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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12
votes
1answer
950 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 ...
10
votes
1answer
275 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 ...
9
votes
2answers
807 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 equations ...
7
votes
2answers
141 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. ...
7
votes
2answers
157 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 \...
5
votes
1answer
213 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{\...
5
votes
3answers
392 views

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 ...
4
votes
1answer
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 ...
4
votes
1answer
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 ...
4
votes
1answer
82 views

Deriving analytic expression for magnetic field & flow lines of bar magnet.

How can we analytically derive the flow-lines of a normal permanent bar-magnet? Physics context & own approach: In classical electromagnetics we have the legendary Maxwell's Equations: $$\begin{...
3
votes
2answers
32 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 ...
3
votes
1answer
55 views

Vector differential equation

In electromagnetism we often have a perpendicular constant magnetic field causing a charge to move in a circle. My question is, how do we formally solve this differential equation which involves a ...
3
votes
2answers
64 views

Can we motivate mathematically why wind turbines almost always have 3 flappers and aeroplane propellers can have any number of flappers?

Firstly I know some might frown upon a question so very broad and applied as this one. It really may not be a well defined mathematical question as some people would prefer on the site. I am okay with ...
3
votes
1answer
134 views

Wave propagation with a complex coefficient $\beta$ in a Robin boundary condition. How does this affect scattering from the boundary?

Can anyone give me an idea of what happens in the following situation involving Robin boundary conditions with a complex coefficient. Lets say we have an incident electromagnetic plane wave $u(x) = e^...
3
votes
2answers
252 views

Electric field lines between surfaces of hollow sphere

I was wondering what was the direction of the electric field between the two surfaces of a hollow sphere with constant charge density $\rho$. With the help of Gauss' Law I got the following absolute ...
3
votes
1answer
48 views

Question on differential equations with $\delta(x)$

In a course of Electrodynamics I came across a function for electric susceptibility $\chi(\tau)$ given by: $$\frac{d^2\chi}{d\tau^2}+\gamma \frac{d\chi}{d\tau}+\omega_0^2\chi=\omega_p^2\delta(\tau)$$ ...
3
votes
1answer
84 views

Does this integral have a simple expression

I am trying to find a simple expression for the integral $\hskip 3cm{\displaystyle 2a\,\text{sin}\,\alpha\int\limits_0^\infty \big(1 - \text{tanh}\,\pi x\,\text{tanh}\,\alpha x}\big)\, dx, \quad a \...
3
votes
1answer
862 views

Proving delta dirac properties

I'm currently studying electromagnetism from Reitz's Foundations on electromagnetic theory, in that book delta dirac is presented as a "function" $\delta$ satisfying: $\delta(x)=0$ for any $x \not=0$...
3
votes
0answers
81 views

Do Maxwell's equations (generalized) apply to _every_ $k$-form on a pseudo-Riemannian manifold?

Given a pseudo-Riemannian $n$-manifold and a $k$-form $F$ on the manifold, I will call its exterior derivative $J=dF$ the source of $F$ and the differential $K=dG$ the dual source of $F$, where $G=​{\...
3
votes
0answers
109 views

Radial fourier transform of gaussians

In this paper is calculated the square modulus of the radial fourier transform of the function $\rho(r)$ $$\left|F(q)\right|^2=\left| \int_{\mathbb{R}^3} e^{i\mathbf{q}\cdot\mathbf{r}}\rho(\mathbf{r})...
3
votes
2answers
1k views

Why do time derivatives change to '$j\omega$' when taking the time derivative of a phasor?

For example, Faraday's law in the time domain is written as $$\nabla \times \vec{\mathbf{E}} = - \frac{\partial\vec{\mathbf{B}}}{\partial t}$$ When using phasor notation, Faraday's law is written as $...
2
votes
1answer
91 views

Equations of motions for a spin interacting with a magnetic field.

Consider the Lagrangian $$L(\theta, \phi) = (-1+\cos \theta) \dot{\phi} + \mathbf{B}\cdot \mathbf{n}$$ Where $\mathbf{B}$ is a constant magnetic field and $\mathbf{n}$ denotes a point on the unit ...
2
votes
1answer
159 views

Deriving units using equation

I have the question "Using the equation and information below, derive the units for the permittivity of free space , $\epsilon_{_0}$. $V = \dfrac 1 {4 \pi \epsilon_{_0}} ~ \dfrac Q r$ $V$ is ...
2
votes
2answers
213 views

What is an unit area vector?

As the title suggests, What is a unit area vector? I've tried googling but unable to arrive at any satisfactory answers. Any help is appreciated.
2
votes
2answers
225 views

Gauss' law and a half-cylinder

The question is: A half cylinder with the square part on the $xy$-plane, and the length $h$ parallel to the $x$-axis. The position of the center of the square part on the $xy$-plane is $(x,y)=(0,...
2
votes
1answer
49 views

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 ...
2
votes
1answer
44 views

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....
2
votes
1answer
32 views

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 \\...
2
votes
2answers
208 views

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 ...
2
votes
1answer
61 views

How would you solve this triple integral?

