All Questions
46
questions
1
vote
2
answers
438
views
Maxwell equations in Special Relativity [closed]
I'm currently studying special relativity, and with it, tensor algebra. I have some difficulties in deriving a tensor differential relation involving the field tensor $F_{\mu\nu}$.
I have the Maxwell ...
- 55
5
votes
2
answers
11k
views
Equivalent form of Bianchi identity in electromagnetism
In electromagnetism, we can write the Bianchi identity in terms of the field strength tensor $F_{\mu \nu}$ as,
$$ \partial_{\lambda} F_{\mu \nu} + \partial_{\mu} F_{\nu \lambda}+ \partial_{\nu} F_{\...
- 1,751
1
vote
1
answer
250
views
Question about tensor form of Maxwell equation [closed]
By variating the Maxwell Lagrangian we get the equation of motion. The remaining two Maxwell equations can be written as
$$\epsilon_{\mu\nu\rho\sigma}\partial^{\rho} F^{\mu\nu} = 0.$$
I have also seen ...
- 233
0
votes
1
answer
147
views
If $T_{kl}=\epsilon_{kil}m_i$, how to show $m_i=0.5\epsilon_{ilk}T_{lk}$? [closed]
In a book I am reading (about magnetic dipole), it is given that $T_{kl}=\epsilon_{kil}m_i$. Then, it says since $T_{kl}=-T_{lk}$, it can be shown that $m_i=0.5\epsilon_{ilk}T_{lk}$. I understand that ...
- 233
1
vote
1
answer
2k
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Rewriting Maxwell's equation in tensor form [closed]
Suppose $F_{ij}=\epsilon_{ijk}B_k $, how to prove the following:
$\partial_iB_i=0$ becomes $\partial_iF_{jk}+\partial_jF_{ki}+\partial_kF_{ij}=0$
$B_iB_i$ becomes $F_{ij}F_{ij}/2$
I can see that it ...
- 233
1
vote
1
answer
170
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Work out components $F^{01}$ and $F^{ij}$ of the antisymmetric tensor $F^{\mu\nu}$ under the Lorentz Transform [closed]
Work out explicitly how the components $F^{0i}$ and $F^{ij}$ of the antysymmetric tensor $F^{\mu\nu}$ introduced in chapter I.6 transform under a Lorentz transformation
This problem is from Zee, ...
- 163
6
votes
3
answers
6k
views
Is this a Lorentz-scalar? How do I tell?
I'm struggling to identify whether a scalar is a Lorentz-scalar. E.g:
$$\partial_i A^i \quad i \in {1,2,3}.$$
How do I determine if this is a Lorentz-scalar or not?
If got the same problem with ...
- 486
0
votes
1
answer
181
views
How to write the Lagrangian in terms of a projection
We know that
$$
L=\frac{1}{2}\left(\partial_{\mu} A_{\nu} \partial^{\mu} A^{\nu}-\partial_{\mu} A_{\nu} \partial^{\nu} A^{\mu}\right)
$$
But how do we write the Lagrangian in the following way:
$$L=...
0
votes
1
answer
505
views
How do you take the derivative with respect to a rank two tensor?
I am learning classical field theory and am trying to find the momentum density of the electromagnetic lagrangian as part of an example of Noether's Theorem. The derivative I am encountering is:
$$
\...
- 935
0
votes
3
answers
4k
views
Tricks for evaluating tensor contractions with Levi-Civita symbol
I am trying to evaluate the Lorentz invariant $\epsilon^{\alpha\beta\gamma\delta}F_{\alpha\beta}F_{\gamma\delta}$, where $F_{\mu\nu}$ is the electromagnetic field tensor,
$$
F_{\mu\nu} = \begin{...
- 935
1
vote
1
answer
182
views
Electric Magnetic duality
In this paper http://arxiv.org/abs/hep-th/9705122 Section 2
We have $$S_A = \frac{1}{4g^2} \int{d^4x F_{\mu\nu}(A)F^{\mu\nu}(A)}$$
where $F_{\mu\nu}(A) = \partial_{[\mu A\nu]}$. Its Bianchi Identity ...
- 215
2
votes
1
answer
319
views
Proof that 4-potential exists from Gauss-Faraday field equation
This is a problem concerning covariant formulation of electromagnetism.
Given
$$\partial_{[\alpha} F_{\beta\gamma]}~=~ 0 $$
how does one prove that $F$ can be obtained from a 4-potential $A$ such ...
- 626
3
votes
1
answer
2k
views
Electromagnetic Tensor in Cylindrical Coordinates
I understand that the Electromagnetic Tensor is given by
$$F^{\mu\nu}\mapsto\begin{pmatrix}0 & -E_{x} & -E_{y} & -E_{z}\\
E_{x} & 0 & -B_{z} & B_{y}\\
E_{y} & B_{z} & ...
- 1,373
1
vote
1
answer
935
views
Derivative of covariant EM tensor
I cannot seem to prove that the derivative of the duel tensor = 0.
$$ \frac{1}{2}\partial_{\alpha}\epsilon^{\alpha \beta \gamma \delta} F_{\gamma \delta} = 0. $$
Writing this out I get (for some ...
- 129
3
votes
3
answers
3k
views
Maxwell Stress Tensor in the absence of a magnetic field
I'm having some trouble calculating the stress tensor in the case of a static electric field without a magnetic field. Following the derivation on Wikipedia,
Start with Lorentz force:
$$\mathbf{F} = ...
- 133