An infinite plane slab of uniform volume charge density occupies the region with ;…

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

The Boundary-Value Problem, Ch The field as a differential equation: Poisson, Laplace and the well-posed problem

Problem

An infinite plane slab of uniform volume charge density

\rho_0 = 2.0\times10^{-6}\,\mathrm{C\,m^{-3}}

occupies the region

-t \le z \le t

with

t = 3.0\,\mathrm{mm}

; outside the slab there is vacuum. By symmetry

V

depends only on

z

. Inside the slab it is proposed that

V(z) = V_0 - \frac{\rho_0}{2\varepsilon_0}z^2, \qquad |z|\le t,

with

V_0

a constant, and outside the slab (for

z>t

) that

V(z) = A - Bz

for constants

A

B

. ((a)) Show that the proposed inside form satisfies Poisson's equation with source

\rho_0

. ((b)) Show that the proposed outside form satisfies Laplace's equation, and find the constant

B

by requiring

dV/dz

(hence the field) to be continuous at

z=t

, where there is no surface charge.

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