Potential of sphere in electric field

[unscientific]

Homework Statement

Part (a): Find potential inside the sphere and outside of the sphere.
Part (b): Find the electric fields in these two cases. Show for the first case it is identical to a conducting sphere in an electric field.

Homework Equations

The Attempt at a Solution

I have found potential inside and outside of sphere by solving the boundary conditions: $\phi_in = \phi_out$ and $\epsilon_{r1} \epsilon_0 \frac{\partial \phi_{in}}{\partial r} - \epsilon_{r2} \epsilon_0 \frac{\partial \phi_{out}}{\partial r} = \sigma$:

\phi_{in} = - \frac{\sigma_0 + 3\epsilon_0\epsilon_{r2}E_0}{\epsilon_0 (\epsilon_{r1} + 2\epsilon_{r2})} r cos \theta

\phi_{out} = \left[ -E_0 r + \frac{\left( E_0 \epsilon_0 (\epsilon_{r1} - \epsilon_{r2} \right) - \sigma_0}{\epsilon_0 (\epsilon_{r1} + 2\epsilon_{r2})} \frac{a^3}{r^2} \right] cos \thetaFor $\sigma_0 = 3\epsilon_0 E_0$:

E_{in} = -\nabla \phi_{in} = \frac{\sigma_0 + 3\epsilon_0\epsilon_{r2}E_0}{\epsilon_0 (\epsilon_{r1} + 2\epsilon_{r2})} cos \theta
= \frac{3E_0 + 3\epsilon_{r2} E_0}{\epsilon_{r1} + 2\epsilon_{r2}} cos \theta

And for outside:
E_{out} = \left[ 1 + \frac{2(\epsilon_{r1} - \epsilon_{r2} - 3)}{\epsilon_{r1} + 2\epsilon_{r2}} \left( \frac{a}{r}\right)^3 \right] E_0 cos \theta

I'm not sure what I'm supposed to look out for?

For $\sigma_0 = -(\epsilon_r -1)\epsilon_0 E_0$:

I'm assuming outside is free space with $\epsilon_{r2} = 1$:

E_{in} = \frac{\sigma_0 + 3\epsilon_r E_0 \epsilon_0}{\epsilon_0 (\epsilon_r + 2} cos \theta
= \frac{1 - \epsilon_r + 3}{2 + \epsilon_r} E_0 cos\theta
= \frac{4 + \epsilon_r}{2 + \epsilon_r} E_0 cos \theta

For outside:
E_{out} = \left[ E_0 + \frac{2\left( E_0 \epsilon_0 (\epsilon_r - 1) - \sigma_0 \right)}{\epsilon_0 (\epsilon_r + 2)} \left( \frac{a}{r}\right)^3 \right] cos \theta
= \left[ 1 + \frac{2(\epsilon_r - 1) + (\epsilon_r - 1)}{\epsilon_r + 2}\left( \frac{a}{r}\right)^3 \right] cos \theta
= \left[ 1 + \frac{3(\epsilon_r - 1)}{\epsilon_r + 2} \left( \frac{a}{r}\right)^3 \right] cos \theta

What's the deal with the conductor in an electric field?
I know that for a conducting sphere, the positive charges migrate to +z while the negative charges migrate to -z and they are all spread on the surface of the sphere.

 

[unscientific]

I have searched and found online that when $\sigma_0 = 3\epsilon_0 E_0$, the field inside would simply me $-E_0 r cos \theta$.

This clearly isn't the case here, I'm not sure why!