PHYS 208

Formulas for Physics 208: Electromagnetism and optics.

Electrostatics
(1)
\begin{align} \epsilon_0 = 8.854 \times 10^{-12} \; \frac{\text{C}^2}{\text{N} \cdot \text{m}^2} \end{align}

Permittivity of Free Space

(2)
\begin{align} e^+ = 1.602 \times 10^{-19} \; \text{C} \end{align}

Charge of a positron

(3)
\begin{align} \vec{F} = \frac{1}{4\pi\epsilon_0} \frac{q_1q_2}{r^2} \hat{r} \end{align}

Magnitude of Force between two charges

(4)
\begin{align} k = \frac{1}{4\pi\epsilon_0} = 9.0\times 10^9 \; \frac{N \cdot m^2}{C^2} \end{align}

Simplification Constant

(5)
\begin{align} \vec{E} = \frac{kQ}{r^2} \hat{r} \end{align}

Electric field vector of a charge

(6)
\begin{align} \vec{F}_0 = q_0 \vec{E} \end{align}

Force on a charge in electric field

(7)
\begin{align} \lambda = \frac{Q}{r} \text{;} \;\; \sigma = \frac{Q}{A} \text{;} \;\; \rho = \frac{Q}{V} \end{align}

Charge densities: 1D, 2D, and 3D respectively

(8)
\begin{align} dQ = \lambda \; dr \text{;} \;\; dQ = \sigma \; dA \text{;} \;\; dQ = \rho \; dV \end{align}

Charge of an infinitesimal unit

(9)
\begin{equation} p = qd \end{equation}

Electric dipole moment

(10)
\begin{align} \vec{p} = q\vec{d} \end{align}

Electric dipole moment as a vector; direction is from negative to positive charge

(11)
\begin{align} \tau = pE \sin \phi \end{align}

Torque on an electric dipole

(12)
\begin{align} \vec{\tau} = \vec{p} \times \vec{E} \end{align}

Torque on an electric dipole

(13)
\begin{align} dW = \tau \; d\phi \end{align}

Work over infinitesimal displacement

(14)
\begin{align} W = \int_{\phi_1}^{\phi_2} \tau(\phi) \; d\phi \end{align}

Work over a finite displacement

(15)
\begin{align} U = -pE \cos \phi \end{align}

Potential Energy

(16)
\begin{align} U = -\vec{p} \cdot \vec{E} \end{align}

Potential Energy

(17)
\begin{align} \Phi_E = \frac{Q}{\epsilon_0} \end{align}

Gauss's Law

(18)
\begin{align} \Phi_E = \oint \vec{E} \cdot d\vec{A} \end{align}

General Gauss's Law

Electric Potential
(19)
\begin{equation} U = q_0Ey \end{equation}

Potential Energy of a charge in a uniform field

(20)
\begin{align} U = \frac{q_0}{4\pi\epsilon_0}\sum_i\frac{q_i}{r_i} \end{align}

Potential energy with multiple charges

(21)
\begin{equation} W_{ab} = U_a - U_b \end{equation}

Work done by the electric field

(22)
\begin{align} U_{tot} = \frac{1}{4\pi\epsilon_0}\sum_{i<j}\frac{q_iq_j}{r_{ij}} \end{align}

Total Potential Energy of a distribution

(23)
\begin{align} V = \frac{U}{q_0} \end{align}

Voltage of a point charge

(24)
\begin{align} V = \frac{1}{4\pi\epsilon_0}\int \frac{1}{r} \; dq \end{align}

Voltage due to a distribution of charge

(25)
\begin{align} V_a - V_b = \frac{W_{ab}}{q_0} \end{align}

Potential from Work

(26)
\begin{align} V_a - V_b = \int_a^b \vec{E} \cdot d\vec{l} \end{align}

Potential from the Electric Field

(27)
\begin{align} C = \frac{Q}{V_{ab}} \end{align}

Capacitance

(28)
\begin{align} C = \frac{\epsilon A}{d} \end{align}

Capacitance of parallel plates separated by a vacuum

(29)
\begin{align} \frac{1}{C_{eq}} = \sum_i \frac{1}{C_i} \end{align}

Capacitors in Series

(30)
\begin{align} C_{eq} = \sum_i C_i \end{align}

Capacitance in Parallel

(31)
\begin{align} U = \frac{Q^2}{2C} = \frac{1}{2}CV^2 = \frac{1}{2}QV \end{align}

