Classical Electrodynamics · Problem 3.20

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

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

Problem

(a) From the results of Problem 3.17 or from first principles show that the potential at a point charge qq between two infinite parallel conducting planes held at zero potential can be written as Φ(z,ρ)=qπϵ0L∑n=1∞sin⁡(nπz0L)sin⁡(nπzL)K0(nπρL)\Phi(z, \rho) = \frac{q}{\pi\epsilon_0 L} \sum_{n=1}^{\infty} \sin\left(\frac{n\pi z_0}{L}\right) \sin\left(\frac{n\pi z}{L}\right) K_0\left(\frac{n\pi\rho}{L}\right) where the planes are at z=0z = 0 and z=Lz = L and the charge is on the zz axis at the point z=z0z = z_0. (b) Calculate the induced surface-charge densities σ0(ρ)\sigma_0(\rho) and σL(ρ)\sigma_L(\rho) on the lower and upper plates. The result for σL(ρ)\sigma_L(\rho) is σL(ρ)=qL2∑n=1∞(−1)nnsin⁡(nπz0L)K0(nπρL)\sigma_L(\rho) = \frac{q}{L^2} \sum_{n=1}^{\infty} (-1)^n n \sin\left(\frac{n\pi z_0}{L}\right) K_0\left(\frac{n\pi\rho}{L}\right) Discuss the connection of this expression with that of Problem 3.19b and 3.19c. (c) From the answer in part b, calculate the total charge QLQ_L on the plate at z=Lz = L. By summing the Fourier series or by other means of comparison, check your answer against the known expression of Problem 1.13 [C. Y. Fong and C. Kittel, Am. J. Phys. 35, 1091 (1967).]

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