Electromagnetism

• Coulomb's Law: $\overrightarrow{F_{21}} = -\overrightarrow{F_{12}} = k\frac{q_1q_2}{r_{12}^3}\overrightarrow{r_{12}}$
• Electric Field: $\overrightarrow{E} \equiv lim_{q\rightarrow 0}\frac{\vec F}{q} = \frac{1}{4\pi\epsilon}*\frac{q}{r^3}\overrightarrow{r_{12}}$
• Dipole moment(dipole field): $p\equiv q*2a$

• Dipole Field: $\vec E = \frac{1}{4\pi\epsilon}*\frac{1}{r^3}[3(\vec p\cdot \vec e_r)\vec e_r - \vec p]$
• Dipole(independent of origin of net charge): $\vec p=q\vec r$
• Potential Energy of Dipole: $U = -\vec p*\vec E + C$
• flux: $\phi_E\equiv \oint \vec E*d\vec S$
• Guass Law: $\phi_E = \frac{q}{\epsilon_0}$
• Conductor E: $E = \frac{\rho}{\epsilon_0}$ conductor is equipotential
• Electric Potential: $V=\frac{1}{4\pi\epsilon_0}*\frac{q}{r} = -\int \vec E$
• Electric Potential Energy: $\Delta U = \Delta W_{ext} = \Delta(\frac{1}{4\pi \epsilon}*\frac{q_1q_2}{r})$
• Self Energry: $W=\int_0^Q Udq$
• Electrostatic Energy: $W=\frac{1}{2}\int\rho UdV$
• Capacitance: $C\equiv \frac{q}{V}$
• $C_p = \sum_i C_i$ $\frac{1}{C_s}=\sum_i\frac{1}{C_i}$
• $W = \frac{1}{2}\frac{q^2}{C} = \frac{1}{2}C(\Delta V)^2 = \int udV$
• Current Density: $\vec J = \sigma_C \vec E$
• $Q = It$
• $\rho = \frac{1}{\sigma}$
• Current: $I = JA =\frac{V}{R}$
• Resistance: $R = \frac{dV}{dI} = \rho\frac{l}{A}$
• Polarization: $\vec P = n<\vec p>$
• polar dielectric: $P = \chi\epsilon_0E$
• susceptibility: $\chi$
• non-polor dielectric: $\vec p=2\alpha E_0$
• Dielectric constant: $\epsilon = \epsilon_0\epsilon_r$, $C_d\equiv\frac{\epsilon A}{d}$
• Electric Displacement: $\vec D = \epsilon_r\epsilon_0\vec E = \epsilon \vec E$
• Constitutive relation: $\vec D = \epsilon_0 \vec E + \vec P = (1+\chi)\epsilon_0\vec E$, $\epsilon_r = 1 + \chi$
• Magnetic Induction: $\vec F = q\vec v*\vec B$
• Ampere's Law: $\vec F_{21} = \frac{\mu_0}{4\pi}\oint\oint\frac{I_2dl_2\times(I_1d\vec l_1\times \vec r_{12})}{r_{12}^3} = \oint I_2d\vec l_2\times\vec B_2$
• Biot-Savart Law: $\vec B = \frac{\mu_0}{4\pi}\oint\frac{Id\vec l\times\vec r}{r^3}$
• Magnetic Force for wires: $\vec F = I\vec l\times \vec B$
• Magnetic Dipole: $\mu = I\vec S$
• Torque: $\vec M = \vec \mu\times\vec B$
• Electromagnetic Force: $\vec F=q(\vec E + \vec v\times\vec B)$
• Cyclotron frequency: $\vec\omega=-\frac{q}{m}\vec B$
• Cyclotron radius: $\rho = \frac{mv_\bot}{qB}$
• Faraday's Law: $\epsilon = -\frac{d}{dt}\Phi$
• Lenz's Law
• Motional Electronmotive Force(emf): moving $\epsilon = -BLv$, $F=ILB$ rotating $\epsilon = \frac{1}{2}BR^2\omega$
• Inductance: $LI=\Phi$
• $\epsilon = -L\frac{dI}{dt}$
• Magnetic Energy: $U_m = \frac{1}{2L}\Phi^2 = \frac{1}{2}LI^2$
• Self-inductance: $L = \frac{2\int u_md\tau}{I^2}$
• Mutual inductance: $L_{12}=L_{21}=M$
• Solenoid: $B=n\mu_0I$
• cross-winded solenoids: $M=\sqrt{L_1L_2}$
• coefficent of coupling: $M=k\sqrt{L_1L_2}$
• In series: $L_s=L_1+L_2(-2M)$
• In parallel: $L_p = \frac{L_1L_2}{L_1+L_2}$
