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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.

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Deriving the Yukawa potential from the field of a screened charge

I am trying to derive the Yukawa potential from the electric field of a screened positive point charge, which is $$ \vec{E}(\vec{r}) = \frac{q}{4\pi\epsilon_0}\frac{e^{-kr}(kr+1)}{r^2}\hat{r}. $$ The ...
lain's user avatar
  • 159
-2 votes
0 answers
20 views

singular positive semi-definite matrix in electromagnetism

anyone knows where he drew this conclusion from?
user900476's user avatar
0 votes
1 answer
103 views

The reason for curl free

I wonder about the reason for the idea of this, would you mind explain for me this can happen in mathematics. Thank you !
Đôn Trần's user avatar
4 votes
0 answers
119 views

The magnetic field of a spinning charged sphere

Evaluate $\displaystyle \int_{0}^{2\pi}\int_{0}^{\pi}\frac{(z_0-R\cos\theta)\sin^2\theta\cos\phi}{[(x_0-R\cos\phi\sin\theta)^2+(y_0-R\sin\phi\sin\theta)^2+(z_0-R\cos\theta)^2]^{\frac{3}{2}}}d\theta d\...
grj040803's user avatar
  • 701
1 vote
0 answers
25 views

2d Fourier Transform Using Weyl Expansion

If we have electric field as $$ \mathbf{E}\left(\mathbf{r}_{\mathrm{d}}, t\right)=\frac{1}{\varepsilon} \int_{\mathcal{V}} d \mathbf{r}^{\prime} \mathbf{K}\left(\mathbf{r}_{\mathrm{d}}-\mathbf{r}^{\...
Hassan Abbas's user avatar
2 votes
3 answers
83 views

How to compute the volume integral for the potential of an arbitrary point outside a uniformly charged ball?

$$\frac{\rho}{4\pi\epsilon_0}\iiint_{D}^{}\frac{1}{\left\| \mathbf{r}-\mathbf{r'} \right \| }dV'$$ $D$ is a ball of radius $R$ $\mathbf{r}$ is the position vector of the point where we want to ...
giannisl9's user avatar
  • 163
0 votes
0 answers
48 views

Electric fields and simply-connected regions

I apologize for the ignorance and the rough English in advance, I have an issue understanding how to match both what happens in physics and what I am seeing in calculus. We learned that if a vector ...
Some random guy's user avatar
2 votes
2 answers
77 views

How to apply integration by parts to simplify an integral of a cross product?

I'm reading a physics paper and am trying to figure out how a certain expression is derived (If interested, see Appendix of the paper, Eq. (A7), (A8)). The authors skip a lot of derivation steps and ...
RawPasta's user avatar
0 votes
0 answers
47 views

Solving 4th order differential equation

I have differential equations such as $$\frac{1}{\lambda^{2}}\psi_{e}'' = \tanh{\psi_{e}+\psi_{h}}$$ $$\psi_{h}'' - \kappa^{2}\psi_{h} = -\alpha^{2}\tanh{\psi_{e}+\psi_{h}}.$$ Boundary conditions are $...
이영규's user avatar
1 vote
0 answers
23 views

Helmholtz - Hodge decomposition in H(curl)

I'm looking for Helmholtz-Hodge type decompositions but for vector fields slighty more regular than $L^2(D,\mathbb{R}^3)$. I'm familiar with the results in the books of Lions and was wondering if ...
Caillou's user avatar
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3 votes
0 answers
91 views

First chern class of magnetic monopole

Example 3.5.5 in the Mirror Symmetry textbook (Hori-Katz-Klemm-et. al) states: Let us compute the first Chern class of the line bundle defined by the $U(1)$ gauge field surrounding a magnetic monopole,...
locally trivial's user avatar
1 vote
3 answers
152 views

$\int_0^R \frac{r^{l+1}}{\sqrt{R^2 - r^2}}\text dr$

I'm trying to solve problem 3.3 from Jackson's Classical Electrodynamics, but I'm encountering some troubles solving $$ \int_0^R \frac{r^{l+1}}{\sqrt{R^2 - r^2}}\text dr, \qquad l = 0,2,4,6,\ldots $$ ...
Peluche's user avatar
  • 135
1 vote
1 answer
100 views

