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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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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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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1answer
31 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
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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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29 views

Calculating the force acting between to magnets [migrated]

I am using neodymium N52 magnets in a project I'm working on, and I have been attempting to calculate the force between them. I just wanted to double check my results with the community and hopefully ...
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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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1k views

Why do time derivatives change to '$j\omega$' when taking the time derivative of a phasor?

For example, Faraday's law in the time domain is written as $$\nabla \times \vec{\mathbf{E}} = - \frac{\partial\vec{\mathbf{B}}}{\partial t}$$ When using phasor notation, Faraday's law is written as $...
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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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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 ...
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37 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 ...
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1answer
42 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
40 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
32 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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156 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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28 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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1answer
60 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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25 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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15 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
40 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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52 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
61 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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1answer
798 views

How to find surface integral of vector field in cylindrical coordinates through a rectangular plane?

Trying to work through drill problem 3.9 from the 8th edition of the textbook "Engineering Electromagnetics by Hayt". this is the problem question: Given the field D = 6ρ sin(0.5φ) aρ + 1.5ρ cos(0.5φ)...
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1answer
35 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
48 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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16 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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3answers
392 views

How do one rigorously prove that the electric potential energy of an conducting sphere with charge $Q$ is $\frac{Q^2}{8\pi\epsilon_0R}$

How do one rigorously prove that the electric potential energy of an conducting sphere with charge $Q$ is $\frac{Q^2}{8\pi\epsilon_0R}$? Is integration the only way? Homogeneous charge distribution ...
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1answer
1k views

Electrostatic Potential Energy integral in spherical coordinates

I'm having trouble with evaluating an integral that arises from attempting to find the total energy of an electrostatic system consisting of two point charges, which involves an integral over all ...
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0answers
32 views

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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27 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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14 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: $(\vec{D_2} - \vec{D_1})\cdot \vec{n} = \sigma$ ...
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1answer
275 views

Adding small correction term to ODE solution

Let $\mathbf{r}(t) = [x(t), y(t), z(t)]$ and $\mathbf{v}(t) = \frac{d}{dt}\mathbf{r}(t)$. I'm trying to solve $$ \frac{d}{dt}\mathbf{v}=\frac{q}{m}(\mathbf{v}\times\mathbf{B}) \tag{1} $$ where $q$ and ...
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1answer
950 views

Electrodynamics in general spacetime

Let $M\cong\mathbb{R}^4_1$ be the usual Minkowski spacetime. Then we can formulate electrodynamics in a Lorentz invariant way by giving the EM-field $2$-form $\mathcal{F}\in\Omega^2(M)$ and ...
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1answer
53 views

Laplacian as limit of Integral Identity

Let $\psi(\vec{r})$ be a scalar field, show that: $$\nabla^2 \psi(\vec{r})=\lim_{\rho \to 0} \frac{3}{\pi \rho^2} \int_\Omega \psi(\vec{r}')-\psi(\vec{r})d\Omega'$$ where $\rho=|\vec{r}-\vec{r'}|$, $...
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1answer
81 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
38 views

How to find the critical point for this coulomb field

Two equal positive charges are at distance $d$, $-d$ from the origin on the $y$ axis. What is the distance on the $x$ axis beyond which a small perturbation in $y$ will move a particle away from the $...
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1answer
55 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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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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0answers
24 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
79 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
213 views

solution to $\square\chi=f$.

For an open set $U \subseteq \mathbb{R}^4$, if $f:U \to \mathbb{R}$ is a "good" (for example, smooth) function, is there a solution to the following equation? $$\left( \Delta - \frac{1}{c^2}\frac{\...
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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 ...
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1answer
110 views

Why do we need both Divergence and Curl to define a vector field?

I was reading Classical Electrodynamics by J.D.Jacskon (section 1.5) where he said: Perhaps some readers know that a vector field can be specified almost completely if its divergence and curl are ...
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2answers
141 views

Apparent paradox when we use the Kelvin–Stokes theorem and there is a time dependency

I am having trouble to understand what is going on with the Maxwell–Faraday equation: $$\nabla \times E = - \frac{\partial B}{\partial t},$$ where $E$ is the electric firld and $B$ the magnetic field. ...
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29 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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7 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 ...
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16 views

Find the differential equation that the function $f$ must verify in order to respect Maxwell's equations, and the relation between parameters A & B

I'm asked to find the differential equation that $f(\theta)$ must verify in order to respect Maxwell's equations and the relation between parameters $A$ and $B$. They give me this equation for the ...
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1answer
56 views

How to integrate Fresnel Integrals? $\int_0^y e^\frac{-j\beta(z)^2}\rho dz$

I am having trouble solving this integration of a spherical fresnel zone with radius y $\displaystyle\int_0^y e^\frac{-j\beta(z)^2}\rho dz$ , where j is complex and $\beta$ and $\rho$ are constants. ...