Questions tagged [electromagnetism]
For questions on Classical Electromagnetism from a mathematical standpoint. This tag should not be the sole tag on a question.
215
questions
-1
votes
0answers
24 views
How does the Electromagnetic Wave Equation not Exhibit a “Type Error”?
According to Wikipedia, the electromagnetic wave equations can be stated as:
But how does this not exhibit a type error? In particualr, it seems to me that the term on the left, for example
$$
\frac{...
2
votes
0answers
38 views
Book suggestion for a Math student taking Electromagnetism [migrated]
I'm hoping to double major in math and physics. I'm currently taking graduate math courses and undergrad physics courses.
I'm having a really hard time with my Electromagnetism class. A lot of the ...
1
vote
1answer
62 views
Derivation of Snells Law using boundary conditions
I asked a similar question on the Physics site but not get the answer I was looking for. Every derivation I have read skips over the details when it comes to deriving Snell's law and $\omega_i = \...
1
vote
1answer
46 views
Homework Question: Setting Up An Integral for Simple Physics Problem
In this problem I am trying to find $E_{x}$(the electric field of at the origin). So far I have the following:
$dE_{x} = -k\frac{dq}{a^2}cos(\theta).$ This is the basic componet of the electric field ...
1
vote
0answers
64 views
Rigorous formulation of electromagnetic field theory for a system of moving charges (non relativistic) using distribution theory
Suppose we have a system of $n$ charged particles with trajectories given by a paths $x_j:\mathbb{R}\to\mathbb{R}^3$ then the Maxwell equations for this system are given first by defining the charge ...
0
votes
1answer
49 views
Electromagnetism & the Gauge Theory
A Gauge Theory obtains from Maxwell's equations from a slight generalization of the target space and geometry: Consider matrix-valued objects instead of scalar-valued objects along with a scalar-...
0
votes
0answers
7 views
coding a nonlinear magnetic equivalent circuit model of an electric machine
I am Scrat. I'm working on a model of a magnetic device based on equivalent circuits. we want to apply magnetic relations into elements of this virtual circuit to simulate magnetic devices by solving ...
0
votes
0answers
44 views
What is wrong here?
What is wrong in turning this expression in spherical coordinates :
$$\int_{0}^{\pi}\int_{0}^{2\pi}\vec{r}\, \cos \Theta \, \sin \Theta \, \mathrm{d} \Theta \, \mathrm{d} \phi $$
to this :
$$\...
8
votes
2answers
141 views
Biot-Savart law on a torus?
In classical electrodynamics, given the shape of a wire carrying electric current, it is possible to obtain the magnetic field configuration $\mathbf{B}$ via the Biot-savart law. If the wire is a ...
4
votes
0answers
93 views
Deriving boundary conditions at a surface of discontinuity: $\int \mathbf{B} \cdot \mathbf{n} \ dS = 0$
I am currently studying Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th edition, by Max Born and Emil Wolf. Chapter 1.1.3 Boundary conditions at ...
1
vote
1answer
32 views
Expression for potential vector of a central field
I know that for the central field
$$
{\bf F(x)}=\alpha\cdot\frac{\bf x}{|{\bf x}|^{3}}=\alpha\cdot\left(\frac{x_{1}}{|{\bf x}|^{3}},\frac{x_{2}}{|{\bf x}|^{3}},\frac{x_{3}}{|{\bf x}|^{3}}\right)
$$
...
2
votes
0answers
63 views
Why is electric potential function in free space infinitely differentiable?
Electric potential function in free space of a continuous charge distribution $\rho'$ distributed over volume $V' \subset \mathbb{R}^3$ is denoted by:
$\psi (x,y,z): \mathbb{R}^3 \setminus{V'} \to \...
1
vote
1answer
35 views
Line integral of point charge.
This question is motivated by a calculation in Section 2.2.4 of Griffiths' book on an Introduction to Electrodynamics, in which he shows that the field of a point charge is curl-free.
