- #1

- 7

- 0

## Homework Statement

The angular momentum density in the electromagnetic field is defined in terms of the momentum density (3.6, BELOW) by

[tex]\textbf{L}_{EM} = \textbf{x}\times\textbf{P}_{EM} = \textbf{x}\times(\textbf{E}\times\textbf{B})/\mu_{0}c^2[/tex]

Show that if the continuity equation for angular momentum is written in the form

[tex]\frac{\partial}{\partial t}(\textbf{L}_{EM})_{i} + \frac{\partial}{\partial x_{j}}(\textbf{M}_{EM})_{ij} = (\textbf{S}_{EM})_{i}[/tex]

then (3.5, BELOW) implies

[tex](\textbf{M}_{EM})_{ij} = \epsilon_{irs}(\textbf{T}_{EM})_{jr}x_{s},[/tex]

[tex]\textbf({S}_{EM})_{i} = -\rho\epsilon_{ijk}x_{j}E_{k}-J_{i}\textbf{x}\cdot\textbf{B}+B_{i}\textbf{x}\cdot\textbf{J}.[/tex]

## Homework Equations

[tex]\frac{\partial}{\partial x_{j}}(\textbf{T}_{EM})_{ij} = \rho E_{i} + (\textbf{J}\times\textbf{B})_{i} + \frac{\partial}{\partial t}(\textbf{P}_{EM})_{i}[/tex] (3.5)

[tex]\textbf{P}_{EM} = \textbf{E}\times\textbf{B}/\mu_{0}c^2[/tex] (3.6)

[tex](\textbf{T}_{EM})_{ij} = -(\frac{1}{2}\varepsilon_{0}|\textbf{E}|^2+\frac{1}{2}|\textbf{B}|^2/\mu_{0})\delta_{ij} + \varepsilon_{0}E_{i}E_{j} + B_{i}B_{j}/\mu_{0}[/tex]

## The Attempt at a Solution

The biggest problem I have is with the tensor algebra. Havent have much practice in it and now all of a sudden I was thrown in a class where tensor algebra is something one should be common with.

This is what I have so far:

I expand Lem and get

[tex](x_{m}E_{i}B_{m} - x_{j}E_{j}B_{i})/\mu_{0}c^2[/tex]

then take the time derivative and using some of Maxwells equations in tensor form i get

[tex]\frac{\partial}{\partial t}(\textbf{L}_{EM})_{i} = (x_{m}E_{i}\epsilon_{mno}\frac{\partial E_{o}}{\partial x_{n}}+x_{j}E_{j}\epsilon_{pqr}\frac{\partial E_{r}}{\partial x_{q}})/\mu_{0}c^2+(x_{m}B_{m}(\epsilon_{mno}\frac{\partial B_{i}}{\partial x_{n}}-\mu_{0}J_{m})-x_{j}B_{i}(\epsilon_{pqr}\frac{\partial B_{r}}{\partial x_{q}}-\mu_{0}J_{j}))/\mu_{0}[/tex]

From here I have no idea how to continue. It is probably something wrong with how i introduce new indices.

The second term d/dxj (Mem)ij I expand and get

[tex]\frac{\partial}{\partial x_{j}}(\textbf{M}_{EM})_{ij} = \epsilon_{irs}(-\frac{\partial}{\partial x_{j}}(\frac{1}{2}\varepsilon_{0}|\textbf{E}|^2+\frac{1}{2}|\textbf{B}|^2/\mu_{0})\delta_{ij} + \frac{\partial}{\partial x_{j}}(\varepsilon_{0}E_{i}E_{j}) + \frac{\partial}{\partial x_{j}}(B_{i}B_{j}/\mu_{0}))x_{s}[/tex]

Will the first term be equal to zero? I dont think there are any indice problems here. The rest is probably taking derivates of products and then use Maxwells equation and many terms will probably then cancel out each other.