Maxwell's Equations

Maxwell’s Equations in Differential Form:

\[\nabla \cdot \mathbf{E}=\frac{\rho}{\epsilon_{0}}\] \[\nabla \cdot \mathbf{B}=0\] \[\nabla \times \mathbf{E}=\color{lightgray}{-\frac{\partial \mathbf{B}}{\partial t}}\] \[\nabla \times \mathbf{B}=\mu_{0} \mathbf{J}\color{lightgray}{+\mu_{0} \epsilon_{0} \frac{\partial \mathbf{E}}{\partial t}}\]

Maxwell’s Equations in Integral Form:

\[\oint \mathbf{E} \cdot d \mathbf{a}=\frac{Q_{\mathrm{enc}}}{\epsilon_{0}}\] \[\oint \mathbf{B} \cdot d \mathbf{a}=0\] \[\oint \mathbf{E} \cdot d \mathbf{l}=\color{lightgray}{-\int \frac{\partial \mathbf{B}}{\partial t} \cdot d \mathbf{a}}\] \[\oint \mathbf{B} \cdot d \mathbf{l}=\mu_{0} I_{\mathrm{enc}}\color{lightgray}{+\mu_{0} \epsilon_{0} \int \frac{\partial \mathbf{E}}{\partial t} \cdot d \mathbf{a}}\]