Electrostatics — Electric Field, Capacitance & Dielectric Breakdown
Gauss's law, Laplace and Poisson equations, FEM for electrostatics, capacitance matrix extraction, and high-voltage insulation design.
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Quick Explainer
What is the Laplace equation and why does it govern electrostatics in free space?
In a region with no free charges, electric potential satisfies Laplace's equation: nabla^2 phi = 0. It comes from combining Gauss's law (zero charge density) with E = -nabla*phi. Laplace solutions have no local maxima or minima inside the domain — fields flow smoothly from boundary to boundary. With charges, Poisson's equation nabla^2 phi = -rho/epsilon applies.
How is a capacitance matrix extracted using FEM?
Set conductor i to unit voltage, all others to zero, solve Laplace, integrate surface charge density on each conductor. C_ij is charge induced on conductor j per unit voltage on i. Repeat for each conductor. This N-solve procedure gives the full capacitance matrix used in circuit simulation tools.