Use Gauss' Law to calculate the electrostatic potential for this cylinder

[Reg_S]

Homework Statement

An infinitely long hollow (non conducting) circular cylinder of radius R fixed at potential V =V•sin(phi) .

Relevant Equations

Using cylinder coordinates with z axis as a symmetric axis , argue V is independent of Z and V(r, -phi)= -V(r, phi)
b) Find electrostatic potential inside and outside of the cylinder

I solved laplacian equation. and got the solution of V(r, phi) = a. +b.lnr + (summation) an r^n sin(n phi +alpha n ) + (summation) bn r ^-n sin( n phi +beta n)

 

[Reg_S]

Please help me how to use BCs and find the constant,

 

[pasmith]

The boundary condition is $V(R, \phi) = V_0\sin \phi$ (please don't use the same symbol for an unknown function and a given constant value). That immediately suggests trying a solution of the form $$V(r, \phi) = V_0f(r)\sin \phi$$ with $f(R) = 1$.

 

[Reg_S]

Thank you, Is it same BCs for inner and outer potential? just using relative term? Can we do (Phi)in = (phi)out at r=R?

 

[pasmith]

The cylinder is non-conducting, so the potential is continuous across it.