Potential energy (charges); Electrostatic force from surface charge:
![$\displaystyle w = \frac{1}{2} \int_v \rho(x) \Phi(x) d^3x,$](img75.svg) |
(32) |
![$\displaystyle \mathbf{F} = \int \frac{\sigma^2}{2\epsilon_0} d\mathbf{a}$](img76.svg) |
(33) |
Poynting vector:
Helmholtz Equation + Dirchlet Equation:
Free current density + conductivity:
![$\displaystyle \mathbf{J} = qN \mathbf{v}_{drift} = \frac{n\mathbf{I}}{a}$](img85.svg) |
(41) |
![$\displaystyle \sigma = \frac{J}{\epsilon}$](img86.svg) |
(42) |
Surface charge induced + Spherical method of images:
Brewster Angle:
![$\displaystyle \theta = \tan^{-1} n$](img92.svg) |
(48) |
Total internal reflection (P-waves) |
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Energy in external magnetic field:
![$\displaystyle U = -\mathbf{mu} \cdot \mathbf{B}$](img93.svg) |
(49) |
![$\displaystyle U = -\mu \mathbf{\sigma} \cdot \mathbf{B}$](img94.svg) |
(50) |
Where iis the Pauli vector |
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Magnetic Forces:
Kirchoff approximation:
![$\displaystyle \psi(r') = \frac{- i k}{2\pi} \frac{e^{ik(r_0 + r')}}{r_0r'}\int_{\sigma}dxdy e^{i\mathbf{q}\cdot\mathbf{x}}$](img100.svg) |
(55) |
![$\displaystyle I = \left\vert\psi(r')\right\vert^2$](img101.svg) |
(56) |
This is the Fourier transform of the aperture shape. Note the similarity to the Born approximation...
![$\displaystyle \sim \frac{e^{ikr}}{r^2}$](img102.svg) |
(57) |
Maxwell-Faraday equation:
Faraday's law of induction.
The voltage induced in a closed circuit is proportional to the rate of change of magnetic flux enclosed.
Biot-Savart law:
![$\displaystyle \mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int \frac{\mathbf{I} \times \hat{r} }{r^2} dl'$](img106.svg) |
(61) |
![$\displaystyle = \frac{\mu_0}{4\pi} I \int \frac{d\mathbf{l}\times \hat{r}}{r^2}$](img107.svg) |
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Polarization, electric dipole moment:
![$\displaystyle \mathbf{P} = \int r' \rho(r')d\tau' = \sum_{i=1} q_i\mathbf{r}_i'$](img108.svg) |
(62) |
![$\displaystyle V_{dipole}(r) = \frac{1}{4\pi\epsilon_0}\frac{\mathbf{P}\cdot\hat{r}}{r^2}$](img109.svg) |
(63) |
![$\displaystyle \mathbf{P} = \mathcal{N}<\mathbf{P}>$](img110.svg) |
(64) |
![$\displaystyle \mathbf{E}_{dip}(\mathbf{r}) = \frac{1}{4\pi\epsilon_0r^3}\big[3(\mathbf{p}\cdot\hat{r})\hat{r} - \mathbf{p}\big]$](img111.svg) |
(65) |
![$\displaystyle \sigma_b = \mathbf{p}\dot\hat{n}$](img112.svg) |
(66) |
![$\displaystyle \rho_b = -\nabla\cdot\mathbf{p}$](img113.svg) |
(67) |
![$\displaystyle \rho_T=\nabla\cdot(\epsilon_0\mathbf{E}-\mathbf{P})$](img114.svg) |
(68) |
![$\displaystyle \epsilon_0 \mathbf{E}-\mathbf{P}=\mathbf{D}$](img115.svg) |
(69) |
Maxwell/Ampere Law (differential + integral forms):
Magnetic field induced around closed loop
![$\propto$](img119.svg)
electric current + displacement current enclosed.
Gauss' Law (Diff+Int):
![$\displaystyle \frac{Q}{\epsilon_0} = \oiint \mathbf{E} \cdot d \mathbf{a} = \iiint \nabla\cdot\mathbf{E}\cdot d\mathbf{V}$](img120.svg) |
(73) |
![$\displaystyle \nabla\cdot\mathbf{E} = \frac{\rho}{\epsilon_0}$](img121.svg) |
(74) |
Electric flux leaving volume
![$\propto$](img119.svg)
charge inside.
Magnetic Dipole:
- vector potential
-
field
- torque
- force
- magnetization
- auxiliary field
Maxwell Stress Tensor:
Energy in a capacitor/inductor:
![$\displaystyle dV = \frac{dq}{C} \rightarrow qdV = dU = \frac{qdq}{C}$](img134.svg) |
(78) |
![$\displaystyle U = \frac{q^2}{2C} = \frac{1}{2} vq = \boxed{\frac{1}{2} cv^2 = U}$](img135.svg) |
(79) |
![$\displaystyle IV = IL\dot{I}=P$](img137.svg) |
(80) |
![$\displaystyle P = LI\frac{dI}{dt}$](img138.svg) |
(81) |
![$\displaystyle \int dPdE = \int L I dI$](img139.svg) |
(82) |
![$\displaystyle \boxed{U=\frac{1}{2} L I^2}$](img140.svg) |
(83) |
Energy of dialectrics, paramagnets:
![$\displaystyle \mathbf{k}\times\mathbf{E} = \mu \omega \mathbf{H}$](img145.svg) |
(87) |
![$\displaystyle \mathbf{k}\times\mathbf{H} = -\epsilon \omega \mathbf{E}$](img146.svg) |
(88) |
![$\displaystyle \mathbf{k} \cdot \mathbf{E} = 0$](img147.svg) |
(89) |
![$\displaystyle \mathbf{k} \cdot \mathbf{H} = 0$](img148.svg) |
(90) |
![$\displaystyle \mathbf{B} = \mp \frac{i\mathbf{E}}{c}$](img149.svg) |
(91) |
![$\displaystyle E_1 = Ecos\theta$](img150.svg) |
(92) |
![$\displaystyle I_1 = Icos^2\theta$](img151.svg) |
|
(boundaries) |
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TODO: PICTURE