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Questions tagged [electromagnetism]

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

80 questions with no upvoted or accepted answers
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88 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=​{\...
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113 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})...
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50 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
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1answer
48 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
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38 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
153 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
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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
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0answers
44 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
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50 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
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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
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0answers
213 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
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233 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
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162 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
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49 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)}...
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0answers
28 views

Trouble with Stokes' Theorem and the line integral of a piece-wise definition of a continuous curve using polar coordinates.

Problem Statement: Given: $\vec B = (\rho cos \phi)\hat \rho+(sin \phi)\hat \phi$ Verify Stokes' Theorem by evaluating: a) $ \oint\limits_c \vec B \bullet d\vec l$, where c represents the closed, ...
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27 views

Spatial curves that obey $z'-r^2 \theta' = \text{const.}$ in cylindrical coordinates

I am interested in a class of (arc-length parametrized) curves $\gamma:\mathbb{R} \to \mathbb{R}^3$ with the following property: If the curve is written in cylindral coordinates $(r,\theta,z)$, it ...
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0answers
17 views

Integral over Green's function of wave equation

the following paper says if $f(\vec{x})=f(x)$ does only depend on $x$ then we have $\int d^3x' \frac{e^{\pm ik_{\phi}|\vec{x}-\vec{x}'|}}{|\vec{x}-\vec{x}'|} f(\vec{x}')= \frac{2\pi i}{k_{\phi}} e^{ ...
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0answers
29 views

Approximating the gradient of a Finite Element solution on nodes

I have been working on a Finite Element implementation that approximates the solution to the following PDE in 3D using tetrahedral elements and piecewise linear basis functions. \begin{equation} \...
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0answers
24 views

Solving the limit in the definition of the curl

Given the definition of a curl: $$(\nabla \times \mathbf{F}) \cdot \mathbf{\hat{n}} \ \overset{\underset{\mathrm{def}}{}}{=} \lim_{A \to 0}\left( \frac{1}{|A|}\oint_{C} \mathbf{F} \cdot d\mathbf{l}\...
1
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1answer
38 views

How to find the magnetic field between two connected current-carrying wires?

How do I find the magnetic field at point $b$, very far from $a$? I know the magnetic field due to 1 current-carrying wire is $$B = \frac{\mu_0 i}{2\pi R}$$ So, does that mean the magnetic field at ...
1
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0answers
29 views

Gauss Law and Potentials

The infinite plane $z = 0$ is earthed and the infinite plane $z=d$ carries a charge of $\sigma$ per unit area. Find the electrostatic potential between the planes. I have tried to compute the ...
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0answers
19 views

Finding vector potential from a given magnetic field

I want to find the vector potential from a given magnetic field in three-dimension in cartesian coordinates for two cases. 1) Where the magnetic field is in any analytical form and 2) When the ...
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0answers
54 views

Vector Calculus - Evaluating $\nabla \times \mathbf{E} = -\frac{1}{c} \partial_t \mathbf{B}$

For the life of me, I cannot remember how to solve equations similar to the cross product equations in Maxwell's equations. I haven't used vector calculus of this level in quite some time and could ...
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0answers
15 views

Interface conditions on electromagnetic fields

Several authors (such as Jackson in his book "Classical Electrodynamics") state the following conditions at an interface between two different media: $(\vec{D_2} - \vec{D_1})\cdot \vec{n} = \sigma$ ...
1
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1answer
92 views

Computing (distributional) gradient of a singular function

This question could well belong better to the physics stackexchange, but I'm hoping that posting it here could give me a more mathematical perspective. I am trying to find the expression for the ...
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0answers
51 views

Calculus of variations ? electrostatic energy problem.

What is the maximum self-energy of an electrostatic distribution subject to the constraints that: the total charge is $1$; and the areal charge density anywhere is either $1$ or $0$. How ...
1
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1answer
27 views

Expansion of 1/R

My problem is the following, I have ${1}/{R}={1}/{|\vec{r}-\vec{r}'(t)|}$ which can be expanded as $1/R=1/r+\vec{r}\cdot\vec{r}'(t)/r^3+...$ How do I do this expansion? This was a part of a ...
1
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0answers
172 views

(Non)-orientable surfaces and (non)-coorientable surfaces (and a little bit of physics)

I (think I) know the difference between orientable and non-orientable topological surfaces. I don't know the difference between co-orientable and non-coorientable surfaces. I must admit that I am not ...
1
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1answer
90 views

Help with Derivation of memristor equation

I know this is not the EE stack exchange. I have tried there, no one is replying. Since this revolves around math, and Chua is laying down the mathematical framework for the memristor, I don't believe ...
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0answers
112 views

Integral of Coulomb interaction over sphere

I am stuck evaluating an integral that appears in a simplified theory of nuclear binding energy. The nucleus is modelled as a sphere of radius $R$ with a continuous charge distribution, and the ...
1
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0answers
674 views

How to Distinguish Between Energy and Coenergy in an Electric Field

I am perusing through my principles of electromechanics textbook and was puzzled by how the electric field energy equation was formulated. For example, the Electric field energy for a capacitor is ...
1
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1answer
144 views

