A duct shaped like a pie-slice has two flat radial walls at and

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 Cylindrical geometry, edges, and what a field does to a dipole

Problem

A duct shaped like a pie-slice has two flat radial walls at

\phi = 0

and

\phi = \beta

, with

\beta = \pi/3

, both earthed. It runs from the axis

s = 0

out to a curved metal wall at

s = b

, on which the potential is held fixed at

V(b,\phi,z) = V_0\sin(\pi z/L)

for all

\phi

[0,\beta]

. The duct is capped at

z = 0

and

z = L

by earthed flat plates. There is no charge inside the duct. ((a)) State which of the three separable coordinate systems matches this boundary, naming the coordinate surfaces that coincide with each piece of the boundary. ((b)) Set up, but do not solve for the coefficients, the general series form of

V(s,\phi,z)

inside the duct that already satisfies the two earthed flat radial walls and the two earthed end caps. ((c)) Decide which family or families of Bessel-type functions belong in the radial factor, and say which of

J_m

Y_m

I_m

K_m

must be discarded by the conditions already imposed, and why.

Browse the problem library