Wire surrounded by a linear dielectric in a uniform E field

[physicsisfun0]

Homework Statement

We have an uncharged, conducting wire with radius a. We surround it by a linear dielectric material, ε_r, which goes out to radius b. We place this in an external electric field, E_o.

Homework Equations

We have electric potential inside (a < s < b)
V_inbetween=Acosφ + (B/s)cosφ
and outside (s > b)
V_outside=-E_oscosφ + (D/s) cosφ

Our boundary conditions are:
V_inbetween=V_outside
ε_rE_sin=ε_rE_sout
when s = a, V = 0
when s = b, V = -E_oscosφ

The Attempt at a Solution

I have solved for the constant and got:
A = - B/a²
B = (-E_ob² + D)/(b²(a^-2 + a^-2)
D = -((ε_rb²E_o + ε_rBE_o + (E_o/a²) + (E_o/a²))b)/(b(a^-2 + b^-2 - ε_r + ε_rb)

Is this correct? Once I know these, I can find electric potential and electric field. From there, how would I go about solving for surface density on the dielectric of a bound and free charge, when s=a and s=b.

 

[kuruman]

It is not clear that you have applied the boundary conditions correctly. Presumably there is vacuum for s > b therefore there ought to be terms with ε₀ coming from the continuity of the normal component of ##\vec D## across the boundary, $$\epsilon_r \frac{\partial V_I}{\partial s}=\epsilon_0 \frac{\partial V_{II}}{\partial s}$$ Also the sum in the expression for coefficient D (Eo/a²))b)/(b(a^-2 + b^-2 - ε_r + ε_rb) is dimensionally inconsistent.