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
$$\begin{equation} V(\rho^\prime, \phi) = \sum_{m = 0}^{\infty} \, [A_m J_m (k\rho^\prime) + B_m N_m (k\rho^\prime)] [C_m \sin m\phi + D_m \cos m\phi] \; ? \end{equation}$$

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?

 

[Nate810]

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.

 

[quicknote]

Hello,

Let me start by addressing your first question. The potential inside and outside of the cylinder can be represented by the sum of Bessel functions (J_m and N_m) and trigonometric functions (sin m\phi and cos m\phi), as shown in the equation you provided. However, it is important to note that this is the general solution and may not apply to every case. The exact solution for this problem would depend on the specific boundary conditions and geometry of the system.

Moving on to your second question, the boundary conditions for this problem would depend on the specific setup and what is being asked for. In general, you would need to know the potential at a specific point or region in order to solve for the potential at other points. The charge density does not directly give information about the radial dependence of the potential, but it can be used to determine the boundary conditions. For example, if the charge density is known at the surface of the cylinder, this can be used to determine the potential at that point and then used as a boundary condition for the rest of the system.

I hope this helps clarify your questions. Remember, the exact solution for this problem would depend on the specific setup and boundary conditions, so it is important to carefully consider these factors when solving for the potential.