This problem has two independent halves

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 Separation of variables in spherical geometry

Problem

This problem has two independent halves, each pairing a spherical-geometry result with a method met earlier. Parts ((a)) to (c) find a dipole by images; parts (d) to (f) compare expansion coefficients across two geometries. A conducting sphere of radius

a = 2.0\,\mathrm{cm}

, centred at the origin and held at zero potential (grounded), sits between two point charges on the

z

axis:

q = +4.0\,\mathrm{nC}

z = +D

and

-q

z = -D

, with

D = 40\,\mathrm{cm}

. Because

D \gg a

, these two charges act on the sphere as an almost uniform applied field of strength

E_0 = 2q/(4\pi\varepsilon_0D^2)

, pointing along

-\hat{\mathbf{z}}

at the sphere. (a) Using the image charge for a grounded sphere (a charge

q

at distance

D

from the centre has an image

-qa/D

at distance

a^2/D

), write down the two image charges and their positions for the two given charges. ((b)) Find the dipole moment

p_{\rm im}

formed by these two image charges. ((c)) Compare

p_{\rm im}

with the value predicted from the polarisability of a conducting sphere,

p = 4\pi\varepsilon_0a^3E_0

, and confirm the two routes agree. ((d)) An infinite duct of rectangular cross-section

0 \le x \le w

(with

w = 2.0\,\mathrm{m}

) has one wall, at

y=0

, held at

V(x,0) = +V_0

for

0 < x < 1.0\,\mathrm{m}

and

V(x,0) = -V_0

for

1.0\,\mathrm{m} < x < 2.0\,\mathrm{m}

, with

V_0 = 60\,\mathrm{V}

, and

V=0

x=0

and

x=w

. Writing

V(x,0) = \sum_{n=1}^{\infty}c_n\sin(n\pi x/w)

, use the orthogonality integral over

x

to find the first two non-zero coefficients

c_n

. ((e)) A sphere of radius

a

is held at

V(\theta) = +V_0

on the upper hemisphere (

0 \le \theta < \pi/2

) and

-V_0

on the lower hemisphere, with the same

V_0 = 60\,\mathrm{V}

. Writing the interior potential as

V(r,\theta) = \sum_{\ell} A_\ell(r/a)^\ell P_\ell(\cos\theta)

, use the orthogonality integral over

\cos\theta

to find

A_1

and

A_3

. ((f)) Both boundary data sets have a discontinuous jump. State, from the sizes found in (d) and (e), how each coefficient sequence decays with its index, and what feature of the data this decay is tied to.

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