Questions tagged [electromagnetism]
For questions on Classical Electromagnetism from a mathematical standpoint. This tag should not be the sole tag on a question.
309
questions
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Using "Maxwell's curl equations" to get $H_y = \dfrac{j}{\omega \mu} \dfrac{\partial{E_x}}{\partial{z}} = \dfrac{1}{\eta}(E^+ e^{-jkz} - E^- e^{jkz})$
I am currently studying the textbook Microwave Engineering, fourth edition, by David Pozar. Chapter 1.4 THE WAVE EQUATION AND BASIC PLANE WAVE SOLUTIONS says the following:
The Helmholtz Equation
In ...
- 4,066
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25
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Integrate $\frac{ e^{i k | \mathbf{r} - \mathbf{r'} |} }{|\mathbf{r} - \mathbf{r'} |}$ in a spherical shell
How can we compute the following triple integral (electromagnetic diffusion in a sphericall shell)?
$ E(\mathbf{r}) = \int_0^{2 \pi} d\phi' \int_0^{\pi} \sin \theta' d\theta' \int_R^{R+h} d r' r'^2 \...
- 13
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1
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30
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Generalizing regular polyhedra by repelling points on a sphere
Find the arrangement of $N$ identical point charges on a sphere. For uniqueness, assume one charge sits on the north pole and another one lies on a fixed latitude of the sphere
Given a circumference ...
- 553
2
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2
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33
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Numerical Solution of nonlinear P-B Equation in unbounded domain for determining the EDL potential distributions around a spherical particle
For my project I am studying a paper, namely "Perturbation solutions for the nonlinear Poisson–Boltzmann equation with a higher order-accuracy Debye–Huckel approximation" by Cunlu Zhao, ...
- 21
1
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36
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Complex integral of $\int_{-\pi}^{\pi}\frac{\cos\theta\,d\theta}{\csc \alpha+\cos\theta}$ [closed]
A current $I_1$ flows in a circular circuit of radius a and a current $I_2$
flows through a very long straight conductor in the same plane of the circular circuit (see
the figure). From the laws of ...
- 37
2
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1
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113
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What does it mean to say that "$h$ is a coordinate measured normal from the surface"? How does this work in practice?
I am currently studying the textbook Microwave Engineering, fourth edition, by David Pozar. Section Fields at a General Material Interface of chapter 1.3 FIELDS IN MEDIA AND BOUNDARY CONDITIONS says ...
- 4,066
2
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1
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183
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Geometric Algebra or Differential Forms for Electromagnetism? [closed]
Electromagnetism (Maxwell's equations) are most often taught using vector calculus.
I have read that both geometric algebra and differential forms are ways to simplify the material.
What are some ...
- 6,472
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0
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11
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Example of a magnetic vector field
I am doing a high school level presentation about maxwells equations. For that I intend to do animations using the python library manim, but that in turn requires me to know the function for a ...
- 33
3
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1
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71
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Calculation of a vector by taking the gradient of the integral of its divergence
I have encountered several times of a special way of calculating a vector from the divergence of the vector. It has at least appeared in the theories of elasticity and electrodynamics.
If I define the ...
- 67
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Is $0$ the null space of the integral operator with kernel $G(r,r') = \frac{\exp(-ik|r-r'|)} {|r-r'|}$?
Let $G(r,r') = \frac{\exp(-ik|r-r'|)} {|r-r'|} $, where $r$ and $r'$ are position vectors in a domain $D$ of $\mathbb R^3$ and $k$ is a positive real constant. Suppose that $h$ is a continuous real ...
- 21
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2
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37
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What is the radial direction?
I'm currently in an electrostatics course; and wanting to rip my hair out.
The question says:
A sphere of radius 𝑎 is polarized such that the polarization at 𝒓 within the sphere is given by
𝑷 = 𝑘𝑟...
CMDR Bobert
1
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1
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64
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What is the correspondence between gauge field terminology and bundle terminology in electromagnetism?
