Potenital of a cylindrical, linear dialectric in a E-Field

[PhotonSSBM]

Homework Statement

A very long cylinder of linear dialectric material is placed in an otherwise uniform electric field $E_0$. Find the resulting field within the cylinder. (The radius is $a$, the susceptibility $X_e$, and the axis is perpendicular to $E_0$)

Homework Equations

Boundary conditions
$V_i = V_o$ at $s=a$

$\epsilon \frac{\partial V_i}{\partial n} = \epsilon_0 \frac {\partial V_o}{\partial n}$ because there is no free charge

$V = -E_0*s*cos(\theta)$ for $r>>a$

The Attempt at a Solution

I've gotten this far

$\epsilon_r[-na^{-n-1}* (Acos(n*\theta) + Bsin(n*\theta))] = na^{n-1} * (Ccos(n*\theta) + Dsin(n*\theta)) -E_0cos(\theta)$

where n is what I'm summing over. I'm unsure of how to restrict the constants of the trig functions to reduce the problem.

 

[TSny]

This equation isn't correct, as written. There should be a summation over $n$ on each side (except for the term $-E_0 \cos\theta ~$, which is not summed over $n$).

For each value of $n$, the coefficient of $\cos(n \theta)$ on the left side must match the coefficient of $\cos(n \theta)$ on the right side. Similarly for $\sin(n \theta)$.

So, what condition on the coefficients do you get when matching $\cos(n \theta)$ for the special case of $n = 1$? What condition on the coefficients do you get when matching $\cos(n \theta)$ for $n \neq 1$?

Similarly, get equations from matching $\sin (n\theta)$.

There is also another boundary condition on $V$ (which you stated in the relevant equations section) that you need to invoke where you can again match coefficients on each side.