Classical Electrodynamics · Problem 3.19

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

Physics — E&M · green-functions-parallel-conducting-planes

Problem

Consider a point charge qq between two infinite parallel conducting planes held at zero potential. Let the planes be located at z=0z = 0 and z=Lz = L in a cylindrical coordinate system, with the charge on the zz axis at z=z0z = z_0, 0<z0<L0 < z_0 < L. Use Green's reciprocation theorem of Problem 1.12 with problem 3.18 as the comparison problem. (a) Show that the amount of induced charge on the plate at z=Lz = L inside a circle of radius aa whose center is on the zz axis is given by QL(a)=−qVΦ(z0,0)Q_L(a) = -\frac{q}{V}\Phi(z_0, 0) where Φ(z0,0)\Phi(z_0, 0) is the potential of Problem 3.18 evaluated at z=z0z = z_0, ρ=0\rho = 0. Find the total charge induced on the upper plate. Compare with the solution (in method and answer) of Problem 1.13. (b) Show that the induced charge density on the upper plate can be written as σ(ρ)=−q2π∫0∞dk sinh⁡(kz0)sinh⁡(kL) kJ0(kρ)\sigma(\rho) = -\frac{q}{2\pi} \int_0^{\infty} dk\, \frac{\sinh(kz_0)}{\sinh(kL)}\, kJ_0(k\rho) This integral can be expressed (see, e.g., Gradshteyn and Ryzhik, p. 728, formula 6.666) as an infinite series involving the modified Bessel functions K0(nπρ/L)K_0(n\pi\rho/L), showing that at large radial distances the induced charge density falls off as (ρ)−1/2e−πρ/L(\rho)^{-1/2} e^{-\pi\rho/L}. (c) Show that the charge density at ρ=0\rho = 0 can be written as the series σ(0)=−q2πL2∑n>0, odd[(n−z0/L)−2−(n+z0/L)−2]\sigma(0) = -\frac{q}{2\pi L^2} \sum_{n>0,\, \text{odd}} \left[(n - z_0/L)^{-2} - (n + z_0/L)^{-2}\right]

Related theory

Browse the problem library