Questions tagged [maxwell-equations]

A set of four equations that define electrodynamics. They comprise the Gauss laws for the electric and magnetic fields, the Faraday law, and the Ampère law. Together, these equations uniquely determine the electric and magnetic fields of a physical system. DO NOT USE THIS TAG for the Maxwell-Boltzmann distribution, or the thermodynamical equations known as Maxwell's relations.

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Hamiltonian classical electrodynamics

After coming across the Lagrangian density of the Maxwell equations $$ \mathcal{L} = -\frac{1}{4\mu_0} F_{\mu\nu}F^{\mu\nu}-J_\mu A^\mu = \frac{\varepsilon_0}{2}||\mathbf{E}||^2-\frac{1}{2\mu_0}||\...
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Propagation delay in current

Can anyone give me a link describing the physics/math behind a changing potential difference in a wire, and the current density that results from this, taking into account inductance of the circuit, ...
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30 views

Calculating the magnetic field around a current-carrying wire of arbitrary length using Maxwell's Equations

Hello I'm trying to use Maxwell's Equations to calculate the electromagnetic fields around different charge/current situations like from a charge using $\nabla \cdot E = \frac{\rho}{\epsilon_0}$ to ...
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Is there a way to derive the Gauss-Faraday Law in the Covariant formulation of classical electromagnetism from the Lagrangian density? [duplicate]

everywhere I look I can only find derivations of the Gauss–Ampère law $\partial_\alpha F^{\alpha\beta}=\mu_0 J^{\beta}$, and this follows quite simply from the variational method with the Lagrangian ...
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How does the electric field of a Reissner-Nordstrom black hole exist outside its event horizon if nothing can escape the event horizon? [duplicate]

I am learning Reissner-Nordstrom black holes and I have learnt that the black hole contains a net charges. The static field due to it ( through the Energy Momentum tensor) exists even outside the ...
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30 views

Electromagnetic Field Tensor and 4 Vector Potential in the presence of magnetic charges and under spherical symmetry

I am a bit confused as to what form does the 4 vector potential and Electromagnetic field tensor $F_{\mu \nu}$ take in the presence of magnetic charges. I got confused in this while looking at the ...
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Do linear theories have infinitely many solutions?

It is said that, in a linear theory, you can add any number of solutions to still find a solution. So, say I initially found three solutions to a linear theory, and I call them a1, a2, and a3. Now I ...
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What is wrong with my counting of electromagnetic field degrees of freedom?

When we go from the field variables $({\vec E},\vec{B})$ to the potentials $(\phi,{\vec A})$, the number of degrees of freedom describing any electromagnetic field is reduced from $6$ ($3$ components ...
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Maxwell's equations and pair production

If a photon can create a particle and an antiparticle pair. then the result is that a changing $E, B$ field creates an actual charge density. why is this not reflected in Gauss law? with a combination ...
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The relationship between scattering width and radar cross section

I have a question regarding Knott's book on radar cross section (RCS). Specifically, I am interested in the relationship between the 3D RCS, $\sigma_\text{3D}$, and the scattering width (2D RCS), $\...
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Relation between quasi-static and fully dynamic $\vec E$ and $\vec H$

Imagine an infinitely long coaxial cable with an inner wire of radius $a$ and outer radius $b$. The space in between the cable is filled with air ($\epsilon=\epsilon_0$). Suppose the inner cable ...
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Meaning of $E$, $ D$, $B$ and $H$-fields

The Lorentz force is $$ F = q(E + v \times B) $$ where $E$ is denoted the electric field and $B$ is denoted the magnetic field. Further, Maxwell's equations are $$\begin{array}{rcl} \nabla \cdot D &...
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Charge distribution Ohm's law

According to Ohm's law $$\textbf{J}=\sigma\textbf{E}$$ where $\textbf{J}$ is the current, $\sigma$ is the electric conductivity, and $\textbf{E}$ is the electric field. Now from the continuity ...
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EM waves in a conductor :dispersion relation

We can write a solution to the wave equation (55) in the usual form: $$ \vec E(\vec r, t)=\vec E_0 e^{i(\vec k\cdot\vec r-\omega t)}\tag{56} $$ But now, if we substitute this into equation (55), we ...
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Are Maxwell's laws mathematically precise?

Electrodynamics makes heavy use of vector calculus, which in turn is about differentiation and integration of scalar and vector fields in $\mathbb{R}^3$. At this point everything seems fine to me, ...
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Field inside an accelerating charged hollow conductor?

