Angular momentum of the EM field of rotating sphere

[Shinobii]

The angular momentum of the electromagnetic field is defined as,

$$
\vec{L_{em}} = \int \vec{l_{em}} d^3r.
$$

To solve this for a rotating sphere I must consider the cases where r < R and r > R.

When I did this problem I thought that there would be two solutions, one for both cases; however, it turns out that there is one solution,

$$
\vec{L_{em}} = \int \vec{l_{em}}_{(r<R)} \, d^3r + \int \vec{l_{em}}_{(r>R)} \, d^3r.
$$

Can anyone tell me why that is? Conceptually I do not understand what is going here.

 

[Shinobii]

Also for the integration, would I integrate the r < R case from $\int_0^R = \int_0^r + \int_r^R$ and the case of r > R, $\int_R^{\infty}$?

Or would I simply just integrate $\int_0^R$ for both cases, without splitting the integral.

 

[Shinobii]

I suppose when calculating the field angular momentum, we do not need to split the r < R integral $\int_0^R$. I also understand now that we are integrating over all space or over the entire field.