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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singular positive semi-definite matrix in electromagnetism
anyone knows where he drew this conclusion from?
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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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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\...
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2d Fourier Transform Using Weyl Expansion
If we have electric field as
$$
\mathbf{E}\left(\mathbf{r}_{\mathrm{d}}, t\right)=\frac{1}{\varepsilon} \int_{\mathcal{V}} d \mathbf{r}^{\prime} \mathbf{K}\left(\mathbf{r}_{\mathrm{d}}-\mathbf{r}^{\...
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3
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How to compute the volume integral for the potential of an arbitrary point outside a uniformly charged ball?
$$\frac{\rho}{4\pi\epsilon_0}\iiint_{D}^{}\frac{1}{\left\| \mathbf{r}-\mathbf{r'} \right \| }dV'$$
$D$ is a ball of radius $R$
$\mathbf{r}$ is the position vector of the point where we want to ...
- 163
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48
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Electric fields and simply-connected regions
I apologize for the ignorance and the rough English in advance, I have an issue understanding how to match both what happens in physics and what I am seeing in calculus.
We learned that if a vector ...
2
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2
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76
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How to apply integration by parts to simplify an integral of a cross product?
I'm reading a physics paper and am trying to figure out how a certain expression is derived (If interested, see Appendix of the paper, Eq. (A7), (A8)). The authors skip a lot of derivation steps and ...
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Solving 4th order differential equation
I have differential equations such as
$$\frac{1}{\lambda^{2}}\psi_{e}'' = \tanh{\psi_{e}+\psi_{h}}$$
$$\psi_{h}'' - \kappa^{2}\psi_{h} = -\alpha^{2}\tanh{\psi_{e}+\psi_{h}}.$$
Boundary conditions are
$...
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Helmholtz - Hodge decomposition in H(curl)
I'm looking for Helmholtz-Hodge type decompositions but for vector fields slighty more regular than $L^2(D,\mathbb{R}^3)$. I'm familiar with the results in the books of Lions and was wondering if ...
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First chern class of magnetic monopole
Example 3.5.5 in the Mirror Symmetry textbook (Hori-Katz-Klemm-et. al) states:
Let us compute the first Chern class of the line bundle defined by the $U(1)$ gauge field surrounding a magnetic monopole,...
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3
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$\int_0^R \frac{r^{l+1}}{\sqrt{R^2 - r^2}}\text dr$
I'm trying to solve problem 3.3 from Jackson's Classical Electrodynamics, but I'm encountering some troubles solving
$$
\int_0^R \frac{r^{l+1}}{\sqrt{R^2 - r^2}}\text dr,
\qquad l = 0,2,4,6,\ldots
$$
...
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1
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Representation of $e^{ikR}/R$ as integral of a Bessel function [closed]
In this paper about the electrodynamcis of a spiral resonator, the authors write
$$\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|}$$
with $R=\sqrt{z^2+...
- 29
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1
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75
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What integral is used to calculate the electric field generated by a continuous charged curve?
I'm studying Multivariable Mathematics, by Ted Shifrin, in which one reads that ''the gravitational force exerted by a continuous mass distribution $\Omega$ with density function $\delta$ is
$$\mathbf{...
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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
$$\frac{\partial}{\partial{t}}(\nabla\times H) = \nabla \times ...
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1
answer
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How to prove this vector identity? [closed]
I've seen this vector identity from the book[1] in page 89,
$$ (\nabla p)\times\nu =0,\ \text{on}\ \partial\Omega,$$
where $\nu $ is the outer normal vector of $\partial \Omega$, $ p \in H_0^1(\Omega)....
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Does this family of curves appearing in the magnetic field of a coil have a name?
While attempting to express the magnetic field induced by a single coil of current (at any point in space, not just on the coil's axis), I tried visualising the set of the infinitesimal contributions $...
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Boundary Conditions on the Magnetic Flux Density (B-field)
My question is similar to this one (Boundary conditions magnetic field) in that it is related to the boundary conditions of the magnetic field (B-field). However, my question focuses on mathematically ...
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67
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Solving a funky differential equation.
I'm currently trying to solve the DE that defines charge in a circuit containing an Inductor, Capacitor, Resistor and (crucially) a Memristor. This needs to be able to work for any variable values and ...
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1
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Show that $\partial_\mu\phi^\ast A^\mu\phi- A_\mu\phi^\ast\partial^\mu\phi=A^\mu\phi\partial_\mu\phi^\ast - A^\mu\phi^\ast\partial_\mu\phi$
The following is loosely related to this question:
[...], the most general renormalisable Lagrangian that is invariant under both Lorentz transformations and gauge transformations is
$$\mathcal{L}=-\...
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1
answer
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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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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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2
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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(...
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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 ...
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0
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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 ...
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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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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
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2
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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 ...
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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
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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 ...
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