# Maxwell's Equations Motivation: Review from Electrostatics in explaining [[Poisson's Equation]] when used with the [[Shockley-Ramo Theorem]]. Note that these four equations are provided here in the differential form, as that was how I had learned them, and which are the ways in which they make most sense to me. ## Gauss's Law In brief, the divergence of an electric field is equal to a constant depending on the volume charge distribution and the medium it is in. $ \nabla \cdot \vec{E} = \rho/\epsilon_0 $ ## Gauss's Law for Magnetism In brief, the divergence of a magnetism field is zero. $ \nabla \cdot \vec{B} = 0 $ ## Faraday's Law of Induction In brief, the curl of an electric field is dependent on the change of the magnetic field over time. Also known as the Maxwell-Faraday equation. $ \nabla \times \vec{E} = -\frac{\partial B}{\partial t} $ ## Ampere's Law In brief, the curl of a magnetic field is dependent on the current density $\left( \vec{J} \right)$ times the magnetic constant $\mu_0 = 4\pi \times 10^{-7} H/m$. There is also a Maxwell addition term used to explain the displacement current issue. $\nabla \times \vec{B} = \mu_0 \left(\vec{J} + \varepsilon_0 \frac{\partial \vec{E}}{\partial t} \right) $