Representation of $e^{ikR}/R$ as integral of a Bessel function [closed]

In this paper about the electrodynamcis of a spiral resonator, the authors write $$\frac{e^{-ikR}}{R}=\int_{0}^{\infty} \frac{xdx}{4\pi\sqrt{x^2-k^2}}J_0(Dx)e^{-\sqrt{x^2-k^2}|z|}$$ with $R=\sqrt{z^2+...
kiterosrp8's user avatar
0 votes
1 answer
75 views

What integral is used to calculate the electric field generated by a continuous charged curve?

I'm studying Multivariable Mathematics, by Ted Shifrin, in which one reads that ''the gravitational force exerted by a continuous mass distribution $\Omega$ with density function $\delta$ is $$\mathbf{...
Henrique Fonseca's user avatar
0 votes
0 answers
61 views

Is $\frac{\partial}{\partial{t}}(\nabla\times H) = \nabla \times \frac{\partial H}{\partial t}$?

While trying to prove a particular equation using Maxwell's equations in electromagnetic theory, there is a step in my textbook that says $$\frac{\partial}{\partial{t}}(\nabla\times H) = \nabla \times ...
Sasikuttan's user avatar
1 vote
1 answer
60 views

How to prove this vector identity? [closed]

I've seen this vector identity from the book[1] in page 89, $$ (\nabla p)\times\nu =0,\ \text{on}\ \partial\Omega,$$ where $\nu $ is the outer normal vector of $\partial \Omega$, $ p \in H_0^1(\Omega)....
Du Xin's user avatar
  • 45
2 votes
0 answers
29 views

Does this family of curves appearing in the magnetic field of a coil have a name?

While attempting to express the magnetic field induced by a single coil of current (at any point in space, not just on the coil's axis), I tried visualising the set of the infinitesimal contributions $...
Sileo's user avatar
  • 165
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0 answers
34 views

Boundary Conditions on the Magnetic Flux Density (B-field)

My question is similar to this one (Boundary conditions magnetic field) in that it is related to the boundary conditions of the magnetic field (B-field). However, my question focuses on mathematically ...
Blue Various's user avatar
0 votes
0 answers
67 views

Solving a funky differential equation.

I'm currently trying to solve the DE that defines charge in a circuit containing an Inductor, Capacitor, Resistor and (crucially) a Memristor. This needs to be able to work for any variable values and ...
Seb's user avatar
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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}=-\...
Sirius Black's user avatar
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 ...
Sirius Black's user avatar
0 votes
1 answer
44 views

Partial differential equation with Faraday's equation

We were asked to find what equation is satisfied by $\Psi(x,y,z,t)$ given that $\textbf{B} = \nabla \times (\textbf{z} \Psi)$ and $\textbf{E} = -\textbf{z} \frac{\partial \Psi}{\partial t}$ while ...
riescharlison's user avatar
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(...
Al.G.'s user avatar
  • 1,490
1 vote
2 answers
77 views

What does $\vec{\nabla}^2 \vec{E} = \vec{\nabla}^2 \left[ f(\vec{k} \cdot \vec{r} - \omega t) \vec{E}_0 \right]$ mean?

$\vec{E} = f(\vec{k} \cdot \vec{r} - \omega t) \vec{E}_0$ with the constant vector field $\vec{E}_0$ I only know the case if I apply the Laplacian operator on a scalar field, in this case it is a ...
CherryBlossom1878's user avatar
1 vote
0 answers
55 views

Writing momentum 4vector as an integral over the EM stress-energy tensor

I have been watching a series of lectures on general relativity by Neil Turok and I have run into a problem. In one of the lectures, the professor writes the momentum 4-vector as a contraction of the ...
Jesse Van Der Kooi's user avatar
0 votes
0 answers
17 views

if divergence of a vector is zero, how to find the spherical coordinate of the vector?

