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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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{...
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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 ...
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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 = \...
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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 ...
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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 ...
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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-...
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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 ...
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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 : $$\...
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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 ...
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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 ...
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32 views

Expression for potential vector of a central field

I know that for the central field $$ {\bf F(x)}=\alpha\cdot\frac{\bf x}{|{\bf x}|^{3}}=\alpha\cdot\left(\frac{x_{1}}{|{\bf x}|^{3}},\frac{x_{2}}{|{\bf x}|^{3}},\frac{x_{3}}{|{\bf x}|^{3}}\right) $$ ...
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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 \...
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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 ...
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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}\\ \...
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Integration by parts of curl in Woltjer’s paper

I am reading A Theorem on Force-Free Magnetic Fields (1958) by L. Woltjer. He used “integrating by parts” twice and I am confused about his results. The first one is in the equation (7). I understand ...
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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 ...
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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 &...
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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 ...
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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{\...
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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}(...
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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 ...
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Higher dimensional FTC in electrostatics : Does it has mathematical rigor?

I have two volumes $V$ and $V'$ in space such that: $∄$ point $P$ $\ni$ $[P \in V ∧ P\in V']$ $V$ is filled with electric charge $q$ $\rho = \dfrac{dq}{dV}$ varies continuously in $V$ $V'$ is filled ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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.
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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, ...
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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}) \...
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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 ...
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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 ...
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1answer
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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} \...
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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 $$\oint\limits_S \vec{E} \cdot d\vec{S} = \frac{q}{\epsilon_0}$$ ...
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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 ...
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115 views

Deriving the wave equation

Given: $$\nabla \times \mathbf H = \frac{4\pi}{c} \mathbf j \ \ \ \ \ \ \ \ \ (1)$$ $$\nabla \times \mathbf E = -\frac{1}{c} \frac{\partial \mathbf H}{\partial t} \ \ \ (2)$$ $$\...
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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 $\...
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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 \...
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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}}...
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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} ...
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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: $$\text{curl}(B)=\text{del}(\phi)$$ ...
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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} ...
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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} = \...
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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 ...
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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, ...
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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, ...
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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}{\...

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