Electromagnetism Cheatsheet

Cheatsheet Content

### Coulomb's Law - **Force between two point charges:** $$F = k \frac{|q_1 q_2|}{r^2}$$ where $k = \frac{1}{4\pi\epsilon_0} \approx 8.987 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$ - **Electric field due to a point charge:** $$\vec{E} = k \frac{q}{r^2} \hat{r}$$ - **Permittivity of free space:** $\epsilon_0 \approx 8.854 \times 10^{-12} \text{ C}^2/(\text{N}\cdot\text{m}^2)$ ### Electric Field and Potential - **Electric field definition:** $\vec{E} = \frac{\vec{F}}{q_0}$ - **Superposition principle:** $\vec{E}_{total} = \sum_i \vec{E}_i$ - **Electric flux:** $\Phi_E = \int \vec{E} \cdot d\vec{A}$ - **Gauss's Law:** $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$ - **Electric potential difference:** $\Delta V = -\int_A^B \vec{E} \cdot d\vec{l}$ - **Electric potential due to a point charge:** $V = k \frac{q}{r}$ - **Relationship between E and V:** $\vec{E} = -\nabla V$ (in 1D, $E = -\frac{dV}{dx}$) - **Electric potential energy:** $U = qV$ or $U = k \frac{q_1 q_2}{r}$ ### Capacitance - **Definition:** $C = \frac{Q}{V}$ - **Parallel plate capacitor:** $C = \frac{\epsilon_0 A}{d}$ - **Energy stored in a capacitor:** $U = \frac{1}{2}QV = \frac{1}{2}CV^2 = \frac{Q^2}{2C}$ - **Capacitors in series:** $\frac{1}{C_{eq}} = \sum_i \frac{1}{C_i}$ - **Capacitors in parallel:** $C_{eq} = \sum_i C_i$ - **Dielectrics:** $C = \kappa C_0$, where $\kappa$ is the dielectric constant ### Current & Resistance - **Electric current:** $I = \frac{dQ}{dt} = nqAV_d$ - **Current density:** $\vec{J} = nq\vec{V}_d$ - **Ohm's Law (microscopic):** $\vec{J} = \sigma \vec{E}$ or $\vec{E} = \rho \vec{J}$ - **Ohm's Law (macroscopic):** $V = IR$ - **Resistance:** $R = \rho \frac{L}{A}$ - **Resistivity temperature dependence:** $\rho = \rho_0 [1 + \alpha(T - T_0)]$ - **Power dissipated by a resistor:** $P = IV = I^2R = \frac{V^2}{R}$ - **Resistors in series:** $R_{eq} = \sum_i R_i$ - **Resistors in parallel:** $\frac{1}{R_{eq}} = \sum_i \frac{1}{R_i}$ ### DC Circuits - **Kirchhoff's Junction Rule:** $\sum I_{in} = \sum I_{out}$ (conservation of charge) - **Kirchhoff's Loop Rule:** $\sum \Delta V = 0$ (conservation of energy) - **RC Circuits (charging):** $Q(t) = Q_{max}(1 - e^{-t/\tau})$, $V_C(t) = V_{max}(1 - e^{-t/\tau})$ - **RC Circuits (discharging):** $Q(t) = Q_0 e^{-t/\tau}$, $V_C(t) = V_0 e^{-t/\tau}$ - **Time constant:** $\tau = RC$ ### Magnetic Fields - **Magnetic force on a moving charge:** $\vec{F}_B = q(\vec{v} \times \vec{B})$ - $|\vec{F}_B| = qvB\sin\theta$ - **Magnetic force on a current-carrying wire:** $\vec{F}_B = I(\vec{L} \times \vec{B})$ - $|\vec{F}_B| = ILB\sin\theta$ - **Torque on a current loop:** $\vec{\tau} = \vec{\mu} \times \vec{B}$ - **Magnetic dipole moment:** $\vec{\mu} = NIA\hat{n}$ - **Biot-Savart Law:** $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}$ - **Permeability of free space:** $\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$ - **Magnetic field of a long straight wire:** $B = \frac{\mu_0 I}{2\pi r}$ - **Magnetic field at center of a current loop:** $B = \frac{\mu_0 I}{2R}$ - **Magnetic field inside a solenoid:** $B = \mu_0 n I$ ($n$ = turns per unit length) - **Ampere's Law:** $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$ ### Faraday's Law & Inductance - **Magnetic flux:** $\Phi_B = \int \vec{B} \cdot