In electromagnetism, the electromagnetic field tensor can be expressed as $$F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu.$$
If we let $A= A_\mu dx^\mu$, since $F= \frac{1}{2} F_{\mu \nu} dx^\...
user955291
1
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1
answer
26
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Calculating the divergence of electric field in standard coordinates
Given an electric field $$ \vec{E(r)} = (c/r^2 ) \hat {r} $$
I want to show that $ \nabla \cdot \vec{E} = 0$ for $ r \ne 0 $ and do the calculation in standard coordinates. For simplicity I'll ...
- 319
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36
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Calculating the average of the square of the magnitude of an electric field
Let the sinusoidal electric field polarised in the $\hat{x}$ direction be $\overline{\mathcal{E}}(x, y, z, t) = \hat{x}A(x, y, z)\cos(\omega t + \phi)$, where $A$ is the amplitude, $\omega$ is the ...
- 4,066
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25
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Proof that force on conductor is equal to minus the derivative of the energy
In many electromagnetism textbooks, we have the following problem:
Given a collection of conductors $C_1,...,C_n$ in vacuum, each with charge $q_1,...,q_n$ and located at positions $\vec{r}_1,...,\vec{...
- 1,342
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0
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52
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Why is $F_{\mu \nu}$ fundamentally geometric in nature?
I am studying gauge theory, and we derive the Lagrangian for electrodynamics by wanting the Lagrangian to be gauge invariant under U(1) symmetry group. That is, invariant under the phase rotation, $$\...
user955291
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30
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Gauss's and Ampere's Law derived from Lagrangian
How can I derive the Gauss's and Ampere's law from the Lagrangian of the electromagnetic field:($\phi$ is the scalar potential and $\mathbf{A}$ is the vector potential )
$$\mathcal{L}=\frac{1}{2}\...
- 123
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46
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Calculate the Solid Angle using Stokes' theorem
The solid angle for the surface S subtended at a point P is:
$$
\Omega=\iint_{S} \frac{\hat{r} \cdot \hat{n}}{r^{2}} d S
$$
where $\hat{r}$ and $\hat{n}$ are unit vectors and $r =|\vec {r}|$ is the ...
- 21
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33
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Irrotational implies conservative without using the path-connection
Let's consider a simply-connected domain $V$ in $\mathbb{R}^{3}$ and a smooth vector field $\mathbf{F}$ in $V$ (please don't answer considering other scenarios). Under this assumption, let's consider ...
- 31
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51
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Can we have vectors with vectors as components?
I was working on my course on Electrodynamics earlier today, when I was tasked with computing the eletric field of a non-trivial charge distribution, and it struck me that I had a field with ...
- 455
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1
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56
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How does $\sqrt{-\omega^2(\epsilon - j\sigma/\omega)\mu}$ having either a positive or negative sign determine $\alpha_0$ and $\beta_0$?
I am told that Maxwell's equations take the form
$$\text{curl} \ \mathbf{E} = - \mu j \omega \mathbf{H}, \ \ \ \ \ \text{curl} \ \mathbf{H} = (\sigma + \epsilon j \omega) \mathbf{E},$$
where $\sigma$ ...
- 4,066
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58
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A laplacian working on an equation containing a laplacian and a gradient
I have an equation as follows:
$$a \Delta \mathbf{u} + \mathbf{\nabla}(\mathbf{\nabla} \cdot \mathbf{u}) = 0$$
in which $a$ is a constant, $\mathbf{u}$ is a vector, $\Delta$ is the Laplacian operator, ...
- 67
1
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1
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43
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Volume Integral : $\int \mathrm d^3\mathbf{r}' \frac{\nabla \cdot \mathbf{M}(\mathbf{r'})}{|\mathbf{r'}-\mathbf{r'}|}$
I am trying to understand the following claims. I would appreciate if you could help me, as I am still unable to understand it.
The problem asks us to find the following integral:
$$I = \int \mathrm d^...
- 113
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41
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Flux of the horizontal Electric field through a hemisphere
Suppose I've a hemisphere and an electric field passing horizontally through this hemisphere. I need to find the flux of this field through this hemisphere.
I can easily consider the electric field to ...
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33
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Magnitude of current
The current of a germanium diode at room temperature is 100uA at a voltage of -1V. Predict the magnitude of the current for voltages of 0.2V and -0.2V at room temperature. Repeat the prediction for ...