The field of a ...
0
votes
0answers
60 views
How can I find equationts for the EM field $E=(E_x,E_y,0)$ and $B=(0,0,B_z)$?
I am trying to find the motion's equation for a charged particle $q$ with mass $m$ in an EM field. First the 2 equations I have are:
$$\frac{\mathrm dU_x}{\mathrm dt}= W_cU_y + \frac{W_cE_x}{B_z}\\
\...
3
votes
0answers
49 views
Integration by parts of curl in Woltjerās paper
I am reading A Theorem on Force-Free Magnetic Fields (1958) by L. Woltjer. He used āintegrating by partsā twice and I am confused about his results.
The first one is in the equation (7). I understand ...
4
votes
1answer
55 views
Uniqueness and symmetry of unbounded Poisson equation
To find the electric potential $\phi: \mathbb{R^3} \to \mathbb{R}$ of a uniformly charged sphere with total charge $Q$ $S=\partial B_0(r)$ of radius $r > 0$ one needs to solve the Poisson ...
0
votes
0answers
51 views
Why is the integral of this term zero?
I came upon a problem in my physics textbook and had a question as to why term was equal to zero.
The equation and its integration :
\begin{align}
B &= u(H+I)
\\
dB&= u dH + udI
\\
\int HdB &...
3
votes
2answers
77 views
Multipole expansion of solution to the Poisson equation
In electrodynamics I have seen the following:
Let $\phi$ be a solution to the Poisson equation $-\Delta \phi= \rho$, and assume that $\rho$ is compactly supported. Then we can expand $\phi$ as the ...
0
votes
0answers
22 views
What is meant by rate of change with respect to volume?
In physics we often come across $$\rho=\dfrac{dq}{dV}$$
Does it mean:
$(i)$ $\displaystyle \lim_{\Delta V \to 0} \dfrac{\Delta q}{\Delta V}$
OR
$(ii)$ $\dfrac{\partial}{\partial z} \left( \dfrac{\...
0
votes
1answer
40 views
Electric field of a charge uniformly distributed on a plane
I am supposed to calculate the electric field $E$ created by a electrical charge $Q>0$ distributed on the surface of a plane. For this I should use
(i) Gauss' theorem $$\int_M \operatorname{div}(...
1
vote
0answers
34 views
Examples of 2nd Order Differential Equations in Electromagnetism
I am looking for examples of second-order ODE's that are similar to the spring, and pendulum, as well as the LRC, LC, RC, and LR circuits and involve electromagnetism. I know how to solve them, and I ...
0
votes
0answers
20 views
Higher dimensional FTC in electrostatics : Does it has mathematical rigor?
I have two volumes $V$ and $V'$ in space such that:
$ā$ point $P$ $\ni$ $[P \in V ā§ P\in V']$
$V$ is filled with electric charge $q$
$\rho = \dfrac{dq}{dV}$ varies continuously in $V$
$V'$ is filled ...
0
votes
1answer
41 views
end-to-end resistance of a truncated cone
Basically the question is the resistance of the whole truncated cone which has top and bottom coal-flaps with radius $r_1$ and $r_2$. I have the $r(x)$ given by a function. I know that I have to ...
2
votes
2answers
66 views
Flow and flow rate… Halp!
I'm so confused...
I think I got the meaning of flux, it's a scalar that indicates the "quantity of a vector field (of field lines)" that crosses a surface of a given area.
So no time relation ...
1
vote
1answer
121 views
How to prove that $u(r)=k \frac{1}{r}$ is the only solution for the integral equation $\int_{V'}\rho'\ u(r)\ dV' = constant$?
Consider a hollow spherical charge with density $\rho'$ continuously varying only with respect to distance from the center $O$.
$V'=$ yellow volume
$k \in \mathbb {R}$
$\forall$ point $P$ inside ...
