Current Electricity: Revision

1. Current & Drift Velocity

Current

I = \dfrac{\Delta Q}{\Delta t} = nqv_dA

Drift speeds are tiny (sub-mm/s); it’s the field that propagates fast, not the electrons.

Ohm's law

V = IR

Resistance from resistivity

R = \rho\dfrac{\ell}{A}

Metals: $\rho$ rises with T. Semiconductors: $\rho$ falls with T.

Series (same I, voltages add): $R_s=R_1+R_2+\cdots$ — always larger than the largest R.
Parallel (same V, currents add): $\dfrac{1}{R_p}=\dfrac{1}{R_1}+\dfrac{1}{R_2}+\cdots$ — always smaller than the smallest R.

Discharging

V = \varepsilon - Ir, \qquad I = \dfrac{\varepsilon}{R+r}

Charging (current forced in reverse)

V = \varepsilon + Ir

Power to external R

P = I^2R = \dfrac{\varepsilon^2 R}{(R+r)^2}

R shrinks toward 0: I rises to its max, but P falls back to 0 (R is now the limiting factor in I²R).
R grows toward ∞: I falls to 0, so P falls to 0 too.
R = r: power peaks, $P_{\max} = \varepsilon^2/4r$ — more current elsewhere does NOT mean more delivered power.

Balance condition (zero galvanometer deflection)

\dfrac{P}{Q} = \dfrac{R}{S}

At balance, no current flows through the galvanometer branch at all.

Full derivations and worked examples: detailed notes →