Questions tagged [electromagnetism]
For questions on Classical Electromagnetism from a mathematical standpoint. This tag should not be the sole tag on a question.
405
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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
$$
\mathbf{E}\left(\mathbf{r}_{\mathrm{d}}, t\right)=\frac{1}{\varepsilon} \int_{\mathcal{V}} d \mathbf{r}^{\prime} \mathbf{K}\left(\mathbf{r}_{\mathrm{d}}-\mathbf{r}^{\...
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How to compute the volume integral for the potential of an arbitrary point outside a uniformly charged ball?
$$\frac{\rho}{4\pi\epsilon_0}\iiint_{D}^{}\frac{1}{\left\| \mathbf{r}-\mathbf{r'} \right \| }dV'$$
$D$ is a ball of radius $R$
$\mathbf{r}$ is the position vector of the point where we want to ...
- 163
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Electric fields and simply-connected regions
I apologize for the ignorance and the rough English in advance, I have an issue understanding how to match both what happens in physics and what I am seeing in calculus.
We learned that if a vector ...
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2
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76
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How to apply integration by parts to simplify an integral of a cross product?
I'm reading a physics paper and am trying to figure out how a certain expression is derived (If interested, see Appendix of the paper, Eq. (A7), (A8)). The authors skip a lot of derivation steps and ...
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Solving 4th order differential equation
I have differential equations such as
$$\frac{1}{\lambda^{2}}\psi_{e}'' = \tanh{\psi_{e}+\psi_{h}}$$
$$\psi_{h}'' - \kappa^{2}\psi_{h} = -\alpha^{2}\tanh{\psi_{e}+\psi_{h}}.$$
Boundary conditions are
$...
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Helmholtz - Hodge decomposition in H(curl)
I'm looking for Helmholtz-Hodge type decompositions but for vector fields slighty more regular than $L^2(D,\mathbb{R}^3)$. I'm familiar with the results in the books of Lions and was wondering if ...
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First chern class of magnetic monopole
Example 3.5.5 in the Mirror Symmetry textbook (Hori-Katz-Klemm-et. al) states:
Let us compute the first Chern class of the line bundle defined by the $U(1)$ gauge field surrounding a magnetic monopole,...
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$\int_0^R \frac{r^{l+1}}{\sqrt{R^2 - r^2}}\text dr$
I'm trying to solve problem 3.3 from Jackson's Classical Electrodynamics, but I'm encountering some troubles solving
$$
\int_0^R \frac{r^{l+1}}{\sqrt{R^2 - r^2}}\text dr,
\qquad l = 0,2,4,6,\ldots
$$
...
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Representation of $e^{ikR}/R$ as integral of a Bessel function [closed]
In this paper about the electrodynamcis of a spiral resonator, the authors write
$$\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|}$$
with $R=\sqrt{z^2+...
- 29
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What integral is used to calculate the electric field generated by a continuous charged curve?
I'm studying Multivariable Mathematics, by Ted Shifrin, in which one reads that ''the gravitational force exerted by a continuous mass distribution $\Omega$ with density function $\delta$ is
$$\mathbf{...
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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
$$\frac{\partial}{\partial{t}}(\nabla\times H) = \nabla \times ...
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How to prove this vector identity? [closed]
I've seen this vector identity from the book[1] in page 89,
$$ (\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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