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 quantum mechanics wave equation deduced by mechanics-optics analogy [closed]

I have a question about quantum mechanics wave equation deduced by the analogy between eikonal equation and fermat least principle (in optics) with, respectively, hamilton jacobi equation and ...
user273366's user avatar
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Boundary Conditions on the Magnetic Flux Density (B-field)

My question is similar to this one (Boundary conditions magnetic field) in that it is related to the boundary conditions of the magnetic field (B-field). However, my question focuses on mathematically ...
Blue Various's user avatar
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Solving a funky differential equation.

I'm currently trying to solve the DE that defines charge in a circuit containing an Inductor, Capacitor, Resistor and (crucially) a Memristor. This needs to be able to work for any variable values and ...
Seb's user avatar
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3 votes
1 answer
187 views

Show that $\partial_\mu\phi^\ast A^\mu\phi- A_\mu\phi^\ast\partial^\mu\phi=A^\mu\phi\partial_\mu\phi^\ast - A^\mu\phi^\ast\partial_\mu\phi$

The following is loosely related to this question: [...], the most general renormalisable Lagrangian that is invariant under both Lorentz transformations and gauge transformations is $$\mathcal{L}=-\...
Sirius Black's user avatar
5 votes
1 answer
115 views

What is the correct sign for the four-vector potential gauge transform; $A_\mu\to A_\mu\pm\partial_\mu\lambda$ and where does this gauge originate? [closed]

I have three questions regarding the following extract(s), I have marked red the parts for which I do not understand for later reference. The convention followed for the Minkowski metric in these ...
Sirius Black's user avatar
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1 answer
39 views

Partial differential equation with Faraday's equation

We were asked to find what equation is satisfied by $\Psi(x,y,z,t)$ given that $\textbf{B} = \nabla \times (\textbf{z} \Psi)$ and $\textbf{E} = -\textbf{z} \frac{\partial \Psi}{\partial t}$ while ...
rikdb's user avatar
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3 votes
1 answer
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Why does $\int_{t_0}^{t_1}(\oint_{\partial\Omega}j)dt=\int\limits _{[t_0,t_1]\times\partial\Omega}dt\wedge j$? Fubini theorem for differential forms?

In an electrodynamics book I came across the following claim for the electric current density (twisted) 2-form $j$ along the boundary of some 3-dimensional volume $\Omega$: $$\int_{t_{0}}^{t_{1}}\left(...
Al.G.'s user avatar
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What does $\vec{\nabla}^2 \vec{E} = \vec{\nabla}^2 \left[ f(\vec{k} \cdot \vec{r} - \omega t) \vec{E}_0 \right]$ mean?

$\vec{E} = f(\vec{k} \cdot \vec{r} - \omega t) \vec{E}_0$ with the constant vector field $\vec{E}_0$ I only know the case if I apply the Laplacian operator on a scalar field, in this case it is a ...
CherryBlossom1878's user avatar
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Writing momentum 4vector as an integral over the EM stress-energy tensor

I have been watching a series of lectures on general relativity by Neil Turok and I have run into a problem. In one of the lectures, the professor writes the momentum 4-vector as a contraction of the ...
Jesse Van Der Kooi's user avatar
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if divergence of a vector is zero, how to find the spherical coordinate of the vector?

The perturbed part of magnetic field is $\mathbf{\delta B}$ where $\mathbf{\delta B} = \delta B_x(x,y), \delta B_y(x,y)$ and $\nabla \cdot \mathbf{\delta B} = 0$. To prove $\mathbf{\delta B} = \delta ...
Mon's user avatar
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Polar coordinates: What unit vectors span the $(r,\theta)$ space? [closed]

Polar coordinates: What unit vectors span the $(r,\theta)$ space? I am thoroughly confused. If in the Cartesian system, the associated orthonormal polar vectors at different points on a circle keep ...
S_M's user avatar
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2 answers
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Book Recommendation: One that has a lot of problems and theory associated with polar coordinates and spherical polar coordinates

I would like to "master" polar coordinates and spherical polar coordinates. In the sense, I would like to become as well versed with them as I am with cartesian coordinates. I have gone ...
S_M's user avatar
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1 answer
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How to evaluate the integral $\int_{-r/2}^{r/2} \int_{-r/2}^{r/2} \frac{1}{x^2+y^2+r^2/4} dx dy$

I came across this integral while trying to evaluate the electrical force exerted by a charged plate in the form of a square with side length $r$. I tried the usual method of first keeping $y$ ...
Alp's user avatar
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14 votes
3 answers
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What is the sum of an infinite resistor ladder with geometric progression?

