Electric Current

$I = \frac{Q}{t} \quad [\text{A}] = \left[\frac{\text{C}}{\text{s}}\right]$

$I = neAv_d \quad [A] = [L^{-3}][C][L^2][LT^{-1}]$

Drift Velocity

$v_d = \frac{eE\tau}{m} = \frac{I}{neA} \quad [LT^{-1}] = \frac{[A]}{[L^{-3}][C][L^2]}$

Current Density

$J = \frac{I}{A} = nev_d = \sigma E \quad [AL^{-2}]$

Ohm's Law

$V = IR \quad [V] = [A][\Omega] = [ML^2T^{-3}A^{-1}]$

Resistance and Resistivity

$R = \frac{\rho l}{A} \quad [\Omega] = [ML^2T^{-3}A^{-2}]$

$\rho = \frac{RA}{l} = \frac{m}{ne^2\tau} \quad [\Omega \cdot m] = [ML^3T^{-3}A^{-2}]$

$\sigma = \frac{1}{\rho} = \frac{ne^2\tau}{m} \quad [S/m] = [M^{-1}L^{-3}T^3A^2]$

Temperature Dependence

$R = R_0(1 + \alpha \Delta T) \quad \alpha \text{ in } [K^{-1}]$

Power

$P = VI = I^2R = \frac{V^2}{R} \quad [W] = [ML^2T^{-3}]$

Electrical Energy

$W = VIt = I^2Rt = \frac{V^2t}{R} \quad [J] = [ML^2T^{-2}]$

EMF and Terminal Voltage

$V = \varepsilon - Ir \text{ (discharging)}; \quad V = \varepsilon + Ir \text{ (charging)}$

$[\varepsilon] = [ML^2T^{-3}A^{-1}] = [V]$

Maximum Power Transfer

$P_{max} = \frac{\varepsilon^2}{4r} \quad \text{when } R = r$

Wheatstone Bridge

$\frac{P}{Q} = \frac{R}{S} \quad \text{at balance (}I_g = 0\text{)}$

Metre Bridge

$\frac{R}{S} = \frac{l}{100 - l} \quad l \text{ in cm}$

Potentiometer — EMF Comparison

$\frac{\varepsilon_1}{\varepsilon_2} = \frac{l_1}{l_2}; \quad \varepsilon = kl \text{ where } k = \frac{V_{wire}}{L}$

Potentiometer — Internal Resistance

$r = R\left(\frac{l_1 - l_2}{l_2}\right)$

Kirchhoff's Sign Convention

$\text{KCL: } \sum I = 0; \quad \text{KVL: } \sum \Delta V = 0$

$\Delta V_{resistor(\text{with current})} = -IR; \quad \Delta V_{resistor(\text{against current})} = +IR$

$\Delta V_{cell(-\to+)} = +\varepsilon; \quad \Delta V_{cell(+\to-)} = -\varepsilon$