d\vec{A}$ - **Faraday's Law of Induction:** $\mathcal{E} = -\frac{d\Phi_B}{dt}$ - **Motional EMF:** $\mathcal{E} = BLv$ (for straight conductor moving perpendicular to B) - **Lenz's Law:** Induced current/EMF opposes the change in magnetic flux that caused it. - **Self-inductance:** $L = \frac{N\Phi_B}{I}$ - **Inductance of a solenoid:** $L = \mu_0 n^2 A l$ - **Induced EMF in an inductor:** $\mathcal{E}_L = -L\frac{dI}{dt}$ - **Energy stored in an inductor:** $U = \frac{1}{2}LI^2$ - **Inductors in series:** $L_{eq} = \sum_i L_i$ - **Inductors in parallel:** $\frac{1}{L_{eq}} = \sum_i \frac{1}{L_i}$ ### RL Circuits - **Current buildup (charging):** $I(t) = I_{max}(1 - e^{-t/\tau})$ - **Current decay (discharging):** $I(t) = I_0 e^{-t/\tau}$ - **Time constant:** $\tau = \frac{L}{R}$ ### AC Circuits (RLC Series) - **Voltage and current (phasor form):** $V(t) = V_{max}\sin(\omega t)$, $I(t) = I_{max}\sin(\omega t - \phi)$ - **Angular frequency:** $\omega = 2\pi f$ - **Capacitive reactance:** $X_C = \frac{1}{\omega C}$ - **Inductive reactance:** $X_L = \omega L$ - **Impedance:** $Z = \sqrt{R^2 + (X_L - X_C)^2}$ - **Phase angle:** $\tan\phi = \frac{X_L - X_C}{R}$ - **Resonance frequency:** $\omega_0 = \frac{1}{\sqrt{LC}}$ - **Average power:** $P_{avg} = I_{rms}V_{rms}\cos\phi = I_{rms}^2 R$ - **RMS values:** $V_{rms} = \frac{V_{max}}{\sqrt{2}}$, $I_{rms} = \frac{I_{max}}{\sqrt{2}}$ ### Maxwell's Equations 1. **Gauss's Law for Electricity:** $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$ - Relates electric field to charge distribution. 2. **Gauss's Law for Magnetism:** $\oint \vec{B} \cdot d\vec{A} = 0$ - Magnetic monopoles do not exist; magnetic field lines form closed loops. 3. **Faraday's Law of Induction:** $\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}$ - A changing magnetic flux induces an electric field (and EMF). 4. **Ampere-Maxwell Law:** $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$ - Electric currents and changing electric flux (displacement current) generate magnetic fields. - **Speed of light in vacuum:** $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \approx 3 \times 10^8 \text{ m/s}$ ### Electromagnetic Waves - **Wave equation:** $\frac{\partial^2 E}{\partial x^2} = \mu_0 \epsilon_0 \frac{\partial^2 E}{\partial t^2}$ - **Speed of EM waves:** $c = \frac{E}{B}$ - **Poynting vector (energy flux):** $\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B})$ - Direction of wave propagation, magnitude is power per unit area. - **Intensity:** $I = S_{avg} = \frac{E_{max}B_{max}}{2\mu_0} = \frac{E_{max}^2}{2\mu_0 c} = \frac{cB_{max}^2}{2\mu_0}$ - **Radiation pressure:** $P_{rad} = \frac{I}{c}$ (perfect absorption), $P_{rad} = \frac{2I}{c}$ (perfect reflection) ### Fundamental Constants - **Elementary charge:** $e \approx 1.602 \times 10^{-19} \text{ C}$ - **Electron mass:** $m_e \approx 9.109 \times 10^{-31} \text{ kg}$ - **Proton mass:** $m_p \approx 1.672 \times 10^{-27} \text{ kg}$ - **Speed of light:** $c \approx 2.998 \times 10^8 \text{ m/s}$ - **Coulomb's constant:** $k = \frac{1}{4\pi\epsilon_0} \approx 8.987 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$ - **Permittivity of free space:** $\epsilon_0 \approx 8.854 \times 10^{-12} \text{ C}^2/(\text{N}\cdot\text{m}^2)$ - **Permeability of free space:** $\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$