Suppose I have a charged, hollow conductor held stationary inside a static electric field. The electric field inside the cavity is shown to be zero from the uniqueness theorem as explained in this ...
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1answer
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Derivation of wavelength in a waveguide

I'm trying to derive this expression for the wave length in a wave guide. I'm following this derivation from the Feynman lectures https://www.feynmanlectures.caltech.edu/II_24.html#Ch24-F4. I do not ...
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1answer
57 views

EM-wave equation in conductors with source terms

The traditional modified Maxwell's equations to express em wave inside conductors that I have come across are: $$ \nabla\cdot\mathbf E = 0 \\\nabla\cdot\mathbf B = 0 \\\nabla\times\mathbf E = -\frac{...
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33 views

Waveguide cut off frequency derivation - Wave equation to Helmholtz equation

I'm trying to derive the cut off frequency for a wave guide. I found a derivation on wikipedia, but I don't understand the first step where we go from the wave equation to the helmholtz equation. Why ...
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How to interpret charge continuity equation for conductors that obey Ohm's law?

For conductors, we propose that the free current density is proportional to the applied Electric field and the constant of proportionality is defined as conductivity. \begin{equation} \textbf{J}_\...
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Condition for steady current and the applicability of Gauss's Law?

The conditions for steady current are often specified as $$\frac{\partial\rho}{\partial t}=0 \,\,\,\,and\,\,\,\frac{\partial\vec{J}}{\partial t}=0 $$ Combining $\frac{\partial\rho}{\partial t}=0$ with ...
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Generalization of Divergence laws to the non-static cases

The laws $\nabla\cdot{\vec{E}}=\frac{\rho}{{\varepsilon}_0}$ and $\nabla\cdot{\vec{B}}=0$ are derived considering the static cases by using Coulomb's law and Biot-Savart's law. Then why are they ...
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Extracellular potential via a point source

You can stimulate neurons extracellularly - that is, you place an electrode at a certain distance. Assuming a punctiform electrode and a homogeneous medium, the extracellular potential (in mV) reduces ...
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70 views

How can force-free magnetic fields have current parallel to magnetic field?

We know that for a force-free magnetic field, $\vec{J} \times \vec{B} = 0$, which means that the current should be parallel to the magnetic field that is creating it. However, from Ampère's law $\vec{\...
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Definition of electromotive force and conductors moving inside a constant magnetic field

This is a followup question to this one. I already found some similar questions in PhysicsSE (as this one and this one) but I didn't found the answer I was looking for. In a conducting loop wire ...
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Maxwell equations and electric field

How can I use the Maxwell equation to prove that electric field is the sum of a conservative parcel of the electric potential gradient and a non-conservative parcel given by the magnetic potential ...
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2answers
31 views

How to compute the field lines of an induced magnetic field inside a capacitor?

Consider a capacitor with a varying voltage applied to it. As the voltage changes over time, the electrical field $\vec{E}$ inside the plates does too. Assumption We assume that the direction of $\...
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3answers
110 views

Different speed of light in two inertial frames and the relativity principle

I'm in a frame in which a medium is at rest, and I observe light move at some speed. Another person observers this medium move at some constant speed, in this case he'll observe a different speed for ...
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Can you write and solve the EM field Lagrangian density without reference to the EM potential? [duplicate]

Is it possible to write the Lagrangian density for the EM field and charges $$L=-\frac{1}{4\mu_0}F^{\mu \nu}F_{\mu \nu}+j^{\mu}A_{\mu}$$ only in terms of the Electromagnetic Tensor and current vector? ...
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1answer
56 views

Maxwell equations in matter and speed of light

I'm in a frame in which a medium is at rest, and I observe light move at some speed. Now this medium moves at some constant speed, in this case I'll observe a different speed for light. We can find ...
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50 views

Is there a circulating $E$ field when a single straight wire cuts a magnetic field?

I am a little confused as to the difference between the Lorentz force and Faraday’s induced emf, specifically in the case of a straight wire moving through a uniform B field. I know the answer is that ...
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Can one explain the attraction of two magnets with Faraday's law?

Supose two ideal magnets, close to each other and both at rest, with a north and a south poles both fixed and well defined. May we admit as well that the total electric charge of both is zero in the ...
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Conditions on $\phi$ and $\boldsymbol{A}$ for when $\boldsymbol{B}$ is uniform

I'm reading "Classical Mechanics" (5ed) by Berkshire and Kibble, in the example for uniform magnetic field on pg.243 (Chapter 10 Lagrangian Mechanics) I came across this A charged particle ...
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Faraday's law for a 3-dimensional conductor plate moving in a uniform magnetic field

I am struggling to understand this supposedly simple problem I found in a highschool textbook. A metallic plate is moving with constant velocity v in a region in which there is a uniform magnetic ...
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27 views

How to measure voltage and current in closed circuit under magnetic induction?

