03Class 12 Physics

Current Electricity formulas

Ohm's law, resistor networks and Kirchhoff's rules for Current Electricity. Redraw messy circuits before writing anything down.

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Current and resistance

Current: $\displaystyle I = \frac{dq}{dt} = neAv_d$
Drift velocity: $\displaystyle v_d = \frac{eE\tau}{m}$
Ohm's law: $\displaystyle V = IR$
Resistance: $\displaystyle R = \frac{\rho L}{A}$
Conductivity: $\displaystyle \sigma = \frac{1}{\rho} = \frac{ne^2\tau}{m}$
Current density: $\displaystyle J = \frac{I}{A} = \sigma E$
Temperature dependence: $\displaystyle \rho_T = \rho_0[1 + \alpha(T - T_0)]$

Terminal voltage: $\displaystyle V = \varepsilon - Ir$
Cells in series: $\displaystyle \varepsilon_{\text{eq}} = \varepsilon_1 + \varepsilon_2, \quad r_{\text{eq}} = r_1 + r_2$
Cells in parallel: $\displaystyle \frac{\varepsilon_{\text{eq}}}{r_{\text{eq}}} = \frac{\varepsilon_1}{r_1} + \frac{\varepsilon_2}{r_2}, \quad \frac{1}{r_{\text{eq}}} = \frac{1}{r_1} + \frac{1}{r_2}$
Kirchhoff's junction rule: $\displaystyle \sum I = 0$
Kirchhoff's loop rule: $\displaystyle \sum V = 0$
Balanced Wheatstone bridge: $\displaystyle \frac{P}{Q} = \frac{R}{S}$

Getting the sign wrong in the loop rule. Fix a direction first, and follow it around the whole loop.
Assuming a wire stretched to twice its length has twice the resistance. Its area also shrinks, so R becomes four times as large.
Using P = I²R and V²/R interchangeably without checking what's constant. Series circuits share current, and parallel circuits share voltage.

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