Maxwell's Equations

Maxwell's equations are a set of four partial differential equations that form the foundation of classical electromagnetism, optics, and electric circuits. They describe how electric and magnetic fields are generated by charges, currents, and by changes of the fields.

1. Gauss's Law

Describes how electric charges produce electric fields. Electric flux through a closed surface is proportional to the enclosed charge.

• Differential form: $\mathrm{\nabla }\cdot \mathbf{E}=\frac{\rho }{{\epsilon }_0}$
• Integral form: ${\oint }_S\mathbf{E}\cdot d\mathbf{A}=\frac{Q_{encl}}{{\epsilon }_0}$

2. Gauss's Law for Magnetism

States that there are no "magnetic charges" (monopoles). Magnetic field lines are continuous loops; they have no beginning or end.

• Differential form: $\mathrm{\nabla }\cdot \mathbf{B}=0$
• Integral form: ${\oint }_S\mathbf{B}\cdot d\mathbf{A}=0$

3. Faraday's Law of Induction

Describes how a time-varying magnetic field creates an electric field. This is the principle behind electric generators and transformers.

• Differential form: $\mathrm{\nabla }\times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$
• Integral form: ${\oint }_C\mathbf{E}\cdot d\mathbf{l}=-\frac{d{\mathrm{\Phi }}_B}{dt}$

4. Ampère's Circuital Law (with Maxwell's Addition)

States that magnetic fields can be generated by electric currents and by changing electric fields (displacement current).

• Differential form: $\mathrm{\nabla }\times \mathbf{B}={\mu }_0(\mathbf{J}+{\epsilon }_0\frac{\partial \mathbf{E}}{\partial t})$
• Integral form: ${\oint }_C\mathbf{B}\cdot d\mathbf{l}={\mu }_0(I_{encl}+{\epsilon }_0\frac{d{\mathrm{\Phi }}_E}{dt})$

Constants & Variables