Electrodynamics

Potential energy (charges); Electrostatic force from surface charge:

$$w = \frac{1}{2} \int_v \rho(x) \Phi(x) d^3x,$$ (32)

$$\mathbf{F} = \int \frac{\sigma^2}{2\epsilon_0} d\mathbf{a}$$ (33)

Poynting vector:

$$\mathbf{s} = \frac{1}{\mu_0}\big(\mathbf{E}\times\mathbf{B}\big)$$ (34)

$$\frac{\partial u}{\partial t} = - \nabla \cdot \mathbf{s}$$ (35)

$$\partial_{\mu}s^{\mu} = 0$$ (36)

$$\partial_{\mu} J^{\mu} = 0$$ (37)

$$\mathbf{\mathcal{P}}_{em} = \frac{1}{c^2} \mathbf{s}$$ (38)

Helmholtz Equation + Dirchlet Equation:

$$(\nabla^2 + k^2) \psi = -\delta(r-r_0)$$ (39)

$$\int dV [ \psi (\nabla^2 + k^2) \phi - \phi (\nabla^2 + k^2) \psi ]$$ (40)

$$\int d\mathbf{s} \cdot(\psi \nabla\phi - \phi \nabla \psi)$$

Free current density + conductivity:

$$\mathbf{J} = qN \mathbf{v}_{drift} = \frac{n\mathbf{I}}{a}$$ (41)

$$\sigma = \frac{J}{\epsilon}$$ (42)

Surface charge induced + Spherical method of images:

$$\sigma = - \epsilon_0 \frac{\partial \phi}{\partial}{\eta}$$ (43)

$$q' = \frac{-a}{y} q$$ (44)

$$\frac{q'}{y'} = \frac{-q}{a}$$ (45)

$$\Phi = \sum_i \phi_i$$ (46)

$$\textbf{TODO: Image Needed Here}$$ (47)

Brewster Angle:

$$\theta = \tan^{-1} n$$ (48) Total internal reflection (P-waves)

Energy in external magnetic field:

$$U = -\mathbf{mu} \cdot \mathbf{B}$$ (49)

$$U = -\mu \mathbf{\sigma} \cdot \mathbf{B}$$ (50)

$$\sigma$$

Magnetic Forces:

$$\mathbf{F}_m = \int dq (\mathbf{v} \times \mathbf{B}) = \int d \mathbf{l }\lambda (\mathbf{v}\times\mathbf{B})$$ (51)

$$\lambda \mathbf{V} = \mathbf{I};$$ (52)

$$\boxed{\int (\mathbf{I}\times\mathbf{B})\cdot d\mathbf{l} = \mathbf{L} }$$ (53)

$$\boxed{\mathbf{F} = \int (\mathbf{J} \times \mathbf{B}) d \tau}$$ (54)

Kirchoff approximation:

$$\psi(r') = \frac{- i k}{2\pi} \frac{e^{ik(r_0 + r')}}{r_0r'}\int_{\sigma}dxdy e^{i\mathbf{q}\cdot\mathbf{x}}$$ (55)

$$I = \left\vert\psi(r')\right\vert^2$$ (56)

This is the Fourier transform of the aperture shape. Note the similarity to the Born approximation...

$$\sim \frac{e^{ikr}}{r^2}$$ (57)

Maxwell-Faraday equation: Faraday's law of induction.

$$\nabla \times \mathbf{E} = \frac{\partial^2\mathbf{B}}{\partial t^2} \cdot d\mathbf{a}$$ (58)

$$\oint \nabla \times \mathbf{E} \cdot d\mathbf{a} = - \int \frac{\partial^2\mathbf{B}}{\partial t^2}\cdot d\mathbf{a}$$ (59)

$$\oint \mathbf{E} \cdot d\mathbf{l} = - \frac{d}{dt} \iint \mathbf{B} \cot d\mathbf{a}$$ (60)

The voltage induced in a closed circuit is proportional to the rate of change of magnetic flux enclosed.

Biot-Savart law:

$$\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int \frac{\mathbf{I} \times \hat{r} }{r^2} dl'$$ (61)

$$= \frac{\mu_0}{4\pi} I \int \frac{d\mathbf{l}\times \hat{r}}{r^2}$$

Polarization, electric dipole moment:

$$\mathbf{P} = \int r' \rho(r')d\tau' = \sum_{i=1} q_i\mathbf{r}_i'$$ (62)

$$V_{dipole}(r) = \frac{1}{4\pi\epsilon_0}\frac{\mathbf{P}\cdot\hat{r}}{r^2}$$ (63)

$$\mathbf{P} = \mathcal{N}<\mathbf{P}>$$ (64)

$$\mathbf{E}_{dip}(\mathbf{r}) = \frac{1}{4\pi\epsilon_0r^3}\big[3(\mathbf{p}\cdot\hat{r})\hat{r} - \mathbf{p}\big]$$ (65)

$$\sigma_b = \mathbf{p}\dot\hat{n}$$ (66)

$$\rho_b = -\nabla\cdot\mathbf{p}$$ (67)

$$\rho_T=\nabla\cdot(\epsilon_0\mathbf{E}-\mathbf{P})$$ (68)

$$\epsilon_0 \mathbf{E}-\mathbf{P}=\mathbf{D}$$ (69)