2
votes
0answers
27 views
Electrical Circuits Modelling and Formal Proofs
Are there any textbooks or articles that formalizes/models electrical circuits (e.g. using graphs, which is probably the most likely approach to be taken) in a precise manner such that we could ...
0
votes
0answers
16 views
Beam propagation in an optical fiber with a $\tanh(\cdot)$ refractive index profile
The differential equation for a optical fiber with a refractive index $n(r)$ is given as
$$\nabla^{2}_{\perp}A(r,\theta)+(k^{2}n(r)^2-\beta^2)A(r,\theta)=0.$$
which is separable in cylindrical ...
1
vote
2answers
35 views
Find the time period ($T$) for an electric field wave: $E=E_0\sin{m t}\sin{2mt}$
Find the time period ($T$) for an electric field wave: $E=E_0\sin{m t}\sin{2mt}$
I thought $T$ is such that, $E(T+t) = E(t)$. As period of given sinusoidal function $E$ is 2$\pi$,
$$ \Rightarrow 2\pi ...
0
votes
0answers
22 views
Consider a wire of length $L$ with linear charge density $\lambda$.
I am a new student, I know the concepts of electric field and it is difficult for me to solve this exercise. Could you help me with this exercise?
Consider a wire of length $L$ with linear charge ...
1
vote
0answers
31 views
How to compute the magnetic field given a circularly polarised electric field?
The question I have is regarding a solution to a later question (Q2). So in order for the question I have to make sense, unfortunately, I must typeset the previous questions.
(Q1)
We may represent a ...
0
votes
2answers
48 views
How should I get the relations eq 3.36 into eq 3.37 in griffiths?
enter image description here
How to get this relations...?? Please give me answers.
0
votes
0answers
37 views
Quantum mechanics. Potential barrier of magnetic field
I am having trouble to think how to solve the following problem:
The plane $x=0$ separates two parts of space: when $x>0$ there is homogeneous magnetic field, which induction vector $B_x$=$B_y$=0, ...
2
votes
0answers
74 views
Restricting a distribution to a non-open subset
If I have an open subset $U \subset \mathbb{R}^n$ and a distribution $\rho \in \mathscr{D}'(U; \mathbb{R})$, i.e. a continuous linear functional $\rho: \mathsf{C}_{\mathsf{c}}^\infty(U;\mathbb{R}) \...
0
votes
0answers
35 views
Deriving $\big|\vec{E}_{AM}^{max}(\vec{r})\big|$ from $\big|\vec{E}_{AM}(\vec{n}, \vec{r})\big|$
I am simulating temporal interference in the brain using tACS and I am currently reading a relevant paper, where they derive a formula to calculate the maximum modulation envelope, seemingly from the ...
1
vote
0answers
14 views
Solution of a modified Poisson-Boltzmann equation
I'm trying to solve a modified Poisson-Boltzmann equation given by
$\frac{d^{2}\phi(z)}{dz^{2}}=2k_{1}\sinh(\phi(z))-k_{2}$,
where $k_{1}$ and $k_{2}$ are constants, and I'm not sure of how to solve ...
1
vote
1answer
16 views
Vector identity proof for dipole magnetic field derivation
The magnetic field strength, $\textbf{B}$, is related to the vector potential, $\textbf{A}$, by $ \textbf{B} = \nabla \times \textbf{A}$.
With $\textbf{A} = \frac{\mu_0}{4\pi} \frac{ \pmb{\mu} \...
0
votes
1answer
49 views
Why is the closed line integral used when stating Gauss's law, instead of the closed surface integral?
Sorry if this is nit-picky, but I'm confused as to how to write Gauss's law.
Both my lecturer and this website state Gauss's law as $$\oint\limits_S \vec{E} \cdot d\vec{S} = \frac{q}{\epsilon_0}$$
...
3
votes
2answers
251 views
'Ugly' Simultaneous equations with 4 variables
I have to solve the following 'Ugly' Simultaneous equations to solve a problem on my textbook of physics. The problem is originally discussed on the thread but, it was unfortunately categorized by ...