I am trying to solve for the equivalent resistance $R_{\infty}$ of an infinite resistor ladder network with geometric progression as in the image below, with the size of the resistors in each section ...
KDP's user avatar
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Interpreting the cohomology class of the Maxwell tensor.

In the introduction to Bott and Tu, “Differential forms in algebraic topology” there is the motivating example of a stationary point charge in $3$-space. The electromagnetic field $\omega$ is a $2$-...
Parth Shimpi's user avatar
3 votes
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An improper integral from Jackson's book involving the modified Bessel function

When deriving the angular distribution of energy for synchrotron radiation one has to evaluate two tricky improper integrals (see [1] below): $$ I_1 \equiv \int_{0}^{\infty} x^2 [K_{2/3}(x)]^2 \, \...
Gabriel Macedo's user avatar
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1 answer
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Analytically solving PDEs on irregular domains in Physics

In many Physics courses you solve PDEs like heat or wave on square, circular, or spherical domains with separation of variables. Are there ways to solve PDEs and Boundary value problems on irregular ...
Masteralien's user avatar
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Calculate Electric Field on the Z-axis from a finite charge wire

I've been trying to find the electric Field on the Z-axis from a non-uniform charge density line charge. The wire is placed on the z-axis from $z=0$ to $z=1$, $E=?$ at $z>1$ and $z<0$ $$ \rho =...
gus2427's user avatar
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1 answer
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I was trying to find field due uniformly charged sheet at a distance h from centre of the square sheet

I assumed the square(side a) sheet to be made up of wires.$$dE=Kdq/r^2$$ The field due to a wire is : Reference $$\frac{K\lambda}{d}\left[\frac{x}{\sqrt{d^2+x^2}}\right]^{(a/2)}_{(-a/2)}=\frac{K\...
Aurelius's user avatar
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Integrals for the the localized pyramid basis functions in Galerkin Method

I tried to show the following relations for the localized pyramid basis function $\phi_{i j}(x, y)=(1-|x| /$ $h)(1-|y| / h),|x|<h,|y|<h$, where $x$ and $y$ are measured from the site $(i, j)$. ...
Márquez Carranza Arturo Ariel's user avatar
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Flux of a vector field on a non-smooth surface? (in terms of electromagnetism)

While studying the famous Ampere's law, I came up with the following vector field $F$ and a surface $S$ lying in $R^3$. (In terms of physics, $F$ is the current density of some current in a circuit, ...
jkuk5046's user avatar
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49 views

Vector Line Integral For Biot Savart Law

How would one go about computing the vector line integral presented in the Biot-Savart law: $$\vec{B}=\int_c\frac{\mu_0I}{4\pi} \frac{d\vec{l}\times\hat{r}}{r^2}$$ I know how to compute vector line ...
JBatswani's user avatar
1 vote
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120 views

Stokes theorem not holding

I have a vector field $\vec{H} = (8z,0,-4x^3)$ Naturally, $\nabla \times \vec{H} = (0,8+12x^2,0)$ Stokes theorem says: $$ \int_s{\nabla \times \vec{H}} \cdot \vec{dS} = \oint_l{\vec{H} \cdot dl} $$ ...
rjpj1998's user avatar
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1 answer
108 views

Stokes theorem 2 sides not matching with Magnetic waves

We have been asked to verify stokes theorem for a magnetic field. We know Stokes theorem states, for any vector field $\vec{H}$: $$\int_S{(\nabla \times \vec{H}) \cdot \vec{dS}} = \oint_L{\vec{H} \...
rjpj1998's user avatar
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1 answer
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Pseudo-vector formal definition

I have a question about the formalization of pseudovectors. Wikipedia (and my electromagnetism professor and all the electromagnetism books) only state that a vector $v$ transforms as $v' = Rv$, while ...
QuantumBrachistochrone's user avatar
3 votes
1 answer
81 views

Solving $2\frac{\partial S}{\partial z}+\left(\frac{\partial S}{\partial r}\right)^2=0$