Maxwell/Ampere Law (differential + integral forms):

$$\nabla\times\mathbf{B} = \mu_0 \mathbf{J} + \frac{1}{c^2}\frac{\partial^2\mathbf{E}}{\partial t^2}$$ (70)

$$\iint \nabla\times\mathbf{B} \cdot d\mathbf{a} = \mu_0 \iint \mat... ...\cdot d\mathbf{a} + \frac{1}{c^2} \frac{d}{dt}\iint \mathbf{E}\cdot d\mathbf{a}$$ (71)

$$\oint \mathbf{B}\cdot d\mathbf{l} = \mu_0 \iint \mathbf{J} \cdot d\mathbf{a} + \frac{1}{c^2} \frac{d}{dt} \iint \mathbf{E} \cdot d\mathbf{a}$$ (72)

Magnetic field induced around closed loop $\propto$ electric current + displacement current enclosed.

Gauss' Law (Diff+Int):

$$\frac{Q}{\epsilon_0} = \oiint \mathbf{E} \cdot d \mathbf{a} = \iiint \nabla\cdot\mathbf{E}\cdot d\mathbf{V}$$ (73)

$$\nabla\cdot\mathbf{E} = \frac{\rho}{\epsilon_0}$$ (74)

Electric flux leaving volume $\propto$ charge inside.

Magnetic Dipole:

• vector potential
  $\mathbf{A} = \frac{\mu_0}{4\pi}\frac{\mathbf{m}\times\mathbf{r}}{r^3}$
• $\mathbf{B}$ field
  $\mathbf{B} = \frac{\mu_0}{4\pi}\bigg(\frac{3\mathbf{r}(\mathbf{m}\cdot\mathbf{r... ...\frac{\mu_0}{4\pi}\frac{1}{r^3}(3(\mathbf{m}\cdot\hat{r})\hat{r} - \mathbf{m})$
• torque
  $\mathbf{m}\times\mathbf{B} = \mathbf{\Gamma}$
• force
  $\mathbf{F} = \nabla(\mathbf{m}\cdot\mathbf{B})$
• magnetization
  $\mathbf{m} = I\int d\mathbf{a} = I \mathbf{a}$
  $\mathbf{M} = \mathcal{N}\mathbf{m}$
• auxiliary field $\mathbf{B} = \mu_0(\mathbf{H} + \mathbf{M})$

Maxwell Stress Tensor:

$$T_{ij} = \epsilon_0\big[E_iE_j - \frac{1}{2}\delta_{ij}E^2 \big] +$$ (75)

$$\frac{1}{\mu_0}\big[B_iB_j - \frac{1}{2} \delta{ij}B^2 \big]$$

$$\frac{\partial^2\mathbf{\mathcal{P}}_{mech}+\mathbf{\mathcal{P}}_{em}}{\partial t^2} = \nabla\cdot\mathbf{T}$$ (76)

$$\mathbf{\mathcal{P}}_{em} = \mu_0\epsilon_0 \mathbf{s}$$ (77)

Energy in a capacitor/inductor:

$$dV = \frac{dq}{C} \rightarrow qdV = dU = \frac{qdq}{C}$$ (78)

$$U = \frac{q^2}{2C} = \frac{1}{2} vq = \boxed{\frac{1}{2} cv^2 = U}$$ (79)

$$(V=L\dot{I})I; IV = IL\dot{I}=P$$ (80)

$$P = LI\frac{dI}{dt}$$ (81)

$$\int dPdE = \int L I dI$$ (82)

$$\boxed{U=\frac{1}{2} L I^2}$$ (83)

Energy of dialectrics, paramagnets:

$$w = \frac{\epsilon_0}{2} \epsilon_r E^2 d\tau = \frac{1}{2}\int \mathbf{D}\mathbf{E}d\tau$$ (84)

$$w = \frac{1}{2\mu_0} \int \mathbf{H}\mathbf{B}d\tau$$ (85)

$$u_{em} = \frac{1}{2} (\mathbf{E}\cdot\mathbf{D} + \mathbf{B}\cdot\mathbf{H})$$ (86)

$\textbf{Electric and Magnetic field relations:}$

$$\mathbf{k}\times\mathbf{E} = \mu \omega \mathbf{H}$$ (87)

$$\mathbf{k}\times\mathbf{H} = -\epsilon \omega \mathbf{E}$$ (88)

$$\mathbf{k} \cdot \mathbf{E} = 0$$ (89)

$$\mathbf{k} \cdot \mathbf{H} = 0$$ (90)

$$\mathbf{B} = \mp \frac{i\mathbf{E}}{c}$$ (91)

$$E_1 = Ecos\theta$$ (92)

$$I_1 = Icos^2\theta$$

$$\mathbf{k}\cdot \mathbf{r} = \mathbf{k''} \cdot \mathbf{r}$$

TODO: PICTURE