Finding the electric field for a shell of charge

Question

Suppose we have charge density defined

$$ \rho(x,y,z) = \begin{cases} 0 & 0 \leq r < a \\ K & a \leq r\leq b\\ 0 & b< r \end{cases} $$

For some constants $K,a,b$

How would we find the electric field for all points in space? Any help I would greatly appreciate! I imagine we argue that the field must be symmetric, but I was having difficulty proceeding from there.

Answer 1

The electric field due to a spherical shell is zero inside the sphere and equal to the field of an equivalent charge at the centre outside the sphere. Thus the field is directed outward everywhere, and its strength is zero for $r\lt a$, $K\frac43\pi(r^3-a^3)/r^2$ for $a\le r\le b$ and $K\frac43\pi(b^3-a^3)/r^2$ for $b\lt r$.