# Questions tagged [electromagnetism]

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

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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 ...
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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 ...
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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 ...
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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. ...
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### 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 ...
1k views

### 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 ...
928 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 ...
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### Showing that $\Phi=\frac1{4\pi\epsilon_0}\int_{r>a}\frac{\rho_1(\mathbf{x})}r\operatorname{dV}$ for potential $\Phi$ and density $\rho_1$

Let $\mathbf{E_1}$ and $\mathbf{E_2}$, with potentials $\phi_1$ and $\phi_2$ respectively, satisfy $\mathbf{E}=-\nabla\phi,\space \mathbf{\nabla}\cdot\mathbf{E}=\frac{\rho}{\epsilon_0}$ with two ...
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### Magnetic force term in Kobe's derivation of Maxwell's equations

I'm trying to follow the paper "Derivation of Maxwell's equations from the gauge invariance of classical mechanics" by Donald Kobe, available here (American Journal of Physics, 1980): https://aapt....
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### Find directions where current is maximal

The current $J_i$ due to an electric field $E_i$ is given by $J_i = σ_{ij} E_j$ , where $σ_{ij} is the conductivity tensor. In a given Cartesian coordinate system,$σ=\begin{pmatrix}2&-1&-1 \\...
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### Setting up and solving this second order nonlinear differential equation

Background I'm trying to model a system where there are two magnets oriented such that they have attraction forces toward each other. One magnet is in a fixed position and the other magnet, $M$, is ...
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### $u = 0$ is the only solution to the homogeneous Helmholtz equation $\Delta u + k^2 u = 0$?

We have that the solution to the inhomogeneous Helmholtz equation $$\Delta u + k^2 u = f$$ can be represented by $$u(x) = \int_{\mathbb{R}^3}G(x - y) f(y) dy$$ where $G$ is the fundamental ...
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### Dipole-Coupling Tensor: Electrostatic Dipole Moments

I've been struggling with this problem today. Here's an image of the question I'm attempting to answer. I'm relatively new to tensor algebra (I've been studying it for about a week or two), and I've ...
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### Cauchy problem and boundary conditions in electromagnetism

Consider a connection on a principal $U(1)$-bundle $A_\mu$ over the flat base manifold $M_4$. The action of the theory is described in terms of the curvatures of such connection coupled to some source ...
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### What is the relation between 2d Fourier Transform and Plane Waves? [closed]

I'm not understanding how the two was related, but I was told that the 2d Fourier Transform decomposes an electromagnetic signal into plane waves. This, however, I am not understanding. I thought it ...
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### Galilean transformation law for electric and magnetic fields

Under Galilean transformations between a frame A and another frame B in which A is moving with constant velocity $\mathbf V$, a velocity $\mathbf v_A$ is frame $A$ is seen as \mathbf v_B = \mathbf ...
I have been reading the forums on how to solve the integral form of ampere's law and I have worked out that the correct way to solve it is to get rid of the dot product by realizing that $|B ∙ dr|$ is ...