0
votes
0answers
115 views
Deriving the wave equation
Given:
$$\nabla \times \mathbf H = \frac{4\pi}{c} \mathbf j \ \ \ \ \ \ \ \ \ (1)$$
$$\nabla \times \mathbf E = -\frac{1}{c} \frac{\partial \mathbf H}{\partial t} \ \ \ (2)$$
$$\...
1
vote
0answers
113 views
dipole in a dielectric sphere
I rather stuck on this homework question. I have been working on it for 3 days. I need a little help.
Context
A point dipole p is placed at the centre of a dielectric sphere with permittivity $\...
0
votes
2answers
43 views
Integral of electric field $E$ is $0$ implies that the field is $0$?
If we consider $\vec{E}$ the electric field in $R^n$ and we have :
$$\int_{R}^{\infty}\vec{E}(\vec{r})d\vec{r}=0$$
where $R$ is in $R^n$ and $\vec{r}$ is in $R^n$
$$\vec{E}(\vec{r})=\frac{1}{4\pi \...
0
votes
0answers
25 views
Electric and magnetic fields in parabolic equation
The split step solution for Parabolic Wave Equation (PWE) applied to the tropospheric propagation problem is:
$$u(x+\Delta x, z)=e^{ikM(z)10^{-6}\Delta x}\mathfrak{F}^{-1}[e^{\frac{-ip^2\Delta x}{2k}}...
0
votes
0answers
117 views
Deriving the wave equation out of $\nabla \times \vec H = \frac{4\pi}{c} \vec J$
I am trying to derive the wave equation presented by Alfven in his 1942 paper
Based on the electrodynamic equations:
$$\nabla \times H = \frac{4\pi}{c}J$$
$$\nabla \times E = -\frac{1}{c} ...
0
votes
0answers
48 views
Biot_Savart law for bounded current systems
I have a PDE I don't know how to solve, which is a result of a physics problem related to the Biot-Savart equation.
The PDE in question is an equation for B:
$$\text{curl}(B)=\text{del}(\phi)$$
...
1
vote
1answer
53 views
How do I show that the two definitions of the curl of a vector field equal each other?
The curl of a 3D vector field is a 3D vector itself and has two definitions - one in integral form and one in differential form.
Definition 1: $$ \operatorname{curl}\vec{F}(x,y,z) \, \cdot \, \hat{n} ...
0
votes
1answer
35 views
The two definitions of the divergence of a vector field?
Now, I am aware that the divergence of a vector field, $\vec{F}$, can be defined in two ways. What I don't understand is why do these equal each other formally?
Definition 1: $$\text{div}\vec{F} = \...
0
votes
0answers
55 views
Image Theory in Electrodynamics
I'm searching for a rigorous mathematical proof of the image theorem for electric/magnetic currents distributions. A proof that, I think, shows that removing the reflecting surface and placing ...
0
votes
1answer
42 views
Mathematical properties of electrodynamic potential
I am faced with a problem that is more mathematical than electrodynamic. However, not having a clearer or shorter title available, I preferred to highlight where the problem came from.
However, ...
1
vote
0answers
34 views
Trouble with Stokes' Theorem and the line integral of a piece-wise definition of a continuous curve using polar coordinates.
Problem Statement:
Given:
$\vec B = (\rho cos \phi)\hat \rho+(sin \phi)\hat \phi$
Verify Stokes' Theorem by evaluating:
a) $ \oint\limits_c \vec B \bullet d\vec l$, where c represents the closed, ...
2
votes
1answer
43 views
Demonstrate divergence and rotational.
Show that
$$DIV(A)=\lim_{\Delta s\rightarrow0}\frac{\displaystyle\int\int_{\Delta v}A\cdot nds}{\Delta v}$$and,
$$ROT(A)\cdot n=\lim_{\Delta s\rightarrow 0}\frac{\displaystyle\oint_{C}A\cdot dr}{\...
