Questions tagged [electromagnetism]
For questions on Classical Electromagnetism from a mathematical standpoint. This tag should not be the sole tag on a question.
269
questions
3
votes
3answers
64 views
Divergence of a radial vector field
I am reading Modern Electrodynamics by Zangwill and cannot verify equation (1.61) [page 7]:
\begin{equation}
\nabla \cdot \textbf{g}(r)=\textbf{g}^{\prime}\cdot \mathbf{\hat{r}},
\end{equation}
where ...
7
votes
0answers
155 views
Finding smooth behaviour of infinite sum
Define
$$E(z) = \sum_{n,m=-\infty}^\infty \frac{z^2}{((n^2 + m^2)z^2 + 1)^{3/2}} = \sum_{k = 0}^\infty \frac{r_2(k) z^2}{(kz^2 + 1)^{3/2}} \text{ for } z \neq 0$$
$$E(0) = \lim_{z \to 0} E(z) = 2 \pi$$...
2
votes
0answers
34 views
Is there a way to represent electrostatics tensors in tensor (possibly tensor product) way?
I'm working with electrostatic interaction tensors, which are defined as follows:
\begin{align}
T &= \frac{1}{r} \\
T^\alpha &\equiv \nabla T = -\frac{r^\alpha}{r^3} \\
T^{\alpha\...
6
votes
1answer
297 views
Representation of the magnetic field in 2D magnetostatics
Consider a magnetostatics problem in $\mathbb{R}^3$. The problem is governed by the following equations
$$\begin{aligned}\text{Maxwell's equations}\quad &\begin{cases}\nabla\times H(x)=J(x)\\\...
0
votes
1answer
24 views
How to solve a pair of coupled Poisson equations with inhomogeneous boundary conditions?
I am trying to make some code that will solve the following 2D Poisson equations:
$$\left(\frac{\partial^2}{\partial x^2}+\frac{\partial}{\partial y^2}\right)P(x,y) = f(x,y),$$
$$\left(\frac{\partial^...
0
votes
0answers
26 views
How to find the hyperboloid points from an electrostatic potential?
Can someone help me with this?
Find the hyperboloid points: $\sigma: x^2+y^2-z^2=1$
In which the electrostatic potential
$$u(X) = \frac{q}{4\pi\varepsilon_{0}}\frac{1}{\left|PX\right|}=\frac{q}{4\pi\...
0
votes
0answers
49 views
Electric field with $\nabla \times \vec{E} = 0$ and $\nabla \cdot \vec{E} = 0$ outside of a conductor circulates a constant electric current
Q: suppose that I know that outside of some conductor circulates a constant electric current , I have $\nabla \times \vec{E} = 0$ and $\nabla \cdot \vec{E} = 0$. How do I prove that $\vec{E} = 0 $ ...
0
votes
0answers
20 views
Differential equation for a vector potential
From Helmholtz’s theorem, any smooth vector field $\mathbf{F}$ that goes to zero at infinite distance can be uniquely decomposed everywhere in the sum of a divergence free component and an ...
1
vote
1answer
31 views
Why must the $B_\theta$ and $B_\phi$ components of the magnetic field be zero at $\theta=0\, \&\, \pi$?
I have been reading this paper and it says (see the last paragraph in the screenshot below) that 'the latter condition requires that $B_\theta$ and $B_\phi$ vanish along the axis $\mu=1,\ -1$'. Why ...
0
votes
1answer
90 views
Why does the electromagnetic tensor in component form coincide with the differential-geometric definition of a $2$-form?
From physics classes, I understand the electromagnetic field strength tensor to be defined as
$$F^{\mu\nu}=\partial^\mu A^\nu-\partial^\nu A^\mu \;,$$
where $\partial^\mu$ is the partial derivative (...
1
vote
0answers
69 views
Representing flux tubes as a pair of level surfaces in R^3
I am trying to see if Vector fields(I am thinking of electric and magnetic fields) without sources(divergence less) can be represented by a pair of functions f and g such that the level surfaces of ...
