Tensor algebra in electromagnetic fields

[billybomb87]

Homework Statement

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

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

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

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

then (3.5, BELOW) implies

$$(\textbf{M}_{EM})_{ij} = \epsilon_{irs}(\textbf{T}_{EM})_{jr}x_{s},$$

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

Homework Equations

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

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

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

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

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

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

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

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

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

Will the first term be equal to zero? I don't 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.

 

[chrispb]

I didn't read all of your work, but just looking at the bottom equation, there must be something wrong with it. You have a free i index on the left hand side, and free r and s indices on the right hand side. Free indices should match.

Once you resolve that problem, I suspect your first term will vanish because you will be contracting something antisymmetric in its indices with something symmetric in its indices.

 

[ulriksvensson]

One thing that really makes your life easier is to write derivative as

$$\frac{\partial Y^{i}}{\partial x^{j}} = \partial_{j}Y^{i}$$

Also, even though it's not written like that in the problem, you should always sum indices appearing upstairs and downstairs, like this

$$\epsilon^{ijk}E_{j}E_{k}$$

In this expression summation is understood over j and k. i is a free upper index. I suspect that you are not dealing with relativity here and then it doesn't matter if the indices are upstairs or downstairs but it makes proofreading a lot easier. For instance if your right hand side has two free indices upstairs, then your left hand side has better have two free indices upstairs as well.

 

[billybomb87]

I didn't read all of your work, but just looking at the bottom equation, there must be something wrong with it. You have a free i index on the left hand side, and free r and s indices on the right hand side. Free indices should match.

Once you resolve that problem, I suspect your first term will vanish because you will be contracting something antisymmetric in its indices with something symmetric in its indices.

If you look closely on the last factor there is an Xs there, does that mean that it isn't a free index?

But still...haven't really understood what I am supposed to do now...please help me

 

[paranormal]

Dear Student,

Thank you for your response. Tensor algebra is indeed a complex and important topic in electromagnetic fields. It is important to have a good understanding of tensor algebra in order to fully comprehend the equations and their implications.

To solve this problem, we can start by expanding the equation for the continuity of angular momentum. We get:

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

Expanding the first term using the definition of angular momentum density, we get:

\frac{\partial}{\partial t}(x_{m}E_{i}B_{m} - x_{j}E_{j}B_{i})/\mu_{0}c^2

Using the product rule and Maxwell's equations, this can be simplified to:

(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

For the second term, we can use the expression for the angular momentum tensor (3.5) and expand it:

\frac{\partial}{\partial x_{j}}(\textbf{M}_{EM})_{ij} = \frac{\partial}{\partial x_{j}}(-(\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}))x_{s}

Using the product rule and simplifying, we get:

\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}) +