Fundamental Constants

$\mu_0 = 4\pi \times 10^{-7} \text{ T·m/A} \quad [\mu_0] = [MLT^{-2}A^{-2}]$

Biot-Savart Law

$dB = \frac{\mu_0}{4\pi} \cdot \frac{I\, dl\sin\theta}{r^2} \quad [B] = [MT^{-2}A^{-1}] \quad \text{SI: Tesla (T)}$

Magnetic Field Configurations

$B_{\text{infinite wire}} = \frac{\mu_0 I}{2\pi d} \quad [MT^{-2}A^{-1}]$

$B_{\text{finite wire}} = \frac{\mu_0 I}{4\pi d}(\sin\alpha + \sin\beta) \quad [MT^{-2}A^{-1}]$

$B_{\text{center of loop}} = \frac{\mu_0 NI}{2R} \quad [MT^{-2}A^{-1}]$

$B_{\text{axis of loop}} = \frac{\mu_0 NI R^2}{2(R^2 + x^2)^{3/2}} \quad [MT^{-2}A^{-1}]$

$B_{\text{solenoid}} = \mu_0 nI \quad \text{where } n = N/L \quad [MT^{-2}A^{-1}]$

$B_{\text{toroid}} = \frac{\mu_0 NI}{2\pi r} \quad [MT^{-2}A^{-1}]$

Ampere's Circuital Law

$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}} \quad [MT^{-2}A^{-1} \cdot L = MLT^{-2}A^{-1}]$

Lorentz Force

$F = qvB\sin\theta \quad [F] = [MLT^{-2}] \quad \text{SI: Newton (N)}$

Circular Motion of Charged Particle

$r = \frac{mv}{qB} \quad [r] = [L] = \text{m}$

$T = \frac{2\pi m}{qB} \quad [T] = [T] = \text{s} \quad \text{(velocity-independent!)}$

$f = \frac{qB}{2\pi m} \quad [f] = [T^{-1}] = \text{Hz}$

Force on Conductor

$F = BIl\sin\theta \quad [MLT^{-2}]$

$\frac{F}{l} = \frac{\mu_0 I_1 I_2}{2\pi d} \quad [MT^{-2}] \quad \text{N/m}$

Torque on Current Loop

$\tau = NIAB\sin\theta = MB\sin\theta \quad [\tau] = [ML^2T^{-2}] \quad \text{N·m}$

$M = NIA \quad [M] = [AL^2] \quad \text{A·m}^2$

Galvanometer Conversions

$S_{\text{shunt}} = \frac{I_g G}{I - I_g} \quad \text{(Ammeter, parallel)}$

$R_{\text{series}} = \frac{V}{I_g} - G \quad \text{(Voltmeter, series)}$

Curie's Law

$\chi = \frac{C}{T} \quad \text{(Paramagnets only; C = Curie constant)}$

Relation between B, H, M

$B = \mu_0(H + M) = \mu_0 \mu_r H \quad \mu_r = 1 + \chi$