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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What is the force corresponding to Lamor's formula for EM radiated power?

The rate at which electromagnetic energy is radiated is given by Lamor's formula. What is the corresponding rate at which momentum is radiated and hence force to this?
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The issues when thinking about the magnetic flux law as being universal in modeling

$$ \varepsilon = -\frac{\partial \Phi}{\partial t} =-\frac{\partial (BA)}{\partial t} $$ Any instance of considering the emf induced in a system, I usually think of the flux law first, I intensely ...
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What are the electric and magnetic fields of an arbitrarily moving point charge assuming instantaneous field propagation for simplicity? [closed]

I could find an equation for the magnetic field. It is called the Biot-Savart law: $$ {\vec{\pmb{B}}} = \frac{\mu_0}{4\pi} \cdot q \cdot \frac{1}{r^2} \cdot \left( {\vec{\pmb{v}}} \times {\hat{\pmb{r}}...
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Repulsive force from charge accumulation in a capacitor plate?

During a charged state of a capacitor (Regardless of the type of capacitor), Simple parallel plate: .png Parallel plate with dielectric material in the gap: ]3 Supercapacitor: How is the ...
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3answers
42 views

Why is it that magnetic fields(or any field)not move in space? [closed]

When I imagine a magnetic field produced by a magnet, or the electric field produced by a charge, I've learned that the fields are stationary, however, their value(across space) changes. If I placed ...
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1answer
109 views

What does Maxwell's equations predict for the propagation of EM waves converging to a point?

Maxwell's equations model EM radiation as propagating away from an accelerating charge. Suppose instead the propagation of this EM radiation is reversed and presented as a source-free boundary ...
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36 views

Using the Lorentz force equation twice to model to different phenomena?

In a rail system, can I apply Lorentz force separately to: Derive the work required to move the charges(motion of current); Derive the kinematics relative to the rod's motion; All stemming from: $$ ...
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17 views

Electromotive force in the presence of non-steady currents

Griffiths's Introduction to Electrodynamics states $$\mathcal E = \oint \mathbf f \cdot d\mathbf l$$ In which $$\mathbf f = \mathbf f_s + \mathbf E$$ Where Griffiths describes the summation as ...
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75 views

In electrodynamics, why do we say $\mathbf J = \sigma \mathbf E$ and not $\mathbf J = \sigma (\mathbf E + \mathbf v \times \mathbf B)$?

Griffiths notes it's because charges have an extremely low $\mathbf v$, so it's essentially an approximation, but aren't charges meant to be electrons? How can they be moving slowly? I usually think ...
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Displacement currents, aren't really currents?

I'm confused with this definition of displacement currents within capacitors via Wikipedia: However it is not an electric current of moving charges, but a time-varying electric field. It's a ...
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1answer
26 views

Relationship between the: Power supply's electric force for current flow, and Lorentz force equation for electric force?

$$ F = qE + qv \times B$$ For the Lorentz force relevant to a current carrying wire, that is caused by the motion of a wire w.r.t to an exterior magnetic field $B$, the second term($qv \times B$) on ...
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Energy-momentum tensor of the electromagnetic field

I have to derive the electromagnetic energy-momentum tensor from Noether's theorem and translation invariance. Due to translation invariance and gauge transformation: $$\delta A_\mu= a^\nu F_{\mu\nu}$$...
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Griffiths Electrodynamics Problem 9.39: How can $\sin(\theta_T)$ be greater than one?

When an electromagnetic wave strikes an interface between two linear media, Snell's law states that $\frac{\sin(\theta_T)}{\cos(\theta_I)} = \frac{n_1}{n_2}$ where $\theta_I$ is the angle of incidence,...
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2answers
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How do we know that electric charges are invariant?

According to tparker at Why charge is Lorentz invariant but relativistic mass is not? So there are two different ways to generalize the mathematical form of Coulomb's law to make it ...
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The Electromagetic Tensor and Minkowski Metric Sign Convention

I am trying to figure out how to switch between Minkowski metric tensor sign conventions of (+, -, -, -) to (-, +, +, +) for the electromagnetic tensor $F^{\alpha \beta}$. For the convention of (+, -,...
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1answer
71 views

Invariance of Maxwell action

I have to show that the Maxwell action $$S=-\frac{1}{4}\int d^4x F^{\mu\nu}F_{\mu\nu}\,$$ is invariant under translation: $\delta_aA_\mu=a^\nu \partial_\nu A^\mu$ with $a^\mu$ as arbitrary and ...
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Diffraction by induced quadrupoles

This is a very interesting problem that I've been struggling to solve for a week, so I decided to ask for some orientation, as I think it could also be of interest for the community. Let's consider a ...
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Effects generated from a supercapacitor placed in a time varying external magnetic field?

The diagram above, showcases the simple outlook of a supercapacitor's interior and combining it with a full circuit loop. If an exterior magnetic field is introduced in all the operating states of a ...
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1answer
72 views

Dipole Inside Cavity of A Spherical Conductor

Consider the following case: There is a short electric dipole placed arbitrarily inside a spherical cavity inside a solid,uncharged conducting sphere We need to find electric field at a point ...
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1answer
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Subarea within a changing magnetic flux?

If I were to introduce a boundary area $\tau$: And after sometime $t$, I introduced a constant magnetic field(let's imagine it spawned suddenly and ignored the change in flux from $t_o$ $\rightarrow$ ...
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30 views

Is the relative number of electric field lines between two charges proportional to the difference between those two charges?

