Questions tagged [electromagnetism]
For questions on Classical Electromagnetism from a mathematical standpoint. This tag should not be the sole tag on a question.
407
questions
19
votes
1
answer
2k
views
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 ...
- 29.9k
14
votes
3
answers
3k
views
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 ...
- 1,111
12
votes
3
answers
397
views
Finding smooth behaviour of infinite sum
Define
$$E(z) = \sum_{n,m=-\infty}^\infty \frac{z^2}{((n^2 + m^2)z^2 + 1)^{3/2}} = \sum_{k = 0}^\infty \frac{r_2(k) z^2}{(kz^2 + 1)^{3/2}} \text{ for } z \neq 0$$
$$E(0) = \lim_{z \to 0} E(z) = 2 \pi$$...
- 2,339
10
votes
2
answers
1k
views
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 ...
user559615
10
votes
2
answers
4k
views
Applying the Fourier transform to Maxwell's equations
I have the following Maxwell's equations:
$$\nabla \times \mathbf{h} = \mathbf{j} + \epsilon_0 \dfrac{\partial{\mathbf{e}}}{\partial{t}} + \dfrac{\partial{\mathbf{p}}}{\partial{t}},$$
$$\nabla \...
- 4,322
10
votes
1
answer
367
views
Adding small correction term to ODE solution
Let $\mathbf{r}(t) = [x(t), y(t), z(t)]$ and $\mathbf{v}(t) = \frac{d}{dt}\mathbf{r}(t)$. I'm trying to solve
$$
\frac{d}{dt}\mathbf{v}=\frac{q}{m}(\mathbf{v}\times\mathbf{B}) \tag{1}
$$
where $q$ and ...
- 1,058
9
votes
2
answers
881
views
Biot-Savart law on a torus?
Background: In classical electrodynamics, given the shape of a wire carrying electric current, it is possible to obtain the magnetic field $\mathbf{B}$ via the Biot-savart law. If the wire is a curve $...
- 2,101
8
votes
1
answer
711
views
A calculus problem from electrostatics
Since this problem consists of multiple parts and one needs to see all of them to understand the problem i'm going to list out all of them:
Consider a uniformly charged spherical shell of radius $R$ ...
- 429
7
votes
1
answer
639
views
A rigorous proof that $\nabla \cdot E = \frac{\rho}{\epsilon_0}$
Suppose that $\rho: \mathbb R^3 \to \mathbb R$ is a function that tells us the electric charge density at each point in space. According to Coulomb's law, the electric field at a point $x \in \mathbb ...
- 52.5k
7
votes
2
answers
176
views
Apparent paradox when we use the Kelvin–Stokes theorem and there is a time dependency
I am having trouble to understand what is going on with the Maxwell–Faraday equation:
$$\nabla \times E = - \frac{\partial B}{\partial t},$$
where $E$ is the electric firld and $B$ the magnetic field. ...
- 1,970
7
votes
1
answer
2k
views
Helmholtz decomposition of a vector field on surface
Does it make sense to do Helmholtz decomposition of a vector field defined on a surface or on a manifold? I am mostly interested in the surface case. I was trying to find a reference for this and ...
- 317
7
votes
1
answer
605
views
Representation of the magnetic field in 2D magnetostatics
Consider a magnetostatics problem in $\mathbb{R}^3$. The problem is governed by the following equations
$$\begin{aligned}\text{Maxwell's equations}\quad &\begin{cases}\nabla\times H(x)=J(x)\\\...
- 392
6
votes
1
answer
2k
views
Geometric Algebra or Differential Forms for Electromagnetism? [closed]
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 ...
- 7,042
6
votes
2
answers
227
views
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 ...
- 977
6
votes
2
answers
131
views
Fubini's theorem for differential forms? Why does $\int_{t_0}^{t_1}(\oint_{\partial\Omega}j)dt=\int\limits_{[t_0,t_1]\times\partial\Omega}dt\wedge j$?
In an electrodynamics book I came across the following claim for the electric current density (twisted) 2-form $j$ along the boundary of some 3-dimensional volume $\Omega$:
$$\int_{t_{0}}^{t_{1}}\left(...
- 1,490