the Lorenz gauge condition always be fullfilled?

Why is the Lorenz gauge condition always possible for classical electromagnetic fields? So far I can only understand the following: If we perform a gauge transformation $A\mapsto A'=A+\mathrm{d}\...
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1answer
106 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 ...