Field Angular Momentum (Thomson Dipole)

Septim

1. The problem statement, all variables and given/known data
I have attached the question as jpg to this post. Typing these were too time consuming and I uploaded the relevant sections as image files, thanks for your understanding.

2. Relevant equations

mu_0*epsilon_0*Poynting Vector = Momentum density

Position vector X Momentum density = Angular momentum density

3. The attempt at a solution
These too are attached in png format.

I have found out that angular momentum density has two components, one in the z direction and one in the x direction. According to the books solution the x component integrates to zero but I was unable to verify this and I am highly skeptical about this topic. Any help would be appreciated. I have attached the relevant information to this post.
1. The problem statement, all variables and given/known data



2. Relevant equations



3. The attempt at a solution

Attached Thumbnails

     

TSny

In constructing your equation (8), note that ##\hat{r}## will generally have a ##\hat{y}## component as well as ##\hat{x}## and ##\hat{z}## components: ##\hat{r} = (\hat{r} \cdot \hat{x}) \hat{x} + (\hat{r} \cdot \hat{y}) \hat{y} + (\hat{r} \cdot \hat{z}) \hat{z}##.

##\hat{r} \cdot \hat{x} = sin\theta cos\phi##, etc.

See what you get for the x and y components of the angular momentum when you integrate over ##\phi##.