All Questions
Tagged with electromagnetism tensors
11
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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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Finding the change of basis matrix for a type (0,1) tensor
I am considering a tensor (in particular, the electric field), defined by
$$E_m = g_{ij}^k c_{k\ell}^{ij}S_{\ell m} $$
Ultimately, this means that the tensor E is a rank 1, type (0,1) tensor, ...
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2
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Formula for the Maxwell Stress tensor in arbitrary coordinates
This question is nearly identical to my last, except this time its the Maxwell stress tensor, not the Cauchy stress tensor. I often see its components written as
$$\sigma_{ij}=\varepsilon_0E_iE_j+\...
- 12.5k
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Is there a way to represent electrostatics tensors in tensor (possibly tensor product) way?
I'm working with electrostatic interaction tensors, which are defined as follows:
\begin{align}
T &= \frac{1}{r} \\
T^\alpha &\equiv \nabla T = -\frac{r^\alpha}{r^3} \\
T^{\alpha\...
- 21
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Why does the electromagnetic tensor in component form coincide with the differential-geometric definition of a $2$-form?
From physics classes, I understand the electromagnetic field strength tensor to be defined as
$$F^{\mu\nu}=\partial^\mu A^\nu-\partial^\nu A^\mu \;,$$
where $\partial^\mu$ is the partial derivative (...
- 157
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Gradient in tensor form
I found a problem which had $$\partial_i (A_i \vec{G})= (\vec{\nabla} .\vec{ A} )\vec{G}+ (\vec{A}.\nabla) \vec{G} $$ but my problem is what does $$\partial_i (A_i \vec{B})$$ even mean? it doesn't ...
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Show that $\nabla . *F = 0$ is a geometric frame independent of ...
Show that $\nabla\cdot\ast F = 0$ (divergence of the dual of the EMF tensor) is a geometric frame independent version of $F_{ab,c} + F_{bc,a} + F_{ca,b} = 0$, where $F$ is the electromagnetic field ...
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Unconstrained quartic optimization
I am working on antenna array pattern synthesis algorithms, and am trying to minimize the following expression with respect to $\mathbf{v}$
$$ \int_\Omega \Big[ \big( \mathbf{v}^H \mathbf{Y}(\vartheta,...
- 11
2
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645
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Find directions where current is maximal
The current $J_i$ due to an electric field $E_i$ is given by $J_i = σ_{ij} E_j$ , where $σ_{ij} is the conductivity tensor. In
a given Cartesian coordinate system,
$σ=\begin{pmatrix}2&-1&-1 \\...
- 638
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357
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Tensor manipulation involving Levi-Cevita tensor
I am attempting to follow a derivation from a physics paper relating to covariant electromagnetism. It is given that,
$$ F^{\mu \nu} = u^{\mu}E^{\nu} - E^{\mu} u^{\nu} + \epsilon^{\mu \nu \alpha \...
- 508
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Showing that no current flows in some direction, given $\sigma_{ij}$ and that $J_i=\sigma_{ij}E_j$.
The current $J_i$ due to an electric field $E_i$ is given by
$J_i=\sigma_{ij}E_j$, where $\sigma_{ij}$ is the conductivity tensor.
In a certain coordinate system, $$(\sigma_{ij})=\begin{pmatrix}
2&...
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