Classical Electrodynamics · Problem 3.17

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

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

Problem

The Dirichlet Green function for the unbounded space between the planes at z=0z = 0 and z=Lz = L allows discussion of a point charge or a distribution of charge between parallel conducting planes held at zero potential. (a) Using cylindrical coordinates show that one form of the Green function is G(x,x′)=4L∑n=1∞∑m=−∞∞eim(ϕ−ϕ′)sin⁡(nπzL)sin⁡(nπz′L)Im(nπLρ<)Km(nπLρ>)G(\mathbf{x}, \mathbf{x}') = \frac{4}{L} \sum_{n=1}^{\infty} \sum_{m=-\infty}^{\infty} e^{im(\phi-\phi')} \sin\left(\frac{n\pi z}{L}\right) \sin\left(\frac{n\pi z'}{L}\right) I_m\left(\frac{n\pi}{L}\rho_<\right) K_m\left(\frac{n\pi}{L}\rho_>\right) (b) Show that an alternative form of the Green function is G(x,x′)=2∑m=−∞∞∫0∞dk eim(ϕ−ϕ′)Jm(kρ)Jm(kρ′)sinh⁡(kz<)sinh⁡[k(L−z>)]sinh⁡(kL)G(\mathbf{x}, \mathbf{x}') = 2 \sum_{m=-\infty}^{\infty} \int_0^{\infty} dk\, e^{im(\phi-\phi')} J_m(k\rho) J_m(k\rho') \frac{\sinh(kz_<)\sinh[k(L-z_>)]}{\sinh(kL)}

Related theory

Browse the problem library