The perturbed part of magnetic field is $\mathbf{\delta B}$ where $\mathbf{\delta B} = \delta B_x(x,y), \delta B_y(x,y)$ and $\nabla \cdot \mathbf{\delta B} = 0$. To prove $\mathbf{\delta B} = \delta ...
Mon's user avatar
  • 37
-1 votes
1 answer
116 views

Polar coordinates: What unit vectors span the $(r,\theta)$ space? [closed]

Polar coordinates: What unit vectors span the $(r,\theta)$ space? I am thoroughly confused. If in the Cartesian system, the associated orthonormal polar vectors at different points on a circle keep ...
S_M's user avatar
  • 419
1 vote
2 answers
176 views

Book Recommendation: One that has a lot of problems and theory associated with polar coordinates and spherical polar coordinates

I would like to "master" polar coordinates and spherical polar coordinates. In the sense, I would like to become as well versed with them as I am with cartesian coordinates. I have gone ...
S_M's user avatar
  • 419
2 votes
1 answer
85 views

How to evaluate the integral $\int_{-r/2}^{r/2} \int_{-r/2}^{r/2} \frac{1}{x^2+y^2+r^2/4} dx dy$

I came across this integral while trying to evaluate the electrical force exerted by a charged plate in the form of a square with side length $r$. I tried the usual method of first keeping $y$ ...
Alp's user avatar
  • 409
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 ...
KDP's user avatar
  • 1,111
1 vote
2 answers
109 views

Interpreting the cohomology class of the Maxwell tensor.

In the introduction to Bott and Tu, “Differential forms in algebraic topology” there is the motivating example of a stationary point charge in $3$-space. The electromagnetic field $\omega$ is a $2$-...
Parth Shimpi's user avatar
3 votes
0 answers
47 views

An improper integral from Jackson's book involving the modified Bessel function

When deriving the angular distribution of energy for synchrotron radiation one has to evaluate two tricky improper integrals (see [1] below): $$ I_1 \equiv \int_{0}^{\infty} x^2 [K_{2/3}(x)]^2 \, \...
Gabriel Macedo's user avatar
0 votes
1 answer
43 views

Analytically solving PDEs on irregular domains in Physics

In many Physics courses you solve PDEs like heat or wave on square, circular, or spherical domains with separation of variables. Are there ways to solve PDEs and Boundary value problems on irregular ...
Masteralien's user avatar
0 votes
0 answers
66 views

Calculate Electric Field on the Z-axis from a finite charge wire

I've been trying to find the electric Field on the Z-axis from a non-uniform charge density line charge. The wire is placed on the z-axis from $z=0$ to $z=1$, $E=?$ at $z>1$ and $z<0$ $$ \rho =...
gus2427's user avatar
0 votes
1 answer
55 views

I was trying to find field due uniformly charged sheet at a distance h from centre of the square sheet

I assumed the square(side a) sheet to be made up of wires.$$dE=Kdq/r^2$$ The field due to a wire is : Reference $$\frac{K\lambda}{d}\left[\frac{x}{\sqrt{d^2+x^2}}\right]^{(a/2)}_{(-a/2)}=\frac{K\...
Aurelius's user avatar
  • 471
0 votes
0 answers
24 views

Integrals for the the localized pyramid basis functions in Galerkin Method

I tried to show the following relations for the localized pyramid basis function $\phi_{i j}(x, y)=(1-|x| /$ $h)(1-|y| / h),|x|<h,|y|<h$, where $x$ and $y$ are measured from the site $(i, j)$. ...
Márquez Carranza Arturo Ariel's user avatar
0 votes
0 answers
26 views

Flux of a vector field on a non-smooth surface? (in terms of electromagnetism)

While studying the famous Ampere's law, I came up with the following vector field $F$ and a surface $S$ lying in $R^3$. (In terms of physics, $F$ is the current density of some current in a circuit, ...
jkuk5046's user avatar
0 votes
0 answers
63 views

Vector Line Integral For Biot Savart Law

How would one go about computing the vector line integral presented in the Biot-Savart law: $$\vec{B}=\int_c\frac{\mu_0I}{4\pi} \frac{d\vec{l}\times\hat{r}}{r^2}$$ I know how to compute vector line ...
JBatswani's user avatar
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: $$ \int_s{\nabla \times \vec{H}} \cdot \vec{dS} = \oint_l{\vec{H} \cdot dl} $$ ...
rjpj1998's user avatar
  • 123
1 vote
1 answer
110 views

