# Stumped on E&M problem

1. Dec 10, 2007

### ayoshi

1. The problem statement, all variables and given/known data
infinitely long conducting cylinder (hollow, so outter radius = b and inner radius = a) which is grounded is in a constant E-field (E=Eo in direction of cylinders' axis) has a dielectric substand "painted" on it, with dielectric constant c and thickness D. Find the electric field everywhere!

2. Relevant equations
Well, for dielectrics the usual BC's are:
E[parallel](inside) = E[parallel](outside)
D[perpendicular](inside) - D[perpendicular](outside) = sigma <-- charge density

Also, the general solution for potential in cylindrical coords is:
V = Ao + Bo*ln(r) + $$\sum(An*r^n +Bn*r^(^-^n^))*(Cn*cos(n*phi)+Dn*sin(n*phi)$$

BC for infinite cylinder:
dV/dz = 0
V = -Eo*r*cos(phi) + constant, as r-->infinity (due to constant E-field.)

3. The attempt at a solution
I'm having some serious trouble with this problem. One thing I do know, however, is that all of the Dn*sin(n*phi) terms contribute nothing by symmetry and so we can kiss the Dn's goodbye. I think the BC's I stated above are correct, but I'm having problems using them. Other questions I have are: i) how does the dielectric effect V, and, ii)despite being grounded, wouldn't the external field effect the cylinder hence making it act like a dipole?

Finally, can we assume the E-field inside is zero since it is a conductor? Or does the dielectric and external E-field change this? The E-field in the region a<r<b should be zero too, shouldn't it?

Any help would be great!