Field Angular Momentum (Thomson Dipole)

[Septim]

Homework Statement

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.

Homework Equations

mu_0*epsilon_0*Poynting Vector = Momentum density

Position vector X Momentum density = Angular momentum density

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.

 

[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 $\phi$.