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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28 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 ...
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1answer
31 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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Vector Quadruple Product [on hold]

I need to get proof and mathematical expression of the following vector product: AxBxCxD= ? where A,B,C,D all are vectors and I need to get answer of this quadruple vector multiplication. It ...
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27 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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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}{\...
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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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1answer
28 views

Help on a hard integral

So, I'm doing an extensive homework of electromagnetism and we are searching for the total electromagnetic angular momentum of the Thomson dipole. In the end, there is one integral we cannot solve. By ...
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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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1answer
13 views

Differentiation and scalar product

In my electromagnetism book there is an equality I am not getting: $\vec{H} \cdot \frac{\partial \vec{H}}{\partial t} = \frac{1}{2} \frac{\partial(\vec{H} \cdot \vec{H})}{\partial t}$. Where does ...
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1answer
29 views

Vector identity used in electromagnetism

Is there a simple proof of this identity or a reference to some textbook where could I find a simple proof of the $(1)$? $$\boldsymbol{\nabla}\times (\boldsymbol{\nabla}\times \mathbf{E})=-\frac{\...
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1answer
29 views

On using the mean value theorem on this surface integral.

In electrostatics, the surface of a conductor $S$ is always at a constant potential $\phi _{0}$, where the aforementioned potential is a scalar function $\phi (x,y,z)$ defined as : $$\phi (\textbf{x}...
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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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1answer
47 views

Help understanding the derivation of Maxwell's equations from Euler-Lagrange equations

I am having trouble with the following points in the derivation of Maxwell's equations on page 10 of these notes. Specifically, I am new to the co-/contra-variant notation used in relativistic forms ...
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1answer
35 views

Establish the dispersion relation ω = ω(k)

Stuck on this question, need help. Answer: w = ck
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22 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}\...
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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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1answer
48 views

Question about expression in Griffiths

So, in Griffiths' E&M book, he comes up with this expression for the magnetic dipole moment, $$A_{\text{dip}}(\textbf{r}) = \frac{\mu_{0}I}{4\pi r^{2}} \oint r'\cos(\alpha) \,d\textbf{l}' = \...
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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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1answer
32 views

Closed loop via integration

if an integral around a closed loop is 0, then why is curl of the electric field not 0? We know that the Work along a loop is 0, but the electric field also does work. Therefore, again shouldn't curl ...
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2answers
53 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 ...
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4answers
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How can Green's Theorem be used to derive Maxwell's equations?

I've learned how to prove Green's Theorem and I read that it contributed to deriving Maxwell's equations. How can Green's Theorem be used to derive any of four Maxwell's equations? What else do I have ...
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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 ...
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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 ...
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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 ...
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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 $\...
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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
33 views

Proof: Any function of the form $f \left(t - \mathbf{a}_n \cdot \dfrac{\mathbf{r}}{c} \right)$ is a solution to the $n$-dimensional wave equation

My electromagnetism (Maxwell's equations) textbook gives the following wave equation for free space: $$\nabla^2 \mathbf{h}(\mathbf{r}, t) - \dfrac{1}{c^2} \dfrac{\partial^2{\mathbf{h}(\mathbf{r}, t)}}...
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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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2answers
269 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 \...
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1answer
62 views

Author's derivation of time-independent form of Maxwell's equations

Laser Electronics, 3rd edition, by Joseph T. Verdeyen, gives the following: To describe an electromagnetic wave, we need two field-intensity vectors, $\mathbf{e}$ and $\mathbf{h}$, which are ...
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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. ...
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36 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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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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1answer
44 views

Rigorous explanation of integration involving delta distribution

In a physics class, I saw the following: The charge density of a uniformly charged circle (charge $Q$) of radius $R$ can be described in cylindrical coordinates using the delta distribution as $$ \...
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71 views

Cross product of unit vector in cylindrical and spherical coordinate system [closed]

For cartesian, the unit vectors are $(ax, ay, az)$ For cylindrical, the unit vectors are $(ar, a\theta, az)$ for spherical, the unit vectors are $(aR, a\theta, a\phi)$ How can one compute cross ...
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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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1answer
37 views

Gauss's law in infinite space

Consider an infinite $3$D space with a charge density $\rho$ and a resulting electric field $E$. Imagine $\forall (x,y,z)\in \mathbb{R}^3, \rho(x,y,z) = \rho_0$(a non-zero constant). In this case, ...
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1answer
49 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: $$\frac{d^2\chi}{d\tau^2}+\gamma \frac{d\chi}{d\tau}+\omega_0^2\chi=\omega_p^2\delta(\tau)$$ ...
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2answers
32 views

Binomial series expansion of a trinomial?

In electrostatics, the potential of a charge $q$ placed on the $z$-axis at $z=a$ is \begin{equation} \phi=\frac{1}{4\pi \epsilon_0}\frac{q}{r_1} \end{equation} where $r_1$ is the distance from the ...
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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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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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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: $(\vec{D_2} - \vec{D_1})\cdot \vec{n} = \sigma$ ...
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Rutherford Scattering - Annular Detector in the Far Field [closed]

I have been tasked to find the rate at which scattered electrons will be detected on an annular detector in the far-field. The exact question I'm working with is: Suppose that 1keV electrons, ...
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1answer
58 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 ...
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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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1answer
108 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{...
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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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1answer
41 views

Normalising angled Earth magnetic field

Me and my team are participating in ESA Astro Pi challenge. Our program will ran on the ISS for 3 hours and we will our results back and analyze them. We want to investigate the connection between ...
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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 ...