I have been trying to solve this PDE $$2\frac{\partial S}{\partial z}+\left(\frac{\partial S}{\partial r}\right)^2=0$$ The solution of this equation corresponding to a spherical wave of radius of ...
Nikhil Mehra's user avatar
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67 views

Divergence theorem with normal component of a curl to a surface

Let $\mathbf{A}$ be a vector function in $\mathbf{R}^3$ and we want to find the normal and tangent components of $\nabla \times \mathbf{A}$ on a smooth and closed surface $\Gamma$. $\mathbf{n}$ is the ...
Francisco Sáenz's user avatar
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60 views

Electric field flux proportional to the field lines generated by (for example) a static charge

Suppose we have a stationary positive charge at a point in space that we call $+Q$. We know by definition that the flow of the electrostatic field is given by, in its simplified form, $$\Phi_S(\vec E)=...
Sebastiano's user avatar
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1 vote
1 answer
86 views

Distance becoming equal to displacement

Consider a charged particle of charge q and mass m being projected from the origin with a velocity u in a region of uniform magnetic field $\mathbf{B} = - B \hat{\mathbf{k}} $ with a resistive force ...
Srish Dutta's user avatar
8 votes
1 answer
689 views

A calculus problem from electrostatics

Since this problem consists of multiple parts and one needs to see all of them to understand the problem i'm going to list out all of them: Consider a uniformly charged spherical shell of radius $R$ ...
Tomy's user avatar
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6 votes
1 answer
108 views

What is the value of $\frac{1}{2}\int_B\int_B\frac{\rho(x,y,z)\rho(x',y',z')}{4\pi\epsilon_0\sqrt{(x-x')^2+(y-y')^2+(z-z')^2}}dxdydzdx'dy'dz'$?

I am reading a book about electromagnetism by Yousuke Nagaoka. Suppose $R$ is a positive real number. Suppose $Q$ is a positive real number. Let $B:=\{(x,y,z)\in\mathbb{R}^3:\sqrt{x^2+y^2+z^2}\leq R\}...
tchappy ha's user avatar
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1 vote
0 answers
58 views

Non-homogeneous wave equation, retarded potentials and causality

Consider the non-homogeneous wave equation in three dimensions with homogeneous initial conditions: $$ \begin{align} & \square f(\underline{x}, t) = g(\underline{x},t), \hspace{3mm} \underline{x} \...
Matteo Menghini's user avatar
4 votes
1 answer
68 views

Linear system $Ax=y$ with partially known $x,y$ and non singular $A$

PHYSICAL INTUITION While proving the equivalence between the Dirichlet problem (i.e. the potential is known on the surface of every conductor) and the mixed problem (i.e. the potential is known on ...
Matteo Menghini's user avatar
0 votes
1 answer
71 views

Why does $\boldsymbol{\nabla} \times \textbf{E}=\textbf{0}$ imply $\boldsymbol{E_2}^{\parallel}=\boldsymbol{E_1}^{\parallel}$?

I am currently studying 'Introduction to Electromagnetism' by David Griffiths, and I was reading about the electric displacement $\boldsymbol{D}$. I decided to try to extract eq. 4.27, which states: $\...
Rasmus Andersen's user avatar
0 votes
2 answers
110 views

Convergence of the infinite series $\sum_{n\in\text{odd}}^{\infty}\frac{z^n}{n}$

This is a follow-up on a previous question I have asked, but since I have made some improvements, I wanted to make a new post. I was studying 'Introduction to Electromagnetism' by David Griffiths and ...
Rasmus Andersen's user avatar
2 votes
0 answers
61 views

Evaluation of Fourier series $\sum_{n=1,3,5...} \left[\frac{1}{n}\text{e}^{-\frac{n \pi x}{a}} \text{sin}(\frac{n \pi y}{a}) \right]$

I was studying electromagnetism and followed 'Introduction to Electromagnetism' by David Griffiths. During his derivation of the solution to Laplace's equation in ch. 3.3, he derives the equation $$V(...
Rasmus Andersen's user avatar
1 vote
1 answer
158 views

Evaluating an Integral with a Dot Product

Lets say I have $\int{ \overrightarrow{B} \cdot\ \overrightarrow{dA} }$ is that equal to $ \int{ B \cdot dA } \cdot\cos(\theta)$ or $\int{(B \cdot\cos(\theta)) \cdot dA} $ For example: a ...
ogginger's user avatar
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0 answers
35 views

