Questions tagged [electromagnetism]

For questions on Classical Electromagnetism from a mathematical standpoint. This tag should not be the sole tag on a question.

For questions on Classical Electromagnetism from a mathematical standpoint. This tag should not be the sole tag on a question. Examples of other tags that might accompany this include (algebra-precalculus), (vector-analysis), and (fourier-analysis).

310 questions
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Electrodynamics in general spacetime

Let $M\cong\mathbb{R}^4_1$ be the usual Minkowski spacetime. Then we can formulate electrodynamics in a Lorentz invariant way by giving the EM-field $2$-form $\mathcal{F}\in\Omega^2(M)$ and reformuling the homogeneous Maxwell equations…
12
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Finding smooth behaviour of infinite sum

Define $$E(z) = \sum_{n,m=-\infty}^\infty \frac{z^2}{((n^2 + m^2)z^2 + 1)^{3/2}} = \sum_{k = 0}^\infty \frac{r_2(k) z^2}{(kz^2 + 1)^{3/2}} \text{ for } z \neq 0$$ $$E(0) = \lim_{z \to 0} E(z) = 2 \pi$$ where $r_2(k)$ is the number of ways of writing…
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Adding small correction term to ODE solution

Let $\mathbf{r}(t) = [x(t), y(t), z(t)]$ and $\mathbf{v}(t) = \frac{d}{dt}\mathbf{r}(t)$. I'm trying to solve $$ \frac{d}{dt}\mathbf{v}=\frac{q}{m}(\mathbf{v}\times\mathbf{B}) \tag{1} $$ where $q$ and $m$ are real constants and…
Mr. G
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10
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Applying the Fourier transform to Maxwell's equations

I have the following Maxwell's equations: $$\nabla \times \mathbf{h} = \mathbf{j} + \epsilon_0 \dfrac{\partial{\mathbf{e}}}{\partial{t}} + \dfrac{\partial{\mathbf{p}}}{\partial{t}},$$ $$\nabla \times \mathbf{e} = - \mu_0…
9
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2 answers

Are there Soliton Solutions for Maxwell's Equations?

Some non-linear differential equations (such as Korteweg–de Vries and Kadomtsev–Petviashvili equations) have "solitary waves" solutions (solitons). Does the set of partial differential equations known as "Maxwell's equations" theoretically admit…
8
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2 answers

Biot-Savart law on a torus?

In classical electrodynamics, given the shape of a wire carrying electric current, it is possible to obtain the magnetic field configuration $\mathbf{B}$ via the Biot-savart law. If the wire is a curve $\gamma$ parametrized as $\mathbf{y}(s)$, where…
7
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2 answers

Apparent paradox when we use the Kelvin–Stokes theorem and there is a time dependency

I am having trouble to understand what is going on with the Maxwell–Faraday equation: $$\nabla \times E = - \frac{\partial B}{\partial t},$$ where $E$ is the electric firld and $B$ the magnetic field. The equation is local, in the sense that any…
Wolphram jonny
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6
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Representation of the magnetic field in 2D magnetostatics

Consider a magnetostatics problem in $\mathbb{R}^3$. The problem is governed by the following equations $$\begin{aligned}\text{Maxwell's equations}\quad &\begin{cases}\nabla\times H(x)=J(x)\\\nabla\cdot B(x)=0\end{cases}\\[3pt] \text{Constitutive…
6
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Helmholtz decomposition of a vector field on surface

Does it make sense to do Helmholtz decomposition of a vector field defined on a surface or on a manifold? I am mostly interested in the surface case. I was trying to find a reference for this and found only a handful of them mostly from…
5
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1 answer

Evaluate Integral $ I:=\int_0^{2\pi} \cos s \,\log (\sqrt{c^2 + a - 2 \cos s}-c) \, \mathrm d s $ for radially magnetized cylinder

When trying to evaluate the magnetic scalar potential $\Phi_m$ of a magnetized cylinder (Magnetization $M$ in $x$-direction, height $Z$, Radius $R$, touching the $xy$-plane from below), I was able to solve two of the three integrals in cylinder…
5
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1 answer

A rigorous proof that $\nabla \cdot E = \frac{\rho}{\epsilon_0}$

Suppose that $\rho: \mathbb R^3 \to \mathbb R$ is a function that tells us the electric charge density at each point in space. According to Coulomb's law, the electric field at a point $x \in \mathbb R^3$ is $$ E(x) = \frac{1}{4 \pi \epsilon_0}…
littleO
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5
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Vector identity used in electromagnetism

Is there a simple proof of this identity or a reference to some textbook where could I find a simple proof of the $(1)$? $$\boldsymbol{\nabla}\times (\boldsymbol{\nabla}\times \mathbf{E})=-\frac{\partial }{\partial t}(\boldsymbol{\nabla}\times…
Sebastiano
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5
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2 answers

Why do time derivatives change to '$j\omega$' when taking the time derivative of a phasor?

For example, Faraday's law in the time domain is written as $$\nabla \times \vec{\mathbf{E}} = - \frac{\partial\vec{\mathbf{B}}}{\partial t}$$ When using phasor notation, Faraday's law is written as $$\nabla \times \vec{\mathbf{E}} = -j\omega \mu…
EarthIsHome
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5
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3 answers

How do one rigorously prove that the electric potential energy of an conducting sphere with charge $Q$ is $\frac{Q^2}{8\pi\epsilon_0R}$

How do one rigorously prove that the electric potential energy of an conducting sphere with charge $Q$ is $\frac{Q^2}{8\pi\epsilon_0R}$? Is integration the only way? Homogeneous charge distribution is assumed, I guess.
onlyme
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4
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Electrostatic Potential Energy integral in spherical coordinates

I'm having trouble with evaluating an integral that arises from attempting to find the total energy of an electrostatic system consisting of two point charges, which involves an integral over all space. The problem is physical, but my bone of…
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