All Questions
46
questions
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31
views
How does Schwinger transform a static $(\phi,0)$ to $(\phi ', A)$ in a moving frame?
Schwinger writes in his paper Electromagnetic mass revisited:
Any spherically symmetrical charge distribution of total charge $e$, at rest, is represented by the potentials
$$\phi \sim ef(r^2),...
- 3,108
0
votes
2
answers
694
views
How to find the mixed tensor, contravariant tensor and tensor trace of $F$?
I have a question in particle physics that ask me to find the mixed tensor, contravariant tensor and tensor trace of $F$:
Our professor didn't teach us that much about the math of tensor, which makes ...
- 55
2
votes
2
answers
374
views
Showing $\partial_{\mu}\tilde{F}^{\mu\nu}=0$ by the antisymmetric properties
The electromagnetic dual tensor is given by
\begin{align}
\tilde{F}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\delta\rho}F_{\delta\rho}
\end{align}
Here, $\epsilon^{\mu\nu\delta\rho}$ is the Levi-Civita ...
- 988
0
votes
1
answer
126
views
Electromagnetic tensor [closed]
How to prove the equality in this?
$$F^{\mu\nu}F_{\mu\nu}=g^{\mu\alpha}g^{\nu\beta}\left(\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}\right)\left(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\...
- 808
2
votes
1
answer
213
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Tensor Differentiation
In the book "Tensors, Relativity and Cosmology" the author derived Maxwell's Equation in covariant form using the EM field tensor Lagrangian $L=-\frac{1}{4}F^{jl}F_{jl}$ (source=0). One of the steps ...
- 1,047
0
votes
1
answer
91
views
Am I proving an identity about Maxwell's "Magnetic" Equations correctly?
Question 3.12(d) of Gravitation (MTW) has me show Maxwell's "magnetic" equations $F_{\alpha \beta , \gamma} + F_{\beta \gamma , \alpha} + F_{\gamma \alpha , \beta} = 0$ can be rewritten as $F_{[\alpha ...
0
votes
1
answer
75
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Show $2(B \cdot \nabla)B = \nabla |B|^2$ when the B-field is curl-less using summation notation
I was able to show for myself that
$$ 2(\mathbf{B} \cdot \mathbf{\nabla})\mathbf{B} = \mathbf{\nabla} |\mathbf{B}|^2$$ when $\mathbf{\nabla} \times \mathbf{B} = 0$, but in order to do this, I had to ...
- 1,302
0
votes
1
answer
2k
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How prove that the contraction of electromagnetic tensor and its dual, $F_{\mu\nu}\tilde F^{\mu\nu}$, is an total derivative? [closed]
I have tried to solve this exercise from Supergravity-Freedman and Van Proeyen (2012),
Excercise 4.10 Show that the quantity $F_{\mu\nu}\tilde F^{\mu\nu}$ is a total derivative, i.e. $$F_{\mu\nu}\...
1
vote
1
answer
126
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Rewrite Lagrangian Density for Electromagnetism
The Lagrangian density for Electromagnetism is given by $$\mathcal{L}=-\frac{1}{4}F_{\mu \nu}F^{\mu \nu},$$ where $F_{\mu \nu} = \partial_{\mu}A_\nu - \partial_{\nu}A_\mu$ is the field strength tensor....
- 58
0
votes
2
answers
955
views
Finding equation of motion of Lagrangian density: What does the location of the indices mean?
We are given the following Lagrangian density:
$$\mathcal{L}=F_{\mu \nu} A^{\mu} \mathcal{J}^{\nu}$$
where $F_{\mu \nu}$ is the electromagnetic field tensor, $ A^{\mu}$ the 4-vector of the vector ...
- 511
-1
votes
1
answer
61
views
Is it clear that $\epsilon_{abcd}F^{ab}(\delta^{\mu}_e\delta^{\nu}_f-\delta^{\mu}_f\delta^{\nu}_e)\eta^{ce}\eta^{df}$ vanishes? [closed]
Is it clear that $$\epsilon_{abcd}F^{ab}(\delta^{\mu}_e\delta^{\nu}_f-\delta^{\mu}_f\delta^{\nu}_e)\eta^{ce}\eta^{df}$$ vanishes without computing explicitly?
Here $\epsilon_{abcd}$ is the totally ...
- 1,982
2
votes
2
answers
2k
views
Deriving Euler-Lagrange for Electrodynamics Lagrangian [duplicate]
For $\mathcal L = -\frac14 F_{\mu\nu}F^{\mu\nu}$ I would appreciate some help evaluating
$$\frac{\partial \mathcal L}{\partial(\partial_{\mu}A_{\nu})}.$$
I've found
$$\frac{\partial \mathcal L}{\...
- 1,982
2
votes
1
answer
382
views
Components of electromagnetic tensor in a moving frame
I need to find the electric and magnetic components of electromagnetic tensor in an inertial frame S' moving in the +x direction with a speed $\beta$ relative to frame S. Electromagnetic tensor in S ...
- 41
0
votes
3
answers
2k
views
Square of the Maxwell Field Tensor
I want to prove that the square of the Maxwell field tensor
$$F_{\mu\nu}F^{\mu\nu}=2(B^2-E^2),$$
but I got $F_{\mu\nu}F^{\mu\nu}=2(-B^2+E^2)$ instead.
Here's what I did:
$$F_{\mu\nu}F^{\mu\nu}=F_{0\nu}...
- 1,161
1
vote
2
answers
639
views
4-Vector Potential Notation
How am I supposed to interpret this notation:
$$F^{uv} = \partial^uA^v-\partial^vA^u$$
I know that $\partial^u = (\frac{1}{c}\frac{\partial}{\partial t},-
\vec\nabla)$
So for example for the ...
- 11