Maxwell's Equations

Maxwell’s Equations are perhaps one of the most robust equations in physics, describing all electromagnetic phenomenon. It holds completely true in anything from classical to quantum physics.

Maxwell’s Correction to Ampere’s Law

Maxwell corrected an inconsistency in Ampere’s Law that prevented the laws of electromagnetism from being tied together.

According to Ampere’s Law, ${\oint }_C\vec{B}\cdot \overrightarrow{ds}={\mu }_0{I}_{encl}$ However, this does not hold true in cases where the current is changing. Maxwell decided to add an additional component to Ampere’s law, called the displacement current, which is added to the real current: ${\oint }_C\vec{B}\cdot \overrightarrow{ds}={\mu }_0(I_{encl}+I_d)$ where $I_d$ is defined as: $I_d={\mathcal{E}}_0\frac{d{\Phi }_E}{dt}$ In this equation, ${\mathcal{E}}_0$ represents the permittivity of free space, and ${\Phi }_E$ is the electric flux, defined as: ${\Phi }_E={\iint }_{SurfaceS}\vec{E}\cdot d\vec{A}$ The displacement current is produced by a changing electric field or current, which produces a magnetic field. The fully modified Ampere’s Law is given by: ${\oint }_C\vec{B}\cdot d\vec{s}={\mu }_0{I}_{encl}+{\mathcal{E}}_0{\mu }_0\frac{d{\Phi }_E}{dt}$

All 4 Maxwell Equations

Gauss’s Law $\oint \vec{E}\cdot d\vec{A}=\frac{Q_{in}}{{\mathcal{E}}_0}$ Gauss’s Law for Magnetism $\oint \vec{B}\cdot d\vec{A}=0$ Faraday’s Law $\oint \vec{E}\cdot d\vec{s}=-\frac{d{\Phi }_m}{dt}$ Ampere-Maxwell Law $\vec{B}\cdot d\vec{s}={\mu }_{0}I+{\mathcal{E}}_0{\mu }_0\frac{d{\Phi }_E}{dt}$

Lorentz Force

Once you have calculated all of these fields, you can use the Lorentz Force Equation: $\vec{F}=q\vec{E}+q\vec{v}\times \vec{B}$

Differential Forms of Maxwell’s Equations

Gauss’ Law $\vec{\nabla }\cdot \vec{E}=\frac{\rho }{{ϵ}_0}$ Faraday’s Law $\vec{\nabla }\times \vec{E}=-\frac{\partial \vec{B}}{\partial t}$ Gauss’ Law $\vec{\nabla }\cdot \vec{B}=0$ Ampere-Maxwell Law $\vec{\nabla }\times \vec{B}={\mu }_0\vec{J}+{\mu }_0{ϵ}_0\frac{\partial \vec{E}}{\partial t}$