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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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1answer
23 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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0answers
23 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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0answers
14 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 ...
1
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1answer
38 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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0answers
35 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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0answers
49 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
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, ...
3
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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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15 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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0answers
22 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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0answers
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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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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, ...
3
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1answer
54 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
22 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 ...
4
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1answer
68 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{...
1
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1answer
71 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 ...
0
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1answer
38 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 ...
3
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2answers
63 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 ...
1
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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 $...
5
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1answer
204 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{\...
1
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1answer
51 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 $\rho=|\vec{r}-\vec{r'}|$, $...
2
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1answer
97 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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0answers
26 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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2answers
139 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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0answers
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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0answers
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 ...
0
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1answer
53 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. ...
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0answers
39 views

Gauge invariance of the Hamiltonian for particle on external electric field

Let us assume we consider a problem of free electrons in an external electric field $$\hat{H}(\mathbf{r})=-\frac{\hbar^2}{2m}\nabla^2-e\Phi(\mathbf{r},t),$$ where $-\nabla \Phi(\mathbf{r},t) =E\...
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0answers
47 views

How to plot the graph of this expression which involves dirac delta function?

I Was Doing a Problem on Electrostatics which required finding the charge density from the given electric field and then plot a graph of the charge density. I was able to find the charge density which ...
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0answers
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Calculus of variations ? electrostatic energy problem.

What is the maximum self-energy of an electrostatic distribution subject to the constraints that: the total charge is $1$; and the areal charge density anywhere is either $1$ or $0$. How ...
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0answers
39 views

Use $\nabla\times(f \vec A)=\nabla f \times \vec A + f (\nabla \times \vec A)$ to rewrite Faraday's law as $\omega \vec B_0=\vec k \times \vec E_0$

We may represent a general electromagnetic plane wave by (real part of the complex exponentials): $$\vec E = \vec E_0\exp(i\vec k \cdot \vec r - i \omega t) \quad\text{&}\quad\vec B = \vec B_0\...
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0answers
35 views

Uniqueness of charge distribution

Some exercises in general physics ask for students to find specific charge distributions on the boundaries of given conductors in various of situations. Usual answers goes like follows; firstly using ...
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2answers
116 views

Evaluate $\int_0^{2\pi}\ln(1+a^2-2a\cos(x))dx\text{ ,with }a>0$ [duplicate]

$$\int_0^{2\pi}\ln(1+a^2-2a\cos(x))dx,\;\;\;\;\text{with }a>0$$ How to evaluate Integral of $\ln(1+a^2-2a\cos x) dx$? where $x$ from $0$ to $2\pi$ and $a>0$, $\ln$ is the natural logarithm.
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1answer
78 views

Why is this length $r d\theta$?

I am trying to find the magnetic field due to a current carrying wire. Why is the length $rd\theta$? One of the radius is $r$ but the other is $(r+dL \cos(\theta))$. And, $\theta$ can be large or ...
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0answers
108 views

Prove Property of Green Function Solution to Laplace Equation in a 2D-square

Let's consider a 2D-square with 4 euqal subsquares containing different dielectrics. Inside the square domain, the unkown potential function $\Phi$ satisfies the Laplace equation: $\nabla^2\Phi=0$ ...
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0answers
12 views

Why does the curl of a function provide this particular amount of information? [duplicate]

In a classical electrodynamics textbook (Griffiths), it is mentioned that even though the electric field function, $E:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$, is a (3D) vector valued function, the ...
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3answers
75 views

Why does the curl of a function provide this particular amount of information?

In a classical electrodynamics textbook (Griffiths), it is mentioned that even though the electric field function, $E:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$, is a (3D) vector valued function, the ...
0
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0answers
18 views

Electric field in Cylinder

I have two infinitely long cylinders having surface charge density p=a,b (b>a). How do I find the electric fields at all points? Should I use Gauss law? If so, how will I need to use surface ...
0
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1answer
102 views

Electric potential in the far region?

I have the electric potential $$\Phi(\vec{x})=\frac{\lambda}{2\varepsilon_0}\log\left(\frac{R+\sqrt{R^2+z^2}}{|z|}\right),$$ whose behavior I have to study in the far and near region, i.e. expand ...
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0answers
25 views

Problem in Divergence theorem.

I just started with Electromagnetics and was studying the divergence theorem. I am stuck not knowing how to take into account the surface integral of a cylindrical system. Question attached below, ...
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3answers
38 views

Unit vector Normal to the plane. [closed]

I am new to electromagnetics. I came across a problem where I had to find a find a unit vector normal to a plane. When points are given, it is the cross product, I understood that much. But, When the ...
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0answers
78 views

Do Maxwell's equations (generalized) apply to _every_ $k$-form on a pseudo-Riemannian manifold?

Given a pseudo-Riemannian $n$-manifold and a $k$-form $F$ on the manifold, I will call its exterior derivative $J=dF$ the source of $F$ and the differential $K=dG$ the dual source of $F$, where $G=​{\...
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1answer
55 views

Why am I not getting the right answer for this integral?

I'm trying to solve the integral below. I'm not getting the right answer no matter what. Can you tell me why my method is wrong? I'm applying the rule for integrating $x^n$ (i.e. $\smash{\frac{x^{n+1}}...
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0answers
19 views

Well posedness of Heterogeneous Helmholtz weak form

I'm trying to model the Heat profile in a stationary microwave oven. In order to obtain the electromagnetic fields, I have to solve the following problem with the Finite Element Method : Find $u$ a ...
0
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1answer
77 views

Using Right-Hand Rule for a Current Running in a Loop

Hello. Using the right hand rule, shouldn't the magnetic force be going into the paper, since velocity is to the left and the force caused by centripetal acceleration is downward? The answer key says ...
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0answers
64 views

Delta distribution and divergence theorem

Let's say we have some vector field $\vec C$ such that $$\operatorname{div}\vec C=-\mu_0\vec j=-\mu_0 I\,\delta(x)\,\delta(y)\,\vec e_z.$$ where $\mu_0$ and $I$ are constants. I am interested in the ...
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2answers
776 views

Are there Soliton Solutions for Maxwell's Equations?

Some non-linear differential equations (such as Korteweg–de Vries and Kadomtsev–Petviashvili equations) have "solitary waves" solutions (solitons). Does the set of partial differential equations ...
2
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0answers
28 views

Charge density of charged conductor with flat side

Given a charged conducting body with a flat side, can the charge density (and hence the normal electric field) be constant on the flat part? According to physics lore, the charge density is greater ...
2
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0answers
108 views

Modelling Diode current with ODE [closed]

I want to write ODE system for simulating following electrical circuit: At each small step dt i just do euler integration. I only know ODE for leaky capacitor: <...