Poisson's Equation

#Physics

$\displaystyle \nabla ^{2} V=-\frac{\rho}{{\varepsilon}_{0}}$

• 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

$\displaystyle \nabla ^{2}V+\partial_{t}(\nabla \cdot \vec{A})=-\frac{\rho}{{\varepsilon}_{0}}$

• Time-dependent version