Stokes theorem 2 sides not matching with Magnetic waves

We have been asked to verify stokes theorem for a magnetic field. We know Stokes theorem states, for any vector field $\vec{H}$: $$\int_S{(\nabla \times \vec{H}) \cdot \vec{dS}} = \oint_L{\vec{H} \...
rjpj1998's user avatar
  • 123
1 vote
1 answer
146 views

Pseudo-vector formal definition

I have a question about the formalization of pseudovectors. Wikipedia (and my electromagnetism professor and all the electromagnetism books) only state that a vector $v$ transforms as $v' = Rv$, while ...
QuantumBrachistochrone's user avatar
3 votes
1 answer
84 views

Solving $2\frac{\partial S}{\partial z}+\left(\frac{\partial S}{\partial r}\right)^2=0$

I have been trying to solve this PDE $$2\frac{\partial S}{\partial z}+\left(\frac{\partial S}{\partial r}\right)^2=0$$ The solution of this equation corresponding to a spherical wave of radius of ...
Nikhil Mehra's user avatar
0 votes
0 answers
89 views

Divergence theorem with normal component of a curl to a surface

Let $\mathbf{A}$ be a vector function in $\mathbf{R}^3$ and we want to find the normal and tangent components of $\nabla \times \mathbf{A}$ on a smooth and closed surface $\Gamma$. $\mathbf{n}$ is the ...
Francisco Sáenz's user avatar
0 votes
0 answers
63 views

Electric field flux proportional to the field lines generated by (for example) a static charge

Suppose we have a stationary positive charge at a point in space that we call $+Q$. We know by definition that the flow of the electrostatic field is given by, in its simplified form, $$\Phi_S(\vec E)=...
Sebastiano's user avatar
  • 7,792
1 vote
1 answer
93 views

Distance becoming equal to displacement

Consider a charged particle of charge q and mass m being projected from the origin with a velocity u in a region of uniform magnetic field $\mathbf{B} = - B \hat{\mathbf{k}} $ with a resistive force ...
Srish Dutta's user avatar
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 $R$ ...
Tomy's user avatar
  • 429
6 votes
1 answer
111 views

What is the value of $\frac{1}{2}\int_B\int_B\frac{\rho(x,y,z)\rho(x',y',z')}{4\pi\epsilon_0\sqrt{(x-x')^2+(y-y')^2+(z-z')^2}}dxdydzdx'dy'dz'$?

I am reading a book about electromagnetism by Yousuke Nagaoka. Suppose $R$ is a positive real number. Suppose $Q$ is a positive real number. Let $B:=\{(x,y,z)\in\mathbb{R}^3:\sqrt{x^2+y^2+z^2}\leq R\}...
tchappy ha's user avatar
  • 8,750
1 vote
0 answers
65 views

Non-homogeneous wave equation, retarded potentials and causality

Consider the non-homogeneous wave equation in three dimensions with homogeneous initial conditions: $$ \begin{align} & \square f(\underline{x}, t) = g(\underline{x},t), \hspace{3mm} \underline{x} \...
Matteo Menghini's user avatar
4 votes
1 answer
70 views

Linear system $Ax=y$ with partially known $x,y$ and non singular $A$

PHYSICAL INTUITION While proving the equivalence between the Dirichlet problem (i.e. the potential is known on the surface of every conductor) and the mixed problem (i.e. the potential is known on ...
Matteo Menghini's user avatar
0 votes
1 answer
73 views

Why does $\boldsymbol{\nabla} \times \textbf{E}=\textbf{0}$ imply $\boldsymbol{E_2}^{\parallel}=\boldsymbol{E_1}^{\parallel}$?

I am currently studying 'Introduction to Electromagnetism' by David Griffiths, and I was reading about the electric displacement $\boldsymbol{D}$. I decided to try to extract eq. 4.27, which states: $\...
Rasmus Andersen's user avatar

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