I have a sphere of radius $2a$ centered at the origin and made of a nonconducting material that has a uniform volume charge density $\rho$. A spherical cavity of radius $a$ eccentric to the right side (touching) with its center aligned with the x-axis is now removed from the sphere. Find the electric field produced by this distribution along the x-axis for $x > 2a$.
I started by establishing expressions for $E$ in both the uniform section and the cavity:
$E_u = \rho 2a/(3\epsilon_0)$ and $E_c = -\rho a/(3\epsilon _0)$ and $E_t$ inside the cavity is $ E_t = \rho a/(3\epsilon_0)$
Is it correct that the $E$ along the x-axis would have the same magnitude as $E_t$?
I'm thinking it's not, but I'm not sure about the other idea I had which is that it would be $E = \rho r/\epsilon_0$. $r$ now being the distance from the center of the solid/enclosing sphere. This is the result of exchanging the $-\rho$ that is required to make the net charge inside the cavity with $\rho$ and adding the $E$-s together again.