(a) Two infinite earthed conducting half-planes meet along a straight edge and form a…
PHYSICS · Subjective · 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) Two infinite earthed conducting half-planes meet along a straight edge and form a wedge-shaped groove whose field region opens through β=2π/3 (the metal fills the remaining 4π/3). A long straight wire carrying line charge λ per unit length runs parallel to the edge, inside the wedge, along its bisector, at perpendicular distance s0 from the edge. Name the coordinate system in which every conducting surface is a level surface of one coordinate, write Laplace's equation for the potential in that system in the source-free parts of the wedge, state every boundary condition (at the two earthed planes, on the axis, and far from the wire), and write the general series form of the solution in the two regions s<s0 and s>s0 that already satisfies the boundary conditions and the requirement that V be continuous at s=s0 everywhere except at the wire itself. Do not evaluate any coefficient. (b) A closed metal can of radius a and height L has its flat top (z=L) and flat bottom (z=0) earthed. Its curved wall, at s=a, is held at V(ϕ,z)=V0cosϕsin(πz/L), where V0 is a constant potential and z is measured from the bottom. State which family of separated radial solutions must appear in the interior (Jm, Ym or the modified functions Im, Km), explain from the location of the non-zero data why the z-dependence is oscillatory here rather than exponential, and say which member of the radial family is removed by the requirement that V stay finite on the axis. Write the resulting form of V(s,ϕ,z), with its one undetermined constant expressed in terms of V0, a and L but not evaluated numerically.