0
votes
0answers
36 views
Electric potential of a semi-spherical shell in the origin
I have to find the eletrical potential in the center of a semi spherical shell of inner radius "a" and outter radius "b" that has volumetric density $\rho=\frac{1}{r}$.
Here's a ...
5
votes
1answer
105 views
Evaluate Integral $ I:=\int_0^{2\pi} \cos s \,\log (\sqrt{c^2 + a - 2 \cos s}-c) \, \mathrm d s $ for radially magnetized cylinder
When trying to evaluate the magnetic scalar potential $\Phi_m$ of a magnetized cylinder (Magnetization $M$ in $x$-direction, height $Z$, Radius $R$, touching the $xy$-plane from below), I was able to ...
0
votes
2answers
49 views
What's the criterion for neglecting terms "much smaller than" other terms?
In a physics exercise relating some electric forces. I found the following equation for the restoring force:
$ F = \frac{kQq}{(y_0 + \Delta y)^2} - \frac{kQq}{y_0^2}$
Where $ \Delta y << y_0 $
...
0
votes
1answer
32 views
Shifting magnetic field axis
Considering the image below, I have magnetometer readings while my magnetometer is oriented along x' and y'. Can I convert these readings to get the equivalent reading if my magnetometer was oriented ...
0
votes
0answers
26 views
Wave equation and skin-depth
This is a follow-up question to one of my unanswered questions
Complex case of the auxiliary equation in ODEs versus PDEs when using separation of variables
In a nutshell, the textbook solved the ...
2
votes
2answers
70 views
Electric potential : numerical value for the triple Integral
The function $\phi:L\to\mathbb{R}$ where $L={\{(x,y)\in\mathbb{R}^2:x^2+y^2=4\}}$ is defined as,
\begin{align*}&\phi(x,y)=\\
&\int_{0}^{\pi}\!\!\!\!\int_{0}^{2\pi}\!\!\!\!\int_{1}^{2}\!\!\frac{...
2
votes
1answer
23 views
How can I expand on this part of this proof concerning magnetic fields?
I am supposed to prove:
$$B_y=\frac{\mu_0}{4\pi}Iaz\int_{0}^{2\pi} \frac{\sin\phi}{(a^2+y^2+z^2-2ay\sin\phi)^{3/2}}d\phi=\frac{\mu_0Ia^2}{4r^3}\biggl(\frac{3yz}{r^2}\biggl)$$
and
$$B_z=\frac{\mu_0}{4\...
0
votes
0answers
11 views
What is the biharmonic operator of a vector ($\nabla^4 \vec{A}$)?
I'm trying to implement a new term into a code that solves electromagnetism, and it is equal to
$$
\frac{\partial \vec{B}}{\partial t} = \nu \nabla^4 \vec{B}
$$
I gather that $\nabla^4$ is called the ...
3
votes
1answer
57 views
Proof that the vector area is the same for all surfaces sharing the same boundary
In the book, Introduction to Electrodynamics by Griffiths (4th edition) in question 1.62 part c, we are asked to prove that the vector area is the same for all surfaces sharing the same boundary. The ...
4
votes
2answers
105 views
Can a small change to the magnetic field result in infinite changes to the vector potential?
Consider a magnetic field, $\mathbf{B}(x,z)$ given by
$$\begin{aligned}
\mathbf{B}(x,z) &= [B_x(x,z),\ B_y(x,z),\ B_z(x,z)] \\
&= \left[-\frac{l}{k}\cos(kx),\ -\sqrt{1-\frac{l^2}{k^2}}\cos(kx),...
0
votes
1answer
32 views
Taylor expansion of $\frac {1}{|x-y|}$with x and y two vectors
This equation comes from a physics script on electrodynamics, saying that this equation comes from a Taylor series expansion.
I understand the first equality, but not the second one. It is really not ...
2
votes
1answer
44 views
Where does this matrix, $\begin{pmatrix}0 & \mathcal{B}_{z} \\-\mathcal{B}_{z} & 0\end{pmatrix}$ come from in the Lorentz force law?