Let's have a long closed loop wire, part of which is a coil to increase locally the inductance. A magnet is moved back and forth inside the coil. A clamp amperimeter far from the coil, in the AC mode, ...
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Completeness of Maxwell's equations

If we consider Maxwell's equations in the form: $\nabla\cdot\overrightarrow{E}=\frac{\rho}{\epsilon_0}$$\nabla\cdot\overrightarrow{B}=0$$\nabla\times\overrightarrow{E}=-\frac{\partial\overrightarrow{B}...
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Complex notation of monochromatic planar waves in Griffith's

The definition of planar waves in Griffith's Electrodynamics textbook is given as: The waves are travelling in the z direction and have no x or y dependence; these are called plane waves, because the ...
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In classical electrodynamics: is the conservation of $J$ and the conservation of field four-momentum entirely independent of one another?

Up to now, I've always seen the conservation of the four-current $J$ and the conservation of field four-momentum as two independent conservation laws: the first coming from the gauge invariance of ...
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Why is the divergence of induced electric field zero?

Why is the divergence of induced electric field zero? If someone says because $\nabla \cdot E=\rho/\epsilon _0$ and for induced fields $\rho=0$ and hence its divergence is zero, then how do we in ...
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Validity of Ampère's law in terms of $H$

We know that the auxiliary magnetic field $\bf{H}$ is $$\mathbf{H}=\frac{1}{\mu_{0}} \mathbf{B}-\mathbf{M}$$ and $$\mathbf{\nabla} \times \mathbf{H}=\mathbf{J}_{f}$$ but this differential equation is ...
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Trouble with enterpreting Faraday's Law

$$ \nabla \times \textbf{E} = -\frac{\partial \textbf{B}}{\partial t} $$ My interpretation of this equation is that: A steady magnetic field will result in an electric field that is $0$. A varying ...
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Ring Cavity as an approximation of the slab waveguide

Consider I have a ring cavity with retangular cross-section - as if I had a long slab waveguide and made both ends meet - with radius much greater than the wavelength. Could I be ok by using the TM/TE ...
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Is there a proof of the uniqueness of the solutions of Maxwell equations with non-fixed charges?

I know that if $\rho(t,\vec x)$, $\vec J(t,\vec x)$, $\vec E(0,\vec x)$ and $\vec B(0,\vec x)$ are given beforehand, with $\nabla \cdot \vec E(0,\vec x) = 4\pi\rho(0,\vec x)$ and $\nabla \cdot \vec B(...
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Understanding tensor and covariance

I'm really struggling to understand the use of tensors when we want to have a covariant equation. From what I understand, if we write an equation using tensors only, then the physics behind it will be ...
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Incident plane wave on metal rings

in this article an incident electromagnetic plane wave on a periodic sequence of square metal rings is considered. The authors say: When the transverse magnetic field faces the vertical strips, it ...
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Gauge-invariance of the Maxwell Lagrangian in Srednicki's book

When it comes to the demonstration of the gauge-invariance of the Lagrangian of the Maxwell-theory Srednicki's book proceeds as follows: $${\cal L} = -\frac{1}{4}F^{\mu\nu}F_{\mu\nu} + J^\mu A_{\mu} \...
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How to identify the multimodes of an optical fiber?

I use Lumerical mode software to simulate a photonic crystal fiber of NANF type. The solver gives me a lot of modes (solutions). I want to know how to know if this fiber is a single mode fiber or ...
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How to calculate Electric field curl in loop by dipole charge?

Let us consider that two charges $q^+$ and $q^-$ are placed at the ends of a rod and attached to its ends, Now the question says if we pass this rod inside a metal loop and take it out, as in Faraday'...
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142 views

Curl of Electric Field due to point charge at origin and Divergence of Magnetic Field due to infinite current carrying wire at origin

$\nabla\times E = 0/(r^2\sin\theta)$ where $\theta$ is the polar angle. Clearly $\nabla \times E = 0$ for all $r$ except $r=0$. But how do we conclude that $\nabla\times E$ at $r=0$? One can surely ...
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Is an EMF observed by a constantly accelerating loop the moment it is stationary to an electric charge?

Suppose I had a positive charge $+q$ fixed at the origin $(x,y,z)=(0,0,0)$. The coordinate system is that of the charge's rest frame. Now suppose I had a circular loop with a radius of 1 whose axis is ...

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