A spherical plasma-diagnostic electrode of radius is isolated with no applied field

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

The Boundary-Value Problem, Ch Separation of variables in spherical geometry

Problem

A spherical plasma-diagnostic electrode of radius

a = 0.12\,\mathrm{m}

is isolated with no applied field. Its sunward hemisphere (

0 \le \theta < \pi/2

) is held at

V_0 = 40\,\mathrm{V}

and its shadow hemisphere (

\pi/2 < \theta \le \pi

) is held at

0\,\mathrm{V}

. Take

V\to 0

r\to\infty

and no other charge nearby. ((a)) Using the orthogonality integral

A_\ell = \dfrac{2\ell+1}{2a^\ell}\displaystyle\int_0^\pi V(a,\theta)P_\ell(\cos\theta)\sin\theta\,d\theta

, find the coefficients

A_0

and

A_1

of the interior series. ((b)) Using only the

\ell=0

and

\ell=1

terms found in (a) (matched to the exterior series), estimate the potential on the axis at

r=3a

\theta=0

. ((c)) Find

A_3

from the same integral, and the size of the

\ell=3

exterior term at that point. Is the two-term estimate of (b) good there, and would two terms estimate the surface field at the pole equally well?

Browse the problem library