The anisotropic conductivity of the Hall configuration.
We will only explore the case of perpendicular electric and magnetic fields, throughout the course, with the convention that: $\boldsymbol{\...
1
vote
0answers
80 views
Biot-Savart law on an exponential spiral
A wire carrying a current $I$ is bent into the shape of an exponential spiral, $r = e^θ$, from $\theta = 0$ to $\theta = 2\pi$ as shown in the figure below.
To complete a loop, the ends of the spiral ...
0
votes
0answers
29 views
Spherical harmonic expansions and magnetic monopoles
The solution to Laplace's equation $\nabla^2 \Phi = 0$ in spherical coordinates for an internal source is,
$$
\Phi(r,\theta,\phi) = \sum_{l=0}^{\infty} \sum_{m=-l}^l a_l^m \frac{1}{r^{l+1}} Y_l^m(\...
0
votes
0answers
28 views
Induced electromotive force
I need help with this task, if anyone had a similar problem it would help me a lot.
The task is:
In addition to a very long conductor in which there is a simple periodic current $i(t)=\sqrt{2}I\cos(wt)...
1
vote
0answers
76 views
Vector spherical harmonics derivatives in spherical coordinates
I am interested in taking the derivative of vector spherical harmonics in the spherical coordinate basis and I was wondering whether anyone had any good resources where to look for help in ...
3
votes
0answers
50 views
Solving cylindrically symmetric non-homogeneous wave equation $\nabla^2\mathbf{A}(s,t)-\frac{\partial^2}{\partial t^2}\mathbf A(s,t)=\mathbf J(s,t)$
I'm trying to solve a non-homogeneous wave equation in cylindrical coordinates
\begin{align}
\nabla^2\mathrm A-\frac{\partial^2\mathrm A}{\partial t^2}=\mathrm J,
\end{align}
where A and J are ...
0
votes
1answer
38 views
What's the easiest way to show rigorously that the integral form of Gauss's law follows from Coulomb's law?
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 ...
0
votes
0answers
41 views
Maxwell equation and divergence theorem contradict?
In my engineering mathematics class, I learned about Gauss' law and form of one Maxwell equation.
I learned that
$∇∙E=ρ/ε$
in Maxwell's equation
$∇∙E=0$
in process of proving Gauss' law.
force of ...
0
votes
0answers
16 views
Integral involving the module of a vector
While working on a proof in electrodynamics, I found the following integral: $\int_{0}^{R} \frac{r^{2}}{|\vec{r}-\vec{a}|^{3}}dr$, where $\vec{a}$ is a constant vector whose module is less than $r$. ...
2
votes
0answers
22 views
How to apply phasor transformation with multiple sinusoidal functions
I am seeing some questions in my textbook involving phasor transformation with multiple functions of cosine or sine in multiplication with each other but they didn't exactly showed how to do it and I ...
4
votes
1answer
135 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 ...
0
votes
0answers
28 views
Charge distribution Ohm's law
According to Ohm's law $$\textbf{J}=\sigma\textbf{E}$$
where $\textbf{J}$ is the current, $\sigma$ is the electric conductivity, and $\textbf{E}$ is the electric field. Now from the continuity ...
0
votes
1answer
38 views
Gradient in tensor form
I found a problem which had $$\partial_i (A_i \vec{G})= (\vec{\nabla} .\vec{ A} )\vec{G}+ (\vec{A}.\nabla) \vec{G} $$ but my problem is what does $$\partial_i (A_i \vec{B})$$ even mean? it doesn't ...
0
votes
0answers
34 views
Electrostatic Potential from Array of Equally Charged Strips
This is question 5.1 from Mathews and Walker , Mathematical Methods of Physics:
it is very difficult for me to find which conformal transformation would simplify the spaced arrangement as in the ...
2
votes
0answers
56 views
Can someone find $\vec{A}$ for this example [found in my TB : Griffith] with this method?
