Classical Electrodynamics · Problem 3.23

Multipoles, Electrostatics of Macroscopic Media, Dielectrics — John David Jackson

Physics — E&M · green-functions-cylindrical-coordinates

Problem

A point charge

q

is located at the point

(\rho', \phi', z')

inside a grounded cylindrical box defined by the surfaces

z = 0

z = L

\rho = a

. Show that the potential inside the box can be expressed in the following alternative forms:

\Phi(\mathbf{x}, \mathbf{x}') = \frac{q}{\pi\epsilon_0 a} \sum_{m=-\infty}^{\infty} \sum_{n=1}^{\infty} \frac{e^{im(\phi-\phi')} J_m\!\left(\frac{x_{mn}\rho}{a}\right) J_m\!\left(\frac{x_{mn}\rho'}{a}\right)}{x_{mn} J_{m+1}^2(x_{mn})\sinh\!\left(\frac{x_{mn}L}{a}\right)} \times \sinh\!\left[\frac{x_{mn}}{a} z_<\right] \sinh\!\left[\frac{x_{mn}}{a}(L - z_>)\right]

\phi(\mathbf{x}, \mathbf{x}') = \frac{q}{\pi\epsilon_0 L} \sum_{m=-\infty}^{\infty} \sum_{n=1}^{\infty} e^{im(\phi-\phi')} \sin\!\left(\frac{n\pi z}{L}\right) \sin\!\left(\frac{n\pi z'}{L}\right) \frac{I_m\!\left(\frac{n\pi\rho_<}{L}\right)}{I_m\!\left(\frac{n\pi a}{L}\right)} \times \left[I_m\!\left(\frac{n\pi a}{L}\right) K_m\!\left(\frac{n\pi\rho_>}{L}\right) - K_m\!\left(\frac{n\pi a}{L}\right) I_m\!\left(\frac{n\pi\rho_>}{L}\right)\right]

\Phi(\mathbf{x}, \mathbf{x}') = \frac{2q}{\pi\epsilon_0 L a^2} \sum_{m=-\infty}^{\infty} \sum_{k=1}^{\infty} \sum_{n=1}^{\infty} \frac{e^{im(\phi-\phi')} \sin\!\left(\frac{k\pi z}{L}\right) \sin\!\left(\frac{k\pi z'}{L}\right) J_m\!\left(\frac{x_{mn}\rho}{a}\right) J_m\!\left(\frac{x_{mn}\rho'}{a}\right)}{\left[\left(\frac{x_{mn}}{a}\right)^2 + \left(\frac{k\pi}{L}\right)^2\right] J_{m+1}^2(x_{mn})}

Discuss the relation of the last expansion (with its extra summation) to the other two.