Questions tagged [electromagnetism]
For questions on Classical Electromagnetism from a mathematical standpoint. This tag should not be the sole tag on a question.
407
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Deriving the Yukawa potential from the field of a screened charge
I am trying to derive the Yukawa potential from the electric field of a screened positive point charge, which is
$$
\vec{E}(\vec{r}) = \frac{q}{4\pi\epsilon_0}\frac{e^{-kr}(kr+1)}{r^2}\hat{r}.
$$
The ...
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0
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20
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singular positive semi-definite matrix in electromagnetism
anyone knows where he drew this conclusion from?
- 171
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1
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103
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The reason for curl free
I wonder about the reason for the idea of this, would you mind explain for me this can happen in mathematics. Thank you !
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119
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The magnetic field of a spinning charged sphere
Evaluate $\displaystyle \int_{0}^{2\pi}\int_{0}^{\pi}\frac{(z_0-R\cos\theta)\sin^2\theta\cos\phi}{[(x_0-R\cos\phi\sin\theta)^2+(y_0-R\sin\phi\sin\theta)^2+(z_0-R\cos\theta)^2]^{\frac{3}{2}}}d\theta d\...
- 681
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2d Fourier Transform Using Weyl Expansion
If we have electric field as
$$
\mathbf{E}\left(\mathbf{r}_{\mathrm{d}}, t\right)=\frac{1}{\varepsilon} \int_{\mathcal{V}} d \mathbf{r}^{\prime} \mathbf{K}\left(\mathbf{r}_{\mathrm{d}}-\mathbf{r}^{\...
- 11
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3
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83
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How to compute the volume integral for the potential of an arbitrary point outside a uniformly charged ball?
$$\frac{\rho}{4\pi\epsilon_0}\iiint_{D}^{}\frac{1}{\left\| \mathbf{r}-\mathbf{r'} \right \| }dV'$$
$D$ is a ball of radius $R$
$\mathbf{r}$ is the position vector of the point where we want to ...
- 163
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0
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48
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Electric fields and simply-connected regions
I apologize for the ignorance and the rough English in advance, I have an issue understanding how to match both what happens in physics and what I am seeing in calculus.
We learned that if a vector ...
2
votes
2
answers
76
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How to apply integration by parts to simplify an integral of a cross product?
I'm reading a physics paper and am trying to figure out how a certain expression is derived (If interested, see Appendix of the paper, Eq. (A7), (A8)). The authors skip a lot of derivation steps and ...
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0
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47
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Solving 4th order differential equation
I have differential equations such as
$$\frac{1}{\lambda^{2}}\psi_{e}'' = \tanh{\psi_{e}+\psi_{h}}$$
$$\psi_{h}'' - \kappa^{2}\psi_{h} = -\alpha^{2}\tanh{\psi_{e}+\psi_{h}}.$$
Boundary conditions are
$...
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0
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23
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Helmholtz - Hodge decomposition in H(curl)
I'm looking for Helmholtz-Hodge type decompositions but for vector fields slighty more regular than $L^2(D,\mathbb{R}^3)$. I'm familiar with the results in the books of Lions and was wondering if ...
- 11
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91
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First chern class of magnetic monopole
Example 3.5.5 in the Mirror Symmetry textbook (Hori-Katz-Klemm-et. al) states:
Let us compute the first Chern class of the line bundle defined by the $U(1)$ gauge field surrounding a magnetic monopole,...
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1
vote
3
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152
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$\int_0^R \frac{r^{l+1}}{\sqrt{R^2 - r^2}}\text dr$
I'm trying to solve problem 3.3 from Jackson's Classical Electrodynamics, but I'm encountering some troubles solving
$$
\int_0^R \frac{r^{l+1}}{\sqrt{R^2 - r^2}}\text dr,
\qquad l = 0,2,4,6,\ldots
$$
...
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vote
1
answer
100
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Representation of $e^{ikR}/R$ as integral of a Bessel function [closed]
In this paper about the electrodynamcis of a spiral resonator, the authors write
$$\frac{e^{-ikR}}{R}=\int_{0}^{\infty} \frac{xdx}{4\pi\sqrt{x^2-k^2}}J_0(Dx)e^{-\sqrt{x^2-k^2}|z|}$$
with $R=\sqrt{z^2+...
- 29
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votes
1
answer
75
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What integral is used to calculate the electric field generated by a continuous charged curve?
I'm studying Multivariable Mathematics, by Ted Shifrin, in which one reads that ''the gravitational force exerted by a continuous mass distribution $\Omega$ with density function $\delta$ is
$$\mathbf{...
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0
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61
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Is $\frac{\partial}{\partial{t}}(\nabla\times H) = \nabla \times \frac{\partial H}{\partial t}$?
