Mathematics Stack Exchange
Mathematics Stack Exchange

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).

362 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…
Daniel Robert-Nicoud
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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…
QCD_IS_GOOD
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10
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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…
The Pointer
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10
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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…
user559615
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…
Quillo
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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…
Felix Crazzolara
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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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7
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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…
AnandJ
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6
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Understanding if the integral expression obtained is correct and if its (incorrect ) mistake in the approach to get that result

The integral was: $$\int_{0}^{\frac{\pi}{2}} \frac{\cos^2x}{(a^2+b^2\sin^2x)^{3/2}}\;dx= \frac{\pi}{2ab^2} (1-\frac{a}{\sqrt{a^2+b^2}}).$$ I encountered this integral while trying to show amperes law working in an EM (electromagntism) problem…
ProblemDestroyer
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Effective resistance in finite grid of resistors

Consider a $m\times n$ grid of one-Ohm resistors. What is the effective resistance of any given edge? I understand how to do the case $m=2$ inductively using the series and parallel laws, but I get stuck in breaking apart the grid for $m,n>2$ to…
zjs
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5
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Formalizing some items in an electrostatics computation

Consider the question attached (Example 3.4 from Zangwill's Modern Electrodynamics, Ch 3). I can follow the solution quite easily using "physics math" but, having just recently finished Spivak's Calculus, I want to formalize the operations performed…
EE18
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How to calculate $\int_0^{2\pi}\frac{\cos(\phi)-R}{1-2R\cos(\phi)+R^2}\cos(n\phi)~d\phi$?

I wish to calculate $$I(R)=\int_0^{2\pi}\frac{\cos(\phi)-R}{1-2R\cos(\phi)+R^2}~\cos(n\phi)~d\phi,$$ where $n\in\mathbb{N}$, $R\in[0,1)$. Based on trial and error from plugging numbers into Wolfram alpha I think the answer is $$I(R)=\begin{cases} 0,…
Peanutlex
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Geometric Algebra or Differential Forms for Electromagnetism?

Electromagnetism (Maxwell's equations) are most often taught using vector calculus. I have read that both geometric algebra and differential forms are ways to simplify the material. What are some advantages or disadvantages of each approach,…
NicNic8
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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…
Lukas Rauber
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