Questions tagged [electromagnetism]
For questions on Classical Electromagnetism from a mathematical standpoint. This tag should not be the sole tag on a question.
19
questions
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
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
4
votes
1
answer
412
views
solution to $\square\chi=f$.
For an open set $U \subseteq \mathbb{R}^4$, if $f:U \to \mathbb{R}$ is a "good" (for example, smooth) function, is there a solution to the following equation?
$$\left( \Delta - \frac{1}{c^2}\frac{\...
user53216
3
votes
1
answer
220
views
Show that $\partial_\mu\phi^\ast A^\mu\phi- A_\mu\phi^\ast\partial^\mu\phi=A^\mu\phi\partial_\mu\phi^\ast - A^\mu\phi^\ast\partial_\mu\phi$
The following is loosely related to this question:
[...], the most general renormalisable Lagrangian that is invariant under both Lorentz transformations and gauge transformations is
$$\mathcal{L}=-\...
- 419
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
10
votes
2
answers
4k
views
Applying the Fourier transform to Maxwell's equations
I have the following Maxwell's equations:
$$\nabla \times \mathbf{h} = \mathbf{j} + \epsilon_0 \dfrac{\partial{\mathbf{e}}}{\partial{t}} + \dfrac{\partial{\mathbf{p}}}{\partial{t}},$$
$$\nabla \...
- 4,322
7
votes
1
answer
639
views
A rigorous proof that $\nabla \cdot E = \frac{\rho}{\epsilon_0}$
Suppose that $\rho: \mathbb R^3 \to \mathbb R$ is a function that tells us the electric charge density at each point in space. According to Coulomb's law, the electric field at a point $x \in \mathbb ...
- 52.5k
5
votes
1
answer
156
views
What is the correct sign for the four-vector potential gauge transform; $A_\mu\to A_\mu\pm\partial_\mu\lambda$ and where does this gauge originate? [closed]
I have three questions regarding the following extract(s), I have marked red the parts for which I do not understand for later reference. The convention followed for the Minkowski metric in these ...
- 419
4
votes
4
answers
11k
views
Why is the divergence of curl expected to be zero?
I was reading the proof of Helmholtz decomposition theorem where I found the relation between the rotational and the irrotational fields are not symmetric. And by that I mean if the divergence of the ...
4
votes
2
answers
145
views
Flux integral of Gauss law
Consider a point charge enclosed by some surface, using spherical coordinates, and taking $\hat a$ to be the unit vector in the direction of the surface element, flux is
$$\oint\vec E\cdot d\vec A = ...
4
votes
2
answers
1k
views
How to compute $(\mathbf{A} \cdot \mathbf{\nabla})\mathbf{B}$?
I'm currently reading Intro to Electrodynamics by Griffiths, and in the maths section, there is the following problem:
"If $\mathbf{A}$ and $\mathbf{B}$ are two vector functions, what does the ...
2
votes
1
answer
1k
views
When does zero divergence imply a vector potential exists?
From electrodynamics we know that $\boldsymbol{\nabla}\mathbf{B}=\mathbf 0$ hence we can introduce a vector potential such that $\mathbf{B}=[\boldsymbol \nabla\times \mathbf{A}]$.
What is the general ...
- 151
1
vote
3
answers
283
views
Why does the curl of a function provide this particular amount of information?
In a classical electrodynamics textbook (Griffiths), it is mentioned that even though the electric field function, $E:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$, is a (3D) vector valued function, the ...
- 429
1
vote
1
answer
871
views
How do we compute Hodge duals?
The motivation for this question is to try to come up with a general expression for $(\star F)_{\mu\nu}$, the $\mu,\nu$ component of the Hodge dual of the Field strength tensor, which is of great ...
- 12.5k
1
vote
0
answers
124
views
Stokes theorem not holding
I have a vector field $\vec{H} = (8z,0,-4x^3)$
Naturally, $\nabla \times \vec{H} = (0,8+12x^2,0)$
Stokes theorem says:
$$
\int_s{\nabla \times \vec{H}} \cdot \vec{dS} = \oint_l{\vec{H} \cdot dl}
$$
...
- 123