Finding the change of basis matrix for a type (0,1) tensor

I am considering a tensor (in particular, the electric field), defined by $$E_m = g_{ij}^k c_{k\ell}^{ij}S_{\ell m} $$ Ultimately, this means that the tensor E is a rank 1, type (0,1) tensor, ...
Luk'yan Vilshansky's user avatar
6 votes
1 answer
132 views

Effective resistance in finite grid of resistors

Consider a $m\times n$ grid of one-Ohm resistors. What is the effective resistance of any given edge? I understand how to do the case $m=2$ inductively using the series and parallel laws, but I get ...
zjs's user avatar
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0 votes
0 answers
23 views

Analytic solution to a definite integral of a magnetic field vector equation

In a research that I am conducting, my main focus of measuring a set-up has diverted to modelling the theoretical working of a magnetic field around a current-inducing solenoid (coil). Given a coil ...
Lex Plantenga's user avatar
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0 answers
43 views

How do scalars like currrent or amplitude add vectorially and give correct results?

I have seen in alternating current that values of current and potential difference in different circuits like LR, CR or LCR circuits are found by adding them like vectors. It also happens with ...
Aurelius's user avatar
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1 vote
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Equilibrium position of $ n $ free charges as polynomials roots

I asked the same question on here but received no answer. The classic problem of the electrostatic equilibrium positions of a linear system of $ n $ free unit charges between two fixed charges is well ...
user967210's user avatar
2 votes
2 answers
241 views

Why can't math software solve the integral $\int\limits_{-a/2}^{a/2}\int\limits_{-a/2}^{a/2}\frac{1}{\sqrt{x^2+y^2+z^2}}dxdy$?

Consider the task of finding the electric field at a height $z$ above the center of a square sheet of side a carrying uniform charge $\sigma$. I am asking this in the math stack exchange because ...
xoux's user avatar
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1 vote
1 answer
55 views

Absorbing constants when determining them through boundary conditions

I am working through example $3.3$ in Griffiths Electrodynamics in section $3.3$ on Separation of Variables. The example involves solving the $2$-dimensional version of Laplace's Equation for the ...
Numerical Disintegration's user avatar
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0 answers
26 views

Implication of Helmholtz decomposition: Fundamental theorem of vector calculus, for causality in electrodynamics - fixing the unphysical

I've been reading Jackson's Electrodynamics chapter 6. I want to believe I now understand the fundamental theorem of vector calculus. A vector field seems to be decomposable into a longitudinal or ...
Kevin Njokom's user avatar
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0 answers
35 views

Lorenz condition and uncoupling pde - general name for techniques of pde uncoupling

I'm reading Jackson's Electrodynamics chapter 6.2. It is possible to reduce Maxwell's equations to $\nabla^2 \phi + \frac{\partial}{\partial t} (\nabla \cdot A) = - \frac{\rho}{\epsilon_0}$ (6.10) $\...
Kevin Njokom's user avatar
4 votes
1 answer
138 views

Finding $\int_{-\infty}^{\infty}\frac{\sin(ax)}{\sqrt{(b-x)^2+c^2}}dx$ [closed]

How do we find $$\int_{-\infty}^{\infty}\dfrac{\sin(ax)}{\sqrt{(b-x)^2+c^2}}dx$$ I have no idea on how to even begin approaching this. Can I get a hint? EDIT: I've got to this integral from a physics ...
EMM's user avatar
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1 vote
1 answer
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Given a triple integral representing the (electric) vector field of a continuous volume charge distribution, how to obtain the potential function?

My question is about the math involved in the concept of electric potential. Though this is a physics concept, it is basically a lot of vector calculus. The derivations below are based on the content ...
xoux's user avatar
  • 4,853
1 vote
0 answers
55 views

Kramers-Kronig computation for real susceptibility

i am trying to get the real part of electric susceptibility using the imaginary part with Kramers-Kronig relation for a Lorentz-Drude model.I chose to ask this question in math stack exchange as im ...
zero's user avatar
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0 answers
25 views

Dirac String force

Assume that the Dirac string is lying along the negative $z$-axis, and is subject to a magnetic field $B$. Assume throughout this question that we are considering a static situation. The force on the ...
Tomy's user avatar
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