Mathematical Methods in the Physical Sciences · Problem 10.3

FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas

Mathematics — Methods · electrostatics-laplace

Problem

Assuming from electricity the equations ∇⋅D=ρ\mathbf{\nabla} \cdot \mathbf{D} = \rho, E=−∇V\mathbf{E} = -\nabla V, D=ϵE\mathbf{D} = \epsilon \mathbf{E} (ϵ=const.\epsilon = \text{const.}), show that in regions where the free charge density ρ\rho is zero, VV satisfies Laplace's equation.

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