While trying to prove a particular equation using Maxwell's equations in electromagnetic theory, there is a step in my textbook that says
$$\frac{\partial}{\partial{t}}(\nabla\times H) = \nabla \times ...
- 535
1
vote
1
answer
60
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How to prove this vector identity? [closed]
I've seen this vector identity from the book[1] in page 89,
$$ (\nabla p)\times\nu =0,\ \text{on}\ \partial\Omega,$$
where $\nu $ is the outer normal vector of $\partial \Omega$, $ p \in H_0^1(\Omega)....
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0
answers
29
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Does this family of curves appearing in the magnetic field of a coil have a name?
While attempting to express the magnetic field induced by a single coil of current (at any point in space, not just on the coil's axis), I tried visualising the set of the infinitesimal contributions $...
- 165
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0
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34
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Boundary Conditions on the Magnetic Flux Density (B-field)
My question is similar to this one (Boundary conditions magnetic field) in that it is related to the boundary conditions of the magnetic field (B-field). However, my question focuses on mathematically ...
- 665
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0
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67
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Solving a funky differential equation.
I'm currently trying to solve the DE that defines charge in a circuit containing an Inductor, Capacitor, Resistor and (crucially) a Memristor. This needs to be able to work for any variable values and ...
- 1
3
votes
1
answer
220
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Show that $\partial_\mu\phi^\ast A^\mu\phi- A_\mu\phi^\ast\partial^\mu\phi=A^\mu\phi\partial_\mu\phi^\ast - A^\mu\phi^\ast\partial_\mu\phi$
The following is loosely related to this question:
[...], the most general renormalisable Lagrangian that is invariant under both Lorentz transformations and gauge transformations is
$$\mathcal{L}=-\...
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votes
1
answer
156
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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 ...
- 419
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votes
1
answer
44
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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 ...
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votes
2
answers
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Fubini's theorem for differential forms? Why does $\int_{t_0}^{t_1}(\oint_{\partial\Omega}j)dt=\int\limits_{[t_0,t_1]\times\partial\Omega}dt\wedge j$?
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(...
- 1,490
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2
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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 ...
1
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0
answers
55
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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 ...
0
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0
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17
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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 ...
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votes
1
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116
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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 ...
- 419
1
vote
2
answers
176
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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 ...
- 419
2
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1
answer
85
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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$ ...
- 409
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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 ...
- 1,111
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vote
2
answers
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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$-...
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0
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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 \, \...
0
votes
1
answer
43
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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 ...
0
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0
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66
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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 $z<0$
$$
\rho =...
- 1
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1
answer
55
views
I was trying to find field due uniformly charged sheet at a distance h from centre of the square sheet
I assumed the square(side a) sheet to be made up of wires.$$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\...
- 471
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0
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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 $(i, j)$. ...
0
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0
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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, ...
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0
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63
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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 ...
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0
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124
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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:
$$
\int_s{\nabla \times \vec{H}} \cdot \vec{dS} = \oint_l{\vec{H} \cdot dl}
$$
...
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1
vote
1
answer
110
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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 $\vec{H}$:
$$\int_S{(\nabla \times \vec{H}) \cdot \vec{dS}} = \oint_L{\vec{H} \...
- 123
1
vote
1
answer
145
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Pseudo-vector formal definition
I have a question about the formalization of pseudovectors. Wikipedia (and my electromagnetism professor and all the electromagnetism books) only state that a vector $v$ transforms as $v' = Rv$, while ...
3
votes
1
answer
84
views
Solving $2\frac{\partial S}{\partial z}+\left(\frac{\partial S}{\partial r}\right)^2=0$
I have been trying to solve this PDE $$2\frac{\partial S}{\partial z}+\left(\frac{\partial S}{\partial r}\right)^2=0$$ The solution of this equation corresponding to a spherical wave of radius of ...
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89
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Divergence theorem with normal component of a curl to a surface
Let $\mathbf{A}$ be a vector function in $\mathbf{R}^3$ and we want to find the normal and tangent components of $\nabla \times \mathbf{A}$ on a smooth and closed surface $\Gamma$. $\mathbf{n}$ is the ...
0
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0
answers
63
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Electric field flux proportional to the field lines generated by (for example) a static charge
Suppose we have a stationary positive charge at a point in space that we call $+Q$. We know by definition that the flow of the electrostatic field is given by, in its simplified form,
$$\Phi_S(\vec E)=...
- 7,792
1
vote
1
answer
93
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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 ...
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votes
1
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711
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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 $R$ ...
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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\}...
- 8,740
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0
answers
65
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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} \...
- 1,179
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votes
1
answer
70
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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 ...
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votes
1
answer
73
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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:
$\...
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