# 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 ${} = 8.854\times 10^{-12} \mathrm{A\cdot s /(V\cdot m)} {}$ $\mu_{0} =$ vacuum permeability ${} = 1.257\times 10^{-6} \mathrm{~V\cdot s /(A\cdot m)} {}$ $\epsilon, \mu =$ material permittivity and permeability ## 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): $\vec{f} = \frac{1}{4\pi \epsilon_{0}} \frac{q_{1}q_{2}}{r^{2}} \hat{r}$ ### Poynting Theorem Power conservation for electromagnetic fields in linear isotropic medium requires $ \nabla\cdot(\vec{E}\times \vec{H})+\frac{ \partial }{ \partial t } (w_{e}+w_{m})+\vec{E}\cdot \vec{J}=0 $ $\vec{J}=$ current density ${} w_{e} \equiv \frac{1}{2}\epsilon_{0}(1+\chi_{e})\lvert \vec{E} \rvert^{2}=\frac{1}{2}\epsilon \lvert \vec{E} \rvert^{2} = {}$ electric energy density $w_{m}= \frac{1}{2}\mu_{0}(1+\chi_{m})\lvert \vec{H} \rvert^{2}=\frac{1}{2}\mu \lvert \vec{H} \rvert^{2}$ = magnetic energy density Or integral form, $ \underbrace{ \oint_{S}\vec{E}\times \vec{H}\cdot da }_{ \text{power flowing out} }+\underbrace{ \frac{d}{dt}\int _{V}(w_{e}+w_{m}) \, dv }_{ \text{change in stored energy} }+\underbrace{ \int _{V}\vec{E}\cdot \vec{J} \, dv }_{ \text{power generated \\ by source } \vec{J} }=0 $ Power flows in the direction of the Poynting Vector $ \vec{S} \equiv \vec{E} \times \vec{H} $ For a uniform plane wave, this results in $\vec{S} =\hat{k} \sqrt{ \frac{\epsilon}{\mu} }\lvert \vec{E} \rvert^{2}$ The complex Poynting vector is similarly defined as $\vec{S}=\vec{E}\times \vec{H}^{\star}$ ## Constitutive Relations (linear, isotropic) $ \vec D=\epsilon_0\vec E+\vec P,\qquad \vec B=\mu_0\left(\vec H+\vec M\right) $$ \vec P=\epsilon_0\chi_e\vec E\ \Rightarrow\ \epsilon=\epsilon_0(1+\chi_e),\qquad n=\sqrt{\frac{\epsilon\mu}{\epsilon_0\mu_0}} $ Loss and dispersion enter as a complex, frequency-dependent $\epsilon(\omega)=\epsilon'-j\epsilon''$ with $\epsilon''>0$ (under $e^{+j\omega t}$). See [[dispersion]]. ## Complex (Phasor) Form and Time Averages Fields are $\mathrm{Re}\{\vec E(\vec r)e^{j\omega t}\}$, with **peak** phasors, and $\dfrac{\partial}{\partial t}\to j\omega$. For two quantities at the same frequency: $ \big\langle a(t)\,b(t)\big\rangle=\frac{1}{2}\mathrm{Re}\{A\,B^*\} $ $ \langle\vec S\rangle=\frac{1}{2}\mathrm{Re}\{\vec E\times\vec H^*\},\qquad \langle w_e\rangle=\frac{1}{4}\epsilon|\vec E|^2,\qquad \langle w_m\rangle=\frac{1}{4}\mu|\vec H|^2 $ The complex Poynting vector $\dfrac{1}{2}\vec E\times\vec H^*$ is **not** the phasor of the real Poynting vector (that oscillates at $2\omega$). Its real part is the time-averaged power flow. Its imaginary part is reactive (stored, sloshing) power. Plane wave intensity: $ I=\frac{|E_0|^2}{2\eta},\qquad \eta=\sqrt{\frac{\mu}{\epsilon}},\qquad \eta_0\approx377\ \Omega $ ## Complex Poynting Theorem From $\nabla\cdot(\vec E\times\vec H^*)=\vec H^*\cdot\nabla\times\vec E-\vec E\cdot\nabla\times\vec H^*$ and the phasor Maxwell equations: $ \nabla\cdot\left(\frac{1}{2}\vec E\times\vec H^*\right)=-\frac{1}{2}\vec E\cdot\vec J^*-2j\omega\left(\langle w_m\rangle-\langle w_e\rangle\right) $ - **Real part**: time-averaged power balance (power flowing out = power delivered by sources − dissipation). - **Imaginary part**: net reactive power, equal to $2\omega$ times (magnetic − electric) stored energy. - Consequences (Haus 1.3): - A lossless isotropic medium must have **real** $\epsilon$ and $\mu$. - In a traveling plane wave, $\langle w_e\rangle=\langle w_m\rangle$. - At resonance in a lossless cavity, the time-averaged electric and magnetic energies are equal. - In a **dispersive** medium, $\dfrac{1}{4}\epsilon|E|^2$ is *not* the stored energy (Haus §11.5). Real Poynting example (Haus 1.1): a plane wave $E_0\cos(\omega t-kz)\hat x$ has $w_e=w_m$ and $\vec S=\dfrac{E_0^2}{\eta}\cos^2(\omega t-kz)\,\hat z$. Check that $\nabla\cdot\vec S+\dfrac{\partial w}{\partial t}=0$.