• Transformation for v
  • $\vec E' = \gamma(\vec E+c\vec\beta\times\vec B)-(\gamma - 1)\vec e_\beta(\vec e_\beta\cdot \vec E)$
  • $\vec B' = \gamma(\vec B-\frac{1}{c}\vec\beta\times\vec E)-(\gamma-1)\vec e_\beta(\vec e_\beta\cdot\vec B)$
• Dipole moment: $\vec \mu_L = I\vec S=-\frac{e}{2m}\vec L=-\mu_B\frac{\vec L}{\hbar} = \gamma\vec L$
• Bohr magneton: $\mu_B=\frac{e\hbar}{2m_e}$
• Spin angular moment: $\vec S$, $(\frac{\vec S}{\hbar})_Z = \pm\frac{1}{2}$
• Spin magnetic moment: $\vec\mu_S = -2\mu_B\frac{\vec S}{\hbar}$
• Total magnetic moment: $\vec \mu=\vec\mu_L+\vec\mu_S=-\frac{e}{2m}(\vec L+2\vec S)$
• Magnetization: $\vec M = \frac{\vec mu}{V}$
• Precession: $\Omega_p = \frac{\mu B}{S} = |\gamma|B$
• Magnetic Field: $\vec H = \frac{\vec{B}-\mu_0\vec M}{\mu_0}$
• linear media and Magnetic Susceptibility: $\vec M = \chi \vec H$
• Permeability: $\vec B=\mu_0(1+\chi)\vec H\equiv\mu_0\mu_r\vec H\equiv\mu\vec H$
• Boundary Condition in Tangent direction: $(\vec H_1-\vec H_2)\vec e_t = J_\sigma$
• Potential energy: $u = -\vec M\cdot\vec B + C$
• Force: $\vec F = (\chi\vec H\cdot\nabla)(\mu\vec H)$
• Susceptibility: $\chi=\frac{1}{3}\frac{n\mu_1^2\mu_0}{k_BT}$
• Curie's Law: $M=C\frac{B_{eff}}{T}$
• Curie temperature: $T_C$, before is ferromagnetic phase
• Curie-Weiss's Law: $M=\frac{C}{T-T_C}H$
• permeability(ferromagnetic): $\vec B=\mu\vec H$
• Magnetic circuit: $\phi = B_1A_1$
• reluctance-magnetomotive force: $\phi(\sum\frac{l}{\mu A})=NI$
• Monopole: $\mu_0g=\oint\vec B\cdot d\vec S$
• Displacement current: $\frac{\partial D}{\partial T}$
• $\vec D=\epsilon_0\vec E + \vec P$
• $\vec H=\frac{1}{\mu_0}\vec B-\vec M$
• Maxwell's Equation
  • $\oint\vec D\cdot d\vec S=q$
  • $\oint\vec B\cdot d\vec S=0$
  • $\oint\vec E\cdot d\vec l=0-\frac{d}{dt}\oint\vec B\cdot d\vec S$
  • $\oint\vec H\cdot d\vec l=I+\frac{d}{dt}\phi\vec D\cdot\vec dS$
• Maxwell's Equation
  • $\nabla\cdot\vec D=\rho_f$
  • $\nabla\cdot\vec B=0$
  • $\nabla\times\vec E=-\frac{\partial\vec B}{\partial\vec t}$
  • $\nabla\times\vec H=\vec J_f + \frac{\partial\vec D}{\partial t}$
• Energy density: $u=\frac{1}{2}\vec E\cdot\vec D + \frac{1}{2}\vec B\cdot\vec H$
• Change rate of energy: $-\frac{dU}{dt} = \int\vec J\cdot\vec EdV$(Joule dissipation)$+\oint(\vec E\times\vec H)\cdot d\vec S$
• Poynting vector(radiation energy current density): $\vec S=\vec E\times\vec H$
• $\overline{S}=\frac{1}{2\mu_0}\vec E_mB_m$
• Power flow: $P=\int\vec S\cdot d\vec A$
• Wave equation(non-conduting media)
  • $\nabla^2\vec E-\mu\epsilon\frac{\partial^2}{\partial t^2}\vec E = 0$
  • $\nabla^2\vec B-\mu\epsilon\frac{\partial^2}{\partial t^2}\vec B = 0$
• Vaccum permeability: $\frac{\mu_0}{4\pi}\equiv 10^{-7}\frac{N}{A^2}$
• Field Vectors
  • $\vec E=\vec E(\vec k\cdot\vec r-ct)$
  • $\vec B=\vec B(\vec k\cdot\vec r-ct)$
• $\vec k\times\vec E=c\vec B$, $E=cB$
• $\vec E = \vec E(\vec r)e^{-i\omega t}$
• $\epsilon_0\mu_0c^2=1$