A thin, charge-free conducting plate occupies a bounded region of the plane, and its…
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Conductors, capacitance and boundary-value electrostatics
Conditions of Validity, Ch The three equations, their characteristics and what makes a problem well posed
Problem
A thin, charge-free conducting plate occupies a bounded region R of the plane, and its electrostatic potential u(x,y) obeys Laplace's equation ∇2u=0 throughout R.
((a)) State, in one sentence, what data must be prescribed on the boundary ∂R (a single closed curve) to determine u throughout R.
((b)) Someone argues: "Specifying u and its outward normal derivative ∂u/∂n along just one small arc of ∂R gives more information than specifying u alone on the whole boundary, so it should determine u at least as well, if not better." Using what the characteristic directions of Laplace's equation tell you about what kind of data it can accept, say whether this proposal is well posed, and explain the error in the reasoning.