A cubical region of side in vacuum holds a uniform charge density

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

The Boundary-Value Problem, Ch The field as a differential equation: Poisson, Laplace and the well-posed problem

Problem

A cubical region of side L=10 cmL=10\,\mathrm{cm} in vacuum holds a uniform charge density ρ0=5.0×10−9 C m−3\rho_0=5.0\times10^{-9}\,\mathrm{C\,m^{-3}}. We try to pose a problem for the potential inside it with Neumann data alone: on five of the six faces the outward normal derivative is given as ∂V/∂n=−8.0 V m−1\partial V/\partial n=-8.0\,\mathrm{V\,m^{-1}} at every point, and on the sixth face ∂V/∂n\partial V/\partial n is to take one uniform value gg. ((a)) Find the only value of gg for which a solution can exist. ((b)) Someone proposes g=−8.0 V m−1g=-8.0\,\mathrm{V\,m^{-1}} as well, so that all six faces match. What can be said about that problem? ((c)) With gg as in (a), does the data fix VV completely? If not, name one further piece of information that would.

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