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1
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64
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Line integral over a small length
In problem 4.7 of Griffiths' "Introduction to electrodynamics, 4th Edition", to find the potential energy of a dipole in an electric field $\vec{E}$ the following step is made:
$$\lim_{\vec{...
- 247
1
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1
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106
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How do we compute Hodge duals?
The motivation for this question is to try to come up with a general expression for $(\star F)_{\mu\nu}$, the $\mu,\nu$ component of the Hodge dual of the Field strength tensor, which is of great ...
- 8,117
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62
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Do these two Green's functions satisfy the Lorenz gauge condition?
I posted this question on Stack Exchange Physics (https://physics.stackexchange.com/questions/679845/do-these-greens-functions-satisfy-the-lorenz-gauge-condition), but repost it here since I didn't ...
- 407
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11
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Partial integration of modified form of Ampere's law
After thinking and searching of/for an answer for an eternity, you are my last hope:
The given equation:
$$
rot(µ^{-1}rot(\vec A))-\epsilon\omega^2*\vec A=\vec J
$$
should be multiplied with a ...
2
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0
answers
55
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Line and Surface Integral with the Dot Product replaced with a Cross Product
Having recently studied magnetostatics, I came across the Biot-Savart law, which is based on the line integal over a current distribution in a curve $C$:
$$\mathbf B(\mathbf r)=\frac{\mu_0}{4\pi}\...
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1
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48
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Is there any condition under which $\nabla\cdot F=0$ implies $F=0$?
On a physics course it was stated that
$$
\nabla\cdot\vec{D}=\rho_f=\nabla\cdot(\varepsilon_0\vec{E})
$$
and then it follows that
$$
\vec{D}=\varepsilon_0\vec{E}
$$
I know this is not generally true, ...
0
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0
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Modelling the magnetic field of a conventional tokamak
Recently I have been reading into Nuclear Fusion and the use of spherical tokamaks. My knowledge of maths and physics is quite limited, only a second year undergraduate level. I was wondering how I ...
- 414
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40
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Perpendicular complex-valued vectors
I'm reading an E&M Textbook and somewhat confused about how to find perpendicular vectors that satisfy the right hand rule, when the coefficients are complex. For example, if $E$ has direction $(\...
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15
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How do you find $\vec{F} = (\vec{p}\cdot \vec{\nabla})\vec{E}$
How do you find
$\vec{F} = (\vec{p}\cdot \vec{\nabla})\vec{E}$ for force on dipole with dipole moment $\vec{p}$ in external electric field $\vec{E}$?
I understand that $\vec{\nabla} (\vec{A}, \vec{B})=...
2
votes
2
answers
70
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$i = \frac{dq}{dt}$ implies $\Delta q = i \Delta t$? Incorrect mathematics used as some kind of hand-wavy justification for an engineering equation?
I am reading an electrical engineering textbook that states that the relationship between current $i$, charge $q$, and time $t$ is
$$i = \dfrac{dq}{dt} \tag{1}$$
Based on this, the authors then state ...
- 4,066
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Deriving force between continuous distributions of two volume charges without using infinitesimals
We know that force between two point charges is:
$$\vec{F}=k\ q\ q'\ \dfrac{\hat{r}}{r^2}\tag1$$
From here how shall we derive the equation for force between continuous distributions of two volume ...
- 1,013
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13
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Single-argument Kronecker delta? Or unit vectors typed with δ?
While looking for a different solution to a problem, I found one that uses deltas in a way I had never seen. The original problem is (From Introduction to Electrodynamics, 1.62):
"Show that $\...
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1
answer
37
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Formula for the Maxwell Stress tensor in arbitrary coordinates
This question is nearly identical to my last, except this time its the Maxwell stress tensor, not the Cauchy stress tensor. I often see its components written as
$$\sigma_{ij}=\varepsilon_0E_iE_j+\...
- 8,117
1
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1
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274
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When does zero divergence imply a vector potential exists?
From electrodynamics we know that $\boldsymbol{\nabla}\mathbf{B}=\mathbf 0$ hence we can introduce a vector potential such that $\mathbf{B}=[\boldsymbol \nabla\times \mathbf{A}]$.
What is the general ...
