Two flat, grounded metal plates lie along and , with , and extend without limit in…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Conductors, capacitance and boundary-value electrostatics

Mathematical Methods for Physics, Ch Separation of variables in Cartesian, cylindrical and spherical geometry

Problem

Two flat, grounded metal plates lie along

y=0

and

y=b

, with

b = 0.5\,\mathrm{m}

, and extend without limit in the

x

direction (

x \ge 0

). In the charge-free gap between them the electrostatic potential

u(x,y)

obeys Laplace's equation

\partial^2u/\partial x^2 + \partial^2u/\partial y^2 = 0

, with both plates at zero potential,

u(x,0) = u(x,b) = 0

for all

x \ge 0

. ((a)) Substitute

u(x,y) = X(x)Y(y)

into Laplace's equation and separate it, writing the two ordinary equations in terms of a single separation constant

\lambda

, choosing the sign of

\lambda

so that the equation for

Y

can satisfy

Y(0)=Y(b)=0

without

Y

being identically zero. ((b)) Find the allowed values

\lambda_1

\lambda_2

\lambda_3

(the three smallest) and the corresponding

Y_n(y)

. ((c)) State which of the two solutions of the equation for

X_n(x)

must be kept for the potential to stay bounded as

x \to \infty

Browse the problem library