1. The problem statement, all variables and given/known data The cylinder has a radius a and is perpendicular to the electric field, E(r)=E(x_hat). It also carries charge Q. The potential is of the form V(r,φ)=A0+A0'ln(r)+∑(n=1 to ∞)((Ancos(nφ)+Bnsin(nφ))rn+(An'cos(nφ)+Bn'sin(nφ))r-n) 2. Relevant equations V=-∫E⋅dl 3. The attempt at a solution The above equation yields V=-Ex=-Ercosφ The two boundary conditions are V(r,φ)=-Ercosφ above the surface of the cylinder and V=0 at r=a. Since -Ercosφ=-ErP1(cosφ), the n on the left side of the equation have to be equal to 1. Starting with the first boundary condition, as r→∞ the r-1 term will be negligible, the A0 will be negligible, and ln(r) term will grow slower than the r term so it will be negligible. Thus, (A1cosφ+B1sinφ)r=-Ercosφ. Solving for A1 gives A1=-E-B1tanφ. The other boundary condition gives A0+A0'ln(a)+(A1cosφ+B1sinφ)a+(A1'cosφ+B1'sinφ)a-1=0. Here I'm stuck. The plan was to solve for B1 then have two equations and two unknowns, but I don't know what to do about the prime letters.