Potential Energy on a Capacitor

(32)
\begin{align} u = \frac{1}{2}\epsilon E^2 \end{align}

Energy density in Electric Field

(33)
\begin{align} \epsilon = K\epsilon_0 \end{align}

Permittivity of a Dielectric

(34)
\begin{align} \sigma_i = \sigma\left(1 - \frac{1}{K}\right) \end{align}

Induced Charge Density on Dielectric

Electromotion
(35)
\begin{equation} I = n|q|v_dA \end{equation}

Current

(36)
\begin{align} I = \frac{dQ}{dt} \end{align}

Current

(37)
\begin{align} J = \frac{I}{A} \end{align}

Current Density

(38)
\begin{align} \vec{J} = nq\vec{v}_d \end{align}

Current Density Vector

(39)
\begin{align} \vec{E} = \rho \vec{J} \end{align}

Electric field in a conducting material; $\rho$ is resistivity

(40)
\begin{align} \rho = \rho_0\left(1+\alpha (T - T_0)\right) \end{align}

Resistivity as a function of Temperature

(41)
\begin{align} R = \frac{\rho L}{A} \end{align}

Resistance from Resistivity

(42)
\begin{equation} V = IR \end{equation}

Ohm's Law

(43)
\begin{align} V_{ab} = \mathcal{E} - Ir \end{align}

Terminal Voltage of emf

(44)
\begin{equation} P = V_{ab}I \end{equation}

Power over a Voltage Drop

(45)
\begin{equation} P = I^2R \end{equation}

Power dissipated by a Resistor

(46)
\begin{align} P_\text{out} = \mathcal{E}I - I^2r \end{align}

Power output of a source

(47)
\begin{align} P_\text{in} = \mathcal{E}I + I^2r \end{align}

Power input to a source by another

(48)
\begin{align} R_\text{eq} = \sum_i R_i \end{align}

Resistors in Series

(49)
\begin{align} \frac{1}{R_\text{eq}} = \sum_i \frac{1}{R_i} \end{align}

Resistors in Parallel

(50)
\begin{align} \sum I = 0 \end{align}

Kirchhoff's Junction Rule

(51)
\begin{align} \sum V = 0 \end{align}

Kirchhoff's Loop Rule

(52)
\begin{align} q(t) = Q_\text{f}\left(1-e^{-t/\tau}\right) \end{align}

Charging a Capacitor in RC Circuit

(53)
\begin{align} i(t) = I_0e^{-t/\tau} \end{align}

Current in RC Circuit

(54)
\begin{align} Q_\text{f} = C\mathcal{E} \end{align}

Final Charge in RC Circuit

(55)
\begin{align} I_0 = \frac{\mathcal{E}}{R} \end{align}

Initial current in RC Circuit

(56)
\begin{align} \tau = RC \end{align}

Time Constant in RC Circuit

Magnetism
(57)
\begin{align} F = |q|vB_\perp \end{align}

Magnitude of Force on charge in Magnetic Field

(58)
\begin{align} \vec{F} = q\vec{v} \times \vec{B} \end{align}

Force on moving charge in Magnetic Field

(59)
\begin{align} \Phi_B = 0 \end{align}

Magnetic Flux through Closed Surface

(60)
\begin{align} \Phi_B = \iint_S \vec{B} \cdot \mathrm{d}\vec{A} \end{align}

Magnetic Flux through any surface

(61)
\begin{align} R = \frac{mv_\perp}{qB} \end{align}

Radius of circular path of a particle in a uniform magnetic field

(62)
\begin{align} f = \frac{\omega}{2\pi} = \frac{qB}{2\pi m} \end{align}

Cyclotron Frequency

(63)
\begin{align} \vec{F} = I\vec{l} \times \vec{B} \end{align}

Force on straight conducting wire

(64)
\begin{align} \vec{\mu} = NI\vec{A} \end{align}

Magnetic Moment

(65)
\begin{align} \tau = \mu B\sin \phi \end{align}

Torque on a conducting loop

(66)
\begin{align} \vec{\tau} = \vec{\mu} \times \vec{B} \end{align}

Torque on conducting loop

(67)
\begin{align} \mu_0 = 4\pi \times 10^{-7} \; \frac{\text{T} \cdot \text{m}}{\text{A}} \end{align}

