Poisson's Equation

${\nabla }^{2}V=-\frac{\rho }{{\epsilon }_0}$

• The laplacian of the voltage
• Given by the definition of voltage and the differential form of Gauss’s law
• The case where the RHS is 0 is called Laplace’s equation

${\nabla }^{2}V+{\partial }_t(\nabla \cdot \vec{A})=-\frac{\rho }{{\epsilon }_0}$

• Time-dependent version