# insulator based gauss law questions

My book is incredibly scarce on insulator based Gauss law questions. Conductors seem to handle themselves pretty simply.

Here's a question I'm working on that isn't part of my book. where the radii are $a,b,c,d$ from smallest to largest and gray regions are insulating spherical shells with charge distributions.

Given $a,b,c,d,q$ Find:-
1)$E(r)$ for all the different positions
2)Graph $E(r)$
3)Find volume charge densities

• I've never done "partial" shaped insulators, my thought process is that since $\Phi = \frac{q_i}{\epsilon_0}$ and $q_i$ is relative to how much of the volume is enclosed by whatever gaussian surface with radius r we make:

$\rho = \frac{q}{V}$

$\rho \cdot V_i = q_i$

$\int^r_a \rho 4 \pi r^2 dr = q_i$

is this the right process?

• Conductors are conveniently discontinuous and easy to handle in separate "radius chunks", but for insulators shouldn't they scramble each other? when I'm dealing with between the outer shell, or beyond the outer shell, how do i handle the fully enclosed inner shell? My gut says it's additive and could be treated as a point charge, but I have no way of being sure.

Thanks for any help.