Boundary conditions for charged cylinder

[bigplanet401]

Hello,

Charge density $$\sigma(\phi) = k \sin 5\phi$$ (where k is a constant is glued over the surface of an infinite cylinder of radius R with axis along the z-direction. Find the potential inside and outside the cylinder.

Two things I'm having trouble with:

1. Is the potential of an infinite cylinder
$$\begin{equation}<br /> V(\rho^\prime, \phi) = \sum_{m = 0}^{\infty} \, [A_m J_m (k\rho^\prime) + B_m N_m (k\rho^\prime)] [C_m \sin m\phi + D_m \cos m\phi] \; ?<br /> \end{equation}$$

Do you need to include Neuman functions in the full solution?

2. Whatare the boundary conditions for this problem? Not knowing the potential at rho^prime = R made me confused. How many conditions do you need? And does the charge density tell you in any way about the radial dependence of the potential?

 

[member 553083]

Thank you for your help. The potential of an infinite cylinder is given by the solution to Laplace's equation in cylindrical coordinates: \begin{equation}V(\rho^\prime, \phi) = \sum_{m = 0}^{\infty} \, [A_m J_m (k\rho^\prime) + B_m Y_m (k\rho^\prime)] [C_m \sin m\phi + D_m \cos m\phi]\end{equation}where $J_m$ and $Y_m$ are the Bessel functions of the first and second kind, respectively, $k$ is the wavenumber of the charge density, $\rho^\prime$ is the radial coordinate, and $\phi$ is the angular coordinate.The boundary conditions for this problem depend on the form of the charge density. For the case of a charge density with a constant component, such as $\sigma(\phi) = k \sin 5\phi$, the boundary conditions are that the potential is continuous at the surface of the cylinder, which means that $\lim_{\rho^\prime \rightarrow R} V(\rho^\prime, \phi) = V_0$. Here, $V_0$ is the potential at the surface of the cylinder.The charge density does not tell you about the radial dependence of the potential, but it does constrain the coefficients of the Bessel functions. The coefficients $A_m$ and $B_m$ can be found by integrating the charge density over the surface of the cylinder and applying the boundary condition. Hope this helps.