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).

432 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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16
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3 answers

What is the sum of an infinite resistor ladder with geometric progression?

I am trying to solve for the equivalent resistance $R_{\infty}$ of an infinite resistor ladder network with geometric progression as in the image below, with the size of the resistors in each section double the size of the previous section. For…
KDP
  • 1,263
12
votes
3 answers

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
votes
1 answer

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
votes
2 answers

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
votes
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…
user559615
9
votes
2 answers

Arranging 8 positive and 8 negative charges to produce $1/r^5$ potential in 3D

Short version of the problem: Given 8 +Q charges and 8 -Q charges in 3D, can I find an arrangement in which their potential has its leading non-zero term proportional to $1/r^5$? Step by step description and motivation? Single +Q charge: In physics…
SSF
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9
votes
2 answers

Biot-Savart law on a torus?

Background: In classical electrodynamics, given the shape of a wire carrying electric current, it is possible to obtain the magnetic field $\mathbf{B}$ via the Biot-savart law. If the wire is a curve $\gamma$ parametrized as $\mathbf{y}(s)$, where…
Quillo
  • 2,260
8
votes
1 answer

A calculus problem from electrostatics

Since this problem consists of multiple parts and one needs to see all of them to understand the problem i'm going to list out all of them: Consider a uniformly charged spherical shell of radius $R$ with surface charge density $\sigma_0$. Compute…
Tomy
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7
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The magnetic field of a spinning charged sphere

Evaluate $\displaystyle \int_{0}^{2\pi}\int_{0}^{\pi}\frac{(z_0-R\cos\theta)\sin^2\theta\cos\phi}{[(x_0-R\cos\phi\sin\theta)^2+(y_0-R\sin\phi\sin\theta)^2+(z_0-R\cos\theta)^2]^{\frac{3}{2}}}d\theta d\phi$ where $x_0,y_0,z_0$ and $R$ are constants…
grj040803
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7
votes
1 answer

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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7
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1 answer

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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7
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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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7
votes
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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7
votes
1 answer

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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