Let us consider a system of two unlike charges and suppose that the magnitude of the positive charge is greater that of the negative charge and call them a and b respectively. If that(asked in the ...
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Ultraviolet catastrophe in a classical world

In the real world, the ultraviolet catastrophe doesn't happen because the quantization of photons modifies the classical behavior of light at frequencies comparable to and higher than the temperature. ...
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Confusion in the derivation of the potential of a magnetic shell

I am reading an old book on electromagnetism (THE MATHEMATICAL THEORY OF ELECTRICITY AND MAGNETISM) and I have some confusion in the following pages: First let me clarify what a "magnetic shell" is: ...
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1answer
37 views

Difference in the direction of propagation of em wave [duplicate]

How are kx-wt and kx+wt in terms of the direction of the wave. I have been stuck at this or an hour, still can not find a definitive answer.
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126 views

Energy momentum tensor of EM field written in symmetric form

I'm reading A. Zee's book, Einstein Gravity in a Nutshell. In problem 7 of chapter IV.2, it is said that the energy momentum tensor of the electromagnetic field \begin{align} T^{\mu\nu}=\eta_{\lambda\...
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Deriving current density from charge density

Sorry for the naive question, but in Jackson's book in Electrodynamics, exercise 9.10, they give the charge density for the transition of hydrogen from 2p to 1s, $$\rho=\frac{2e}{\sqrt{6}a_0^4}re^{-...
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85 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-...
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How can direction of electric field due to a moving charge be from the present position of charge?

According to Maxwell's Equations, the electromagnetic waves in vacuum travel at the speed of light $c$. While solving Maxwell's equations using Lorenz gauge conditions (or basically evaluating scalar ...
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536 views

Induced voltage of a coil inside a coil

I have the length of the first coil, the number of turns in both, the current through the first coil and the cross sectional area of the coil inside. I want to find the induced voltage. I know i ...
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1answer
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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 ...
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The Meaning of Electromagnetic 'News' in Griffiths Book

In the Introduction to Electrodynamics book, by David J. Griffiths, 4th edition, page 60, the author makes the following statement: "it is not the position, velocity, and acceleration of Q right now ...
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24 views

Electric field of a line of Charge on it axis using electric potential

We have a line of charge with length $L$ and charge density $\lambda$ and we want to find its Electric field on a point $p$ with distance $d$ from one end of it using potential. I drew a picture like ...
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How to calculate Average Electric Field for a microscopic filed in dielectric?

In Griffith's book in The chapter of Electric Fields in Matter (The field inside dielectric) , He has considered the Sphere of Radius R and for the inner dielectric element He used the field as $$\...
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1answer
536 views

Maxwell Stress Tensor at material boundaries

I am trying to grasp the meaning of the Maxwell Stress tensor $T_i^j$ at material boundaries. Concretely, I am trying to calculate the force between two waveguides. The results are given in an article ...
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1answer
376 views

Symmetries of the Hamiltonian of a charged particle in a uniform magnetic field

Consider the Hamiltonian of a charged particle of charge $q$ in a uniform magnetic field $\textbf{B}=B\hat{\textbf{z}}$ is given by $$H=\frac{(\textbf{p}-q\textbf{A})^2}{2m}$$ where $\textbf{p}$ is ...
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5answers
300 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 ...
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18 views

Constants of motion of an electron in a harmonic electromagnetic field in free space

I have encountered a question in Classical Electrodynamics, as below: In free space, an electron, initially at rest at $z=0$, is subjected to an intense laser field $\vec E=\hat x A \cos(\omega t-...
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451 views

Multipole expansion of the electromagnetic field

In Jackson's Classical Electrodynamics, section 9.7, he develops the multipole expansion of the electromagnetic fields in terms of the vector spherical harmonics and the spherical Bessel and Hankel ...
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1answer
70 views

Problem in understanding Differential form of Gauss's Law

I am well aware of the integral form of Gauss's Law and the mathematical deduction through which it is reduced to the differential form. But I think I have a flaw in my understanding of divergence. ...
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Why can infinite planes be approximated as Gaussian surfaces?

A little background: I'm an undergraduate studying Electrodynamics, currently in Chapter 8 of Griffiths. A question I came across (8.4 part a for those curious) asks for a calculation of the force ...
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1answer
129 views

Can a charge moving in an open trajectory qualify as current?

It is sometimes said that a point charge is equivalent to an electric current. If it were a steady current, I should be able to find it from Ampere’s law or Biot-Savart’s law. Even if the current is ...
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Feynman's proof for Liénard-Wiechert's potential of a moving charge

Feynman's proof utilizes a geometrical and fundamental integration argument. I like it, except this bit: What makes me unconfortable somehow is that in (c) we are counting in some of the charge we ...
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Coulomb forces in String Theory

In QED electrostatic forces are mediated by field theoretic effects known as virtual "particles" if I am not mistaken, but I don't know how string theory explains electrostatic interactions between ...
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1answer
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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 $...
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59 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: $$\oint_{loop} \vec{E} \cdot d\vec{l}=\int_S -\frac{\partial{\vec{B}}}{\partial{t}} \cdot d\vec{S}$$ ...
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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{\...
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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?
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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 ...
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1answer
50 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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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 ...