All Questions
Tagged with electromagnetism mathematical-physics
16
questions
15
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4
answers
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Electromagnetic field and continuous and differentiable vector fields
We have notions of derivative for a continuous and differentiable vector fields. The operations like curl,divergence etc. have well defined precise notions for these fields.
We know electrostatic and ...
- 1,578
14
votes
1
answer
6k
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How do you go from quantum electrodynamics to Maxwell's equations?
I've read and heard that quantum electrodynamics is more fundamental than maxwells equations. How do you go from quantum electrodynamics to Maxwell's equations?
- 344
3
votes
1
answer
294
views
Formal Connection Between Symmetry and Gauss's Law
In the standard undergraduate treatment of E&M, Gauss's Law is loosely stated as "the electric flux through a closed surface is proportional to the enclosed charge". Equivalently, in differential ...
- 130
15
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3
answers
7k
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Electromagnetism for mathematicians
I am trying to find a book on electromagnetism for mathematicians (so it has to be rigorous).
Preferably a book that extensively uses Stokes' theorem for Maxwell's equations
(unlike other books that ...
8
votes
2
answers
583
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Calculating the potential on a surface from the potential on another surface
The question is short: If a charge (or mass) distribution $\rho$ is enclosed by surface $S_1$, I can calculate the electrostatic (or gravitational) potential on that surface by integrating $\rho(r') \ ...
- 6,273
6
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2
answers
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Proof of Ampère's law from the Biot-Savart law for tridimensional current distributions
Let us assume the validity of the Biot-Savart law for a tridimensional distribution of current:$$\mathbf{B}(\mathbf{x})=\frac{\mu_0}{4\pi}\int_V \mathbf{J}(\mathbf{y})\times\frac{\mathbf{x}-\mathbf{y}}...
4
votes
4
answers
3k
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How to interpret the continuity conditions in the PDEs (for example, Maxwell equations) originated in physics?
I am currently working on PDEs in physics, mostly Maxwell equations. I am a mathematics graduate student, and this question has been haunting me for years.
In PDE theory, or more specifically the ...
- 203
4
votes
1
answer
577
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Can one force the octupole moments of a charge distribution (neutral and with vanishing dipole moment) to vanish using a suitable translation?
In a previous question, I noted that if you have a charge distribution with nonzero charge, then it is possible to choose an origin (at the centre of charge) which makes its dipole moment vanish, and ...
- 134k
28
votes
1
answer
1k
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Electric charges on compact four-manifolds
Textbook wisdom in electromagnetism tells you that there is no total electric charge on a compact manifold. For example, consider space-time of the form $\mathbb{R} \times M_3$ where the first factor ...
- 401
9
votes
2
answers
1k
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Vector Potential for Magnetic field when the field is not in simply-connected region
According to Poincare's Lemma, if $U\subset \mathbb{R}^n$ is a star-shaped set and if $\omega$ is a $k$-form defined in $U$ that is closed, then $\omega$ is exact, meaning that there's some $(k-1)$-...
- 36.4k
16
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2
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Why isn't the path integral defined for non-homotopic paths?
Context
In the Aharonov Bohm effect, there is a solenoid which creates a magnetic field. Since the electron cannot be inside the solenoid, the configuration space is not simply connected.
Question
I'...
- 12.4k
15
votes
1
answer
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Why is the Hodge dual so essential?
It seems unnatural to me that it is so often worthwhile to replace physical objects with their Hodge duals. For instance, if the magnetic field is properly thought of as a 2-form and the electric ...
- 1,452
7
votes
2
answers
1k
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Facing a paradox: Earnshaw's theorem in one dimension
Consider a one-dimensional situation on a straight line (say, $x$-axis). Let a charge of magnitude $q$ be located at $x=x_0$, the potential satisfies the Poisson's equation $$\frac{d^2V}{dx^2}=-\frac{\...
- 26.8k
7
votes
0
answers
443
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1-form formulation of quantized electromagnetism
In a perpetual round of reformulations, I've put quantized electromagnetism into a 1-form notation. I'm looking for references that do anything similar, both to avoid reinventing the wheel and perhaps ...
- 9,948
3
votes
1
answer
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Proof of equality of the integral and differential form of Maxwell's equation
Just curious, can anyone show how the integral and differential form of Maxwell's equation is equivalent? (While it is conceptually obvious, I am thinking rigorous mathematical proof may be useful in ...
- 203