Permeability of Free Space

(68)
\begin{align} c^2 = \frac{1}{\epsilon_0\mu_0} \end{align}

Electromagnetic Relationship

(69)
\begin{align} \vec{B} = \frac{\mu_0}{4\pi}\frac{q\vec{v} \times \hat{r}}{r^2} \end{align}

Magnetic Field induced by moving charge

(70)
\begin{align} \vec{B} = \frac{\mu_0}{4\pi} \int \frac{I\mathrm{d}\vec{l} \times \hat{r}}{r^2} \end{align}

Law of Biot and Sarvat

(71)
\begin{align} B = \frac{\mu_0I}{2\pi r} \end{align}

Field around very long wires

(72)
\begin{align} B = \frac{\mu_0NI}{2a} \end{align}

Field at center of solenoid

(73)
\begin{align} \mu_0I = \oint \vec{B} \cdot \mathrm{d}\vec{l} \end{align}

Amphere's Law

(74)
\begin{align} \mu = \mu_0K_m \end{align}

Permeability of Magnetic Material

Inductance
(75)
\begin{align} \mathcal{E} = -N\frac{\mathrm{d}\Phi_B}{\mathrm{d}t} \end{align}

Induced emf by change in flux

(76)
\begin{align} \mathcal{E} = \oint (\vec{v} \times \vec{B}) \cdot \mathrm{d}\vec{l} \end{align}

General induced emf

(77)
\begin{align} \oint \vec{E} \cdot \mathrm{d}\vec{l} = -\frac{\mathrm{d}\Phi_B}{\mathrm{d}t} \end{align}

Faraday's Law

(78)
\begin{align} \mathcal{E}_1 = -M\frac{\mathrm{d}i_2}{\mathrm{d}t} \end{align}

Induced emf by mutual inductance

(79)
\begin{align} M = \frac{N_2\Phi_2}{i_1} \end{align}

Mutual inductance

(80)
\begin{align} \mathcal{E} = -L\frac{\mathrm{d}i}{\mathrm{d}t} \end{align}

Induced emf by self inductance

(81)
\begin{align} L = \frac{N\Phi}{i} \end{align}

Self Inductance

(82)
\begin{align} U = \frac{1}{2}LI^2 \end{align}

Potential Energy of an Inductor

(83)
\begin{align} u = \frac{B^2}{2\mu} \end{align}

Energy Density of Inductor

(84)
\begin{align} i(t) = I_\text{f}\left(1-e^{-t/\tau}\right) \end{align}

Current in an RL Circuit

(85)
\begin{align} \frac{\mathrm{d}i}{\mathrm{d}t} = \frac{\mathcal{E}}{L}e^{-t/\tau} \end{align}

Current change in RL Circuit

(86)
\begin{align} I_\text{f} = \frac{\mathcal{E}}{R} \end{align}

Final current in RL circuit

(87)
\begin{align} \tau = \frac{L}{R} \end{align}

RL Time Constant

(88)
\begin{align} q(t) = Q\cos(\omega t + \phi) \end{align}

Charge in an LC Circuit

(89)
\begin{align} i(t) = -\omega Q\sin(\omega t + \phi) \end{align}

Current in an LC Circuit

(90)
\begin{align} \omega = \sqrt{\frac{1}{LC}} \end{align}

Oscillation frequency of LC Circuit

(91)
\begin{align} \frac{\mathrm{d}^2q}{\mathrm{d}t^2} + \frac{q}{LC} = 0 \end{align}

Equation for an LC Circuit

Electromagnetism
(92)
\begin{align} \vec{E} = \hat{j}E\cos(kx-\omega t) \end{align}

Electric propagation

(93)
\begin{align} \vec{B} = \hat{k}B\cos(kx-\omega t) \end{align}

Magnetic propogation

(94)
\begin{equation} E = cB \end{equation}

Electromagnetic Proportionality

(95)
\begin{align} c = \lambda f \end{align}

Wavelength and Frequency

(96)
\begin{align} k = \frac{2\pi}{\lambda} \end{align}

Wave Number (in m-1)

(97)
\begin{align} \omega = 2\pi f \end{align}

Angular frequency (in Hz)

(98)
\begin{align} n = \sqrt{KK_m} \end{align}

Refractive Index

(99)
\begin{align} v = \frac{c}{n} \end{align}

Speed of Light in a medium

(100)
\begin{align} u = \epsilon_0E^2 \end{align}

Energy density of electromagnetic wave

(101)
\begin{align} \vec{S} = \frac{1}{\mu_0}\vec{E} \times \vec{B} \end{align}

Poynting Vector

(102)
\begin{align} P = \oint \vec{S} \cdot \mathrm{d}\vec{A} \end{align}

Power of a wave

(103)
\begin{align} I = \frac{1}{2}S_\text{max} = \frac{1}{2}\epsilon_0cE^2 \end{align}