Tensor manipulation involving Levi-Cevita tensor

I am attempting to follow a derivation from a physics paper relating to covariant electromagnetism. It is given that, $$ F^{\mu \nu} = u^{\mu}E^{\nu} - E^{\mu} u^{\nu} + \epsilon^{\mu \nu \alpha \...
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0answers
181 views

area of arbitrary surface element

I am a physics student with a minimal background in differential geometry and I am trying to determine an area element on an arbitrary surface. Suppose we have a surface parameterized by a function $z=...
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0answers
78 views

Convolution type integral with two variables

Is there some canonical way to approach integrals of type $$ I(k,q) = \int {\rm d}^{3} s~ e^{i k \cdot s} f\left(|s|\right)g\left(|q-s|^2\right), $$ where $s$, $k$ and $q$ are momentum vectors, and ...
1
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0answers
209 views

“Reciprocity” theorems, Green's second identity, and ways to convert elliptic PDEs to integral equations

A standard physics-textbook approach to derive an integral equation for the electrostatic potential is to use Green's second identity. The electrostatic potential $\Phi$ satisfies Poisson's equation, $...
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1answer
38 views

How do I show that the two definitions of the curl of a vector field equal each other?

The curl of a 3D vector field is a 3D vector itself and has two definitions - one in integral form and one in differential form. Definition 1: $$ \operatorname{curl}\vec{F}(x,y,z) \, \cdot \, \hat{n} ...
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0answers
33 views

Image Theory in Electrodynamics

I'm searching for a rigorous mathematical proof of the image theorem for electric/magnetic currents distributions. A proof that, I think, shows that removing the reflecting surface and placing ...
0
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1answer
36 views

Mathematical properties of electrodynamic potential

I am faced with a problem that is more mathematical than electrodynamic. However, not having a clearer or shorter title available, I preferred to highlight where the problem came from. However, ...
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0answers
19 views

Cycloid motion of charged particle in electromagnetic field

The question is from Schaum's Theoretical Mechanics. The electric field is given by $\underline E=E\hat k$ The magnetic field is given by $\underline B=-B\hat j$ Prove that the motion of a ...
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15 views

Triple infinite summation of a 3D Fourier series for Madelung Potential

I'm trying to evaluate the equation below excluding the case when $n_x=n_y=n_z=0$. I know this equation converges everywhere except where x,y, and z are all multiples of $2\pi$. I've attempted ...
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42 views

Question regarding substitution in integral

Suppose I want to make the substitution $\mathbf{k}\to-\mathbf{k}$ in the integral $$\int\mathrm{d}^3\mathbf{k}\,\boldsymbol{\alpha}(\mathbf{k},t)e^{-i\mathbf{k}\cdot\mathbf{r}}$$ where the domain of ...
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0answers
9 views

Diamond spring. Induced average current an direction

A diamond spring of side l is placed in a uniform magnetic field B. When pulling from opposite vertices, the spring deforms in a time interval delta t. Assuming that the total resistance of the ...
0
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1answer
41 views

Position and velocity problem

A particle's position is described by Cartesian co-ordinates $x\,\textbf{i} + y\,\textbf{j}$. It moves under the influence of a magnetic field $\textbf{B}=B\,\textbf{k}$ for $x>0$, and $\textbf{B}=...
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1answer
27 views

voltage matrix doesn't have a solution

Circuit Problem Follow the link to the circuit that I need to solve for all the resistor voltage drops. I have to make 5 linearly independent equations in order to solve for all 5 unknown voltages. ...
0
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0answers
38 views

Navier-Stokes smoothness problem and Gauge Theory

Recently, I came across this paper where the author describes an analogy between electrodynamics and fluid dynamics. He develops a one-to-one correspondence between the equations of electrodynamics ...
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0answers
23 views

Question with some vector integral identities

I'm working on the topic about the electromagnetic angular momentum and I found a reference which provides me an interesting decomposition. We know that these fields vanish at the infinity and that $\...
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0answers
29 views

Fourier Transform of Poisson's equation, then taking it back to real space.

I'm a bit stuck on a homework problem and could use some guidance. The problem asks to use a specific potential in Poisson's equation ($ \nabla^2\Phi = -\rho/\epsilon_0 $), Fourier transform it, ...
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0answers
26 views

What is the rule of integration of heaviside step function

I am trying to calculate an exterior multipole moment for a disc in the xy-plane and part of the integral involve a Heaviside function, i.e.: $$\int_{0}^{\infty}r^{l + 1}\Theta(R-r)dr \tag{1}$$ I ...
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0answers
35 views

Rotationally invariant Green's functions for the three-variable Laplace equation in all known coordinate systems

Green's function for the three-variable Laplace equation in Cartesian coordinates is $$\frac{1}{|\mathbf{r}-\mathbf{r'}|} = \frac{1}{\sqrt{(x-x')^2+(y-y')^2+(z-z')^2}}$$ It may be written in ...
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11 views

finite diference time domain on maxwells equations vs finite difference on magnetic and electric field with wave equations

So I'm just curious you can either write down Maxwell's equations for E and B, or just write wave equations with sources (assuming non zero charge density and current density). With the FDTD you have ...