Example 10.2 of 3rd edition Griffith [electrodynamics]
click here to read this question
So I thought to convert I into $\vec{J}$ as follows :
$$\vec{J}(\vec{r},t)= I_o\theta(t)\delta(x)\delta(y)\hat{z}...
0
votes
1answer
36 views
Vector Analysis cross product and dot product
Three vectors $\vec A,\vec B,\vec C$ Compare the value of these 4 questions.(use the parallelepiped)
a) $(\vec A\times \vec C)\cdot\vec B$
b) $\vec A\cdot(\vec C\times\vec B)$
c) $(\vec A\times\vec B)\...
0
votes
2answers
64 views
calculating the area of a circle by integrating over straight lines
The Problem
Imagine having a circle of radius $R$. We all know the area of this circle is $A=\pi R^2$ and the circumference is $2\pi R$. I would like to know why this approach for calculating the area ...
0
votes
0answers
19 views
Electric field with Maxwell's 1st equation
With a total charge Q within a radius R the problem is to find the electric field at a radius r>R and r<R. The charge density is within the radius R proportional to 1 / r and zero outside (...
1
vote
1answer
95 views
Simplifying Complex Propagation Constant of an Electromagnetic Plane Wave
I am attempting to study for an exam and encountered a problem involving an electromagnetic plane wave with a provided electric field and propagation constant, but despite setting up the solution ...
1
vote
0answers
87 views
What is a real world application of complex matrices?
During my CS undergrad I had 2 semesters of linear algebra, and I really enjoyed solving matrices; as I progressed further into my degree and my field, I realized that there are quite a few real-life ...
2
votes
0answers
37 views
Show that $\nabla . *F = 0$ is a geometric frame independent of ...
Show that $\nabla\cdot\ast F = 0$ (divergence of the dual of the EMF tensor) is a geometric frame independent version of $F_{ab,c} + F_{bc,a} + F_{ca,b} = 0$, where $F$ is the electromagnetic field ...
1
vote
2answers
27 views
Show that $\frac{dF}{dt}=\frac{\partial F}{\partial t}+\nabla F\cdot \frac{d\vec{r}}{dt}$
where F is a differentiable function of $x, y,z, t$ and $x, y, z$ differentiable functions of $t$, Show that $\frac{dF}{dt}=\frac{\partial F}{\partial t}+\nabla F\cdot \frac{d\vec{r}}{dt}$
We define $...
1
vote
1answer
60 views
Origin of Legendre's constant term.
I'm that student who needs to know where does something comes from. I have been studying Differential Equations and Electrodynamics (I'm a physics student), and I was wondering why we (in physics) use ...
-1
votes
1answer
60 views
How to approach this question?
I have been trying this, but I do not understand how to get the gradient of the triple integration.
If a region $V$ bounded by a surface $S$ has a continuous charge (or mass) distribution of density $...
1
vote
0answers
42 views
References for magnetic Sobolev spaces
I am working on PDE. Recently I have been studying magnetic Sobolev spaces. While the theory is clear to me, having very little knowledge about physics, I have almost no idea how these spaces help ...
0
votes
1answer
26 views
Hybrid field mode in electromagnetic cylindrical wave guide
If we have a boundary-value problem that consists of a circular cylindrical waveguide (axis along $\hat{z}$, say), with some arbitrary boundary condition (so not necessarily the usual conductive wall) ...
0
votes
0answers
17 views
Differentiation in spherical coordinates
I have a vector potential as shown below.
$$\vec A =I^e d\ell \dfrac{e^{-ir}}{r} \vec a_z$$
Where $d\ell$ is
$$d\ell= \sqrt {dr^2+r^2d\phi^2+dz^2}$$
In order to get magnetic field, I should evaluate ...
1
vote
0answers
43 views
Unconstrained quartic optimization
I am working on antenna array pattern synthesis algorithms, and am trying to minimize the following expression with respect to $\mathbf{v}$
$$ \int_\Omega \Big[ \big( \mathbf{v}^H \mathbf{Y}(\vartheta,...