Intensity of electromagnetic wave

(104)
\begin{align} p_\text{rad} = \frac{I}{c} \end{align}

Radiation pressure on absorbent material

(105)
\begin{align} p_\text{rad} = \frac{2I}{c} \end{align}

Radiation pressure on reflective material

(106)
\begin{align} E_y = -2E\sin kx \sin \omega t \end{align}

Electric standing wave

(107)
\begin{align} B_z = -2B\cos kx \cos \omega t \end{align}

Magnetic standing wave

(108)
\begin{align} \omega = \sqrt{\frac{1}{LC}} \end{align}

Oscillation frequency of LC Circuit

(109)
\begin{align} \frac{\mathrm{d}^2q}{\mathrm{d}t^2} + \frac{q}{LC} = 0 \end{align}

Equation for an LC Circuit

(110)
\begin{align} \theta_r = \theta_a \end{align}

Angle of Reflection

(111)
\begin{align} n_a\sin\theta_a = n_b\sin\theta_b \end{align}

Snell's Law

(112)
\begin{align} I = I_\text{max}\cos^2\phi \end{align}

Malus's Law

(113)
\begin{align} \tan \theta_p = \frac{n_b}{n_a} \end{align}

Brewster's Law

Optics
(114)
\begin{align} r_2-r_1 = \left\{m\lambda \quad |\quad m\in\mathbb{Z}\right\} \end{align}

Constructive Interference

(115)
\begin{align} r_2-r_1 = \left\{\left(m+\frac{1}{2}\right)\lambda \quad |\quad m\in\mathbb{Z}\right\} \end{align}

Destructive Interference

(116)
\begin{align} y_m = \left\{\frac{Rm\lambda}{d} \quad |\quad m\in\mathbb{Z}\right\} \end{align}

Two-slit experiment band distance approximation

(117)
\begin{align} \phi = k(r_2-r_1) \end{align}

Two-wave interference phase difference

(118)
\begin{align} E_P = 2E\left|\cos\frac{\phi}{2}\right| \end{align}

Two-wave interference amplitude

(119)
\begin{align} I = I_0 \cos^2\frac{\phi}{2} \end{align}

Two-wave interference intensity

(120)
\begin{align} n = \sqrt{KK_m} \end{align}

Refractive Index

(121)
\begin{align} v = \frac{c}{n} \end{align}

Speed of Light in a medium

(122)
\begin{align} u = \epsilon_0E^2 \end{align}

Energy density of electromagnetic wave

(123)
\begin{align} \vec{S} = \frac{1}{\mu_0}\vec{E} \times \vec{B} \end{align}

Poynting Vector

(124)
\begin{align} P = \oint \vec{S} \cdot \mathrm{d}\vec{A} \end{align}

Power of a wave

(125)
\begin{align} I = \frac{1}{2}S_\text{max} = \frac{1}{2}\epsilon_0cE^2 \end{align}

Intensity of electromagnetic wave

(126)
\begin{align} p_\text{rad} = \frac{I}{c} \end{align}

Radiation pressure on absorbent material

(127)
\begin{align} p_\text{rad} = \frac{2I}{c} \end{align}

Radiation pressure on reflective material

(128)
\begin{align} E_y = -2E\sin kx \sin \omega t \end{align}

Electric standing wave

(129)
\begin{align} B_z = -2B\cos kx \cos \omega t \end{align}

Magnetic standing wave

(130)
\begin{align} \omega = \sqrt{\frac{1}{LC}} \end{align}

Oscillation frequency of LC Circuit

(131)
\begin{align} \frac{\mathrm{d}^2q}{\mathrm{d}t^2} + \frac{q}{LC} = 0 \end{align}

Equation for an LC Circuit

(132)
\begin{align} \theta_r = \theta_a \end{align}

Angle of Reflection

(133)
\begin{align} n_a\sin\theta_a = n_b\sin\theta_b \end{align}

Snell's Law

(134)
\begin{align} I = I_\text{max}\cos^2\phi \end{align}

Malus's Law

(135)
\begin{align} \tan \theta_p = \frac{n_b}{n_a} \end{align}

Brewster's Law

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