I was given to find the total charge inside a cube, side equal $a$ with one corner at the origin, produced by this charge distribution: $$\rho=\frac{\epsilon_0E_0xyz}{a^4}\exp\left(\frac{-(x+y+z)}{a}\...
2
votes
1answer
62 views

Need help understanding a particular proof regarding integration to find the net electric field

I have the following question regarding Electromagnetism. I have placed the question here instead of Physics Stack exchange since it is specifically the mathematics and not the physics concepts that I ...
2
votes
1answer
314 views

Showing that no current flows in some direction, given $\sigma_{ij}$ and that $J_i=\sigma_{ij}E_j$.

The current $J_i$ due to an electric field $E_i$ is given by $J_i=\sigma_{ij}E_j$, where $\sigma_{ij}$ is the conductivity tensor. In a certain coordinate system, $$(\sigma_{ij})=\begin{pmatrix} 2&...
2
votes
1answer
205 views

$u = 0$ 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 ...
2
votes
1answer
70 views

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 ...
2
votes
0answers
37 views

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 ...
2
votes
1answer
44 views

Dirichlet problem on a wire: a co-dimension 2 boundary condition

A wire may be thought of as a smooth compact curve $C \subset \mathbb{R}^3$ with boundary two endpoints. Suppose we are given a smooth $\phi: C \to \mathbb{R}$ (a potential on the wire), then can $\...
2
votes
1answer
111 views

Why do we need both Divergence and Curl to define a vector field?

I was reading Classical Electrodynamics by J.D.Jacskon (section 1.5) where he said: Perhaps some readers know that a vector field can be specified almost completely if its divergence and curl are ...
2
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0answers
37 views

Uniqueness of charge distribution

Some exercises in general physics ask for students to find specific charge distributions on the boundaries of given conductors in various of situations. Usual answers goes like follows; firstly using ...
2
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0answers
139 views

Prove Property of Green Function Solution to Laplace Equation in a 2D-square

Let's consider a 2D-square with 4 euqal subsquares containing different dielectrics. Inside the square domain, the unkown potential function $\Phi$ satisfies the Laplace equation: $\nabla^2\Phi=0$ ...
2
votes
0answers
12 views

Why does the curl of a function provide this particular amount of information? [duplicate]

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 ...
2
votes
0answers
29 views

Charge density of charged conductor with flat side

Given a charged conducting body with a flat side, can the charge density (and hence the normal electric field) be constant on the flat part? According to physics lore, the charge density is greater ...
2
votes
0answers
124 views

Modelling Diode current with ODE [closed]

I want to write ODE system for simulating following electrical circuit: At each small step dt i just do euler integration. I only know ODE for leaky capacitor: <...
2
votes
0answers
42 views

Find wave function satisfying Schroedinger Equation $[\frac{1}{2m}(\frac{\hbar}{i}\nabla-\frac{e} {c}A(r))^2+V(r)]\psi^{'}(r) = E\psi^{'}(r)$

Given an "ordinary" wave function $\psi(r)$ that satisfies the "ordinary" stationary Schrodinger equation $[-\frac{\hbar^2}{2m}\Delta+V(r)]\psi(r)= E\psi(r)$, I want to construct a wave function $\psi^...
2
votes
0answers
77 views

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 ...
2
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0answers
48 views

Does every field with radial dependence $r^{-2}$ violate Gauss' law for magnetism?

We know that a magnetic field in the form $$\vec B = k \frac{\hat r}{r^2} \tag{1}$$ where $k$ is a constant, violates Gauss' law $$\vec \nabla \cdot \vec B = 0 \tag{2}$$ Indeed, we have $$\vec \...
2
votes
0answers
35 views

How to prove that $\int \varphi \, d\ell=-\int \operatorname{grad}\varphi\wedge dS $.

Let $\varphi$ be a scalar function having a continuous gradient throughout a region $R$ of space, let $\ell$ be the boundary curve of any surface $S$ lying entirely within the region $R$, then prove ...
2
votes
0answers
205 views

Find the potential at the center of the sphere.

The inside of a grounded spherical metal shell of radius R is filled with charge of uniform density ρ. Find the potential at the center. My attempt: $$V=\frac{1}{4\pi \epsilon_0} \int_{V} \frac{\rho}...
2
votes
0answers
230 views

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 ...
2
votes
0answers
156 views

Using Line Integrals to solve Amperes Law

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 ...
2
votes
0answers
47 views

Solving a 1D integral with system of equations for retarded electromagnetic fields

I need to solve the following integral to calculate the effect of retarded electromagnetic fields on a test charge: $\int\limits_0^\zeta\frac{(\psi-(1+x)\sin(\psi+\alpha))(\frac{\psi^2}{2\beta^2(1+x)}...