# Maxwell's Equations | | Integral form | Differential form | Phasor Form | | ------- | ----------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------ | ------------------------------------------------------- | | E-Gauss | $\iint_{S}E\cdot dA = \frac{Q_{\text{enc}}}{\epsilon_{0}}$ | ${} \vec{\nabla}\cdot \vec{E}=\frac{\rho}{\epsilon_{0}} {}$ | ${} \nabla\cdot \vec{E}=\frac{\rho}{\epsilon} {}$ | | M-Gauss | ${} \iint_{S}B\cdot dA=0 {}$ | ${} \vec{\nabla}\cdot \vec{B}=0 {}$ | ${} \nabla\cdot \vec{H}=0 {}$ | | Faraday | ${} \oint _{C}E\cdot dl = -\frac{ \partial \Phi_{B} }{ \partial t } {}$ | ${} \vec{\nabla}\times \vec{E}=-\frac{ \partial \vec{B} }{ \partial t } {}$ | ${} \nabla \times E = -j\omega \mu H {}$ | | Ampere | ${} \oint _{C}B\cdot dl= \mu_{0}\left( I_{\text{enc}}+\epsilon_{0} \frac{ \partial \Phi_{E} }{ \partial t } \right) {}$ | ${} \vec{\nabla} \times \vec{B} = \mu_{0}\left( \vec{J}+\epsilon_{0} \frac{ \partial \vec{E} }{ \partial t } \right) {}$ | ${} \nabla \times \vec{H}=j\omega\epsilon E+\vec{J} {}$ | $\vec{E} =$ Electric field $\left[ \text{V}/\text{m} \right]=[\text{N}/\text{C}]$ ${} \vec{B} =\mu_{0}\vec{H}= {}$ Magnetic field $\left[ \text{T} \right]$ $\vec{J}=$ Current density ${} [\text{A}/\text{m}^{2}] {}$ $\rho =$ total electric charge density $\epsilon_{0} =$ vacuum permittivity $= 8.854\times 10^{-12} \text{ F/m}$ $\mu_{0} =$ vacuum permeability $= 1.257\times 10^{-6} \text{ H/m}$ $\epsilon, \mu =$ material permittivity and permeability for optics ## Lorentz's Law $ \vec{f} = q(\vec{E}+\vec{v}\times \vec{B}) $ $\vec{f} =$ Force $[\text{N}]$ $\vec{v} =$ Charge Velocity $[\text{m/s}]$ $q =$ Charge $[\text{C}]$ Coulomb's Law (Guass+faraday for two point charges): $F = \frac{1}{4\pi \epsilon_{0}} \frac{q_{1}q_{2}}{r^{2}} \hat{r}$