- 141
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0
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34
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Finding Surface current density using time varying electromagnetic field
PEC to lossless boundary:
I have an understanding that in this case the surface current density can be found using $$Js=n_{12}×H_2$$
I was given an electric field, to which i then found the magnetic ...
- 9
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1
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60
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Prove $\nabla(\frac{\vec{r}}{r}.\vec{r'})=-\frac{\vec{r}}{r}\times\left(\frac{\vec{r}}{r}\times\frac{\vec{r'}}{r}\right)$
How to prove the following,
$$\nabla\left(\frac{\vec{r}}{r}.\vec{r'}\right)=-\frac{\vec{r}}{r}\times\left(\frac{\vec{r}}{r}\times\frac{\vec{r'}}{r}\right)$$
where $\vec{r}$ and $\vec{r'}$ are two ...
- 69
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1
answer
63
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Variational derivation of *all* covariant Maxwell's equations?
If I suppose there exists a 4-"vector potential" $A\in\Omega^1(U)$ such that the Faraday 2-form satisfies $F = dA$ (which is equivalent to assuming the homogeneous Maxwell's equations $dF=0$ ...
- 6,755
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0
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15
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Field components of incident wave
Incident vector is written as
$$ \vec{E}_i = E_0\left( \hat{x} \cos \left( \theta_i \right) -\hat{z}\sin\left( \theta_i \right) \right) e^{-j k_1 \left( x\sin \theta_i + z\cos \theta_i \right) }$$
My ...
- 137
4
votes
0
answers
117
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How to solve $\sum_{n=-\infty}^\infty\frac{y^2}{[(x-n\pi)^2+y^2]^{3/2}}$?
I need to solve this sum:
$$\sum_{n=-\infty}^\infty\frac{y^2}{[(x-n\pi)^2+y^2]^{3/2}}.$$
Do you have any ideas for how I could do this?
I know that this sum:
$$\sum_{n=-\infty}^\infty\frac{y}{(x-n\pi)^...
- 809
0
votes
1
answer
57
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Wave equation under Galilean transformation
In Jackson's book on classical electrodynamics (3rd ed, ch 11, p. 516), he mentions how a wave equation for a field $\psi(\bf{x}^{'},t^{'})$ is transformed under Galilean shift, defined as $\mathbf{x}^...
- 1,209
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1
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85
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Why is the Kaluza-Klein ansatz the natural choice?
In Kaluza-Klein theory we can choose a parametrisation for the 5-dimensional metric: $$d\hat{s}^2 \equiv \hat{g}_{ab} dx^a dx^b = g_{\mu\nu}dx^\mu dx^\nu + \phi^2(dz + A_\mu dx^\mu)^2 $$
where $g_{\mu\...
- 51
0
votes
0
answers
22
views
Curl of current density over distance when deriving the Biot-Savart Law
I'm reading this Physics Exchange post on deriving the Biot-Savart Law from Maxwell's Equations. However, this step is confusing me:
"Now we need only calculate B=∇×A. But"
$$\nabla\times\...
- 121
3
votes
3
answers
288
views
Divergence of a radial vector field
I am reading Modern Electrodynamics by Zangwill and cannot verify equation (1.61) [page 7]:
\begin{equation}
\nabla \cdot \textbf{g}(r)=\textbf{g}^{\prime}\cdot \mathbf{\hat{r}},
\end{equation}
where ...
- 51
12
votes
3
answers
356
views
Finding smooth behaviour of infinite sum
Define
$$E(z) = \sum_{n,m=-\infty}^\infty \frac{z^2}{((n^2 + m^2)z^2 + 1)^{3/2}} = \sum_{k = 0}^\infty \frac{r_2(k) z^2}{(kz^2 + 1)^{3/2}} \text{ for } z \neq 0$$
$$E(0) = \lim_{z \to 0} E(z) = 2 \pi$$...
- 2,193
2
votes
0
answers
40
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Is there a way to represent electrostatics tensors in tensor (possibly tensor product) way?
I'm working with electrostatic interaction tensors, which are defined as follows:
\begin{align}
T &= \frac{1}{r} \\
T^\alpha &\equiv \nabla T = -\frac{r^\alpha}{r^3} \\
T^{\alpha\...
- 21