Section 1: Microscopic Picture of Current

Electric current I = Q/t = neAvd (SI unit: ampere). The microscopic formula I = neAvd connects four measurable quantities: carrier density n, charge e, area A, and drift velocity vd. Drift velocity is extremely small (~10^{-4} m/s) because electrons undergo frequent collisions with lattice ions (every τ ~ 10^{-14} s), acquiring only a tiny net velocity. Random thermal speed (~10^{5} m/s) is much larger but produces no net current. Current flows "instantaneously" because the electric field — not the electrons themselves — propagates at ~c when a circuit is closed.

Current density J = I/A = nevd = σE is the vector (microscopic) form of Ohm's law. Conductivity σ = $ne^{2}$τ/m and resistivity ρ = 1/σ.

Section 2: Ohm's Law, Resistance, and Resistivity

Ohm's law V = IR applies to ohmic conductors at constant temperature. Resistance R = ρl/A: longer wire or narrower cross-section → higher R. Resistivity ρ is a material constant (independent of shape). Temperature dependence: R = $R_{0}$(1 + α$\Delta T$). For metals: α > 0 (R increases). For semiconductors: α < 0 (R decreases). For alloys (manganin, constantan): α ≈ 0 (R stable).

Section 3: Combinations and Power

Series: R_eq = ΣRᵢ; same current; voltage divides. Parallel: 1/R_eq = Σ(1/Rᵢ); same voltage; current divides. Resistor rules are opposite to capacitor rules. Power P = VI = $I^{2}$R = $V^{2}$/R (W). For series use $I^{2}$R; for parallel use $V^{2}$/R. For n identical resistors on same battery: P_parallel/P_series = $n^{2}$. Example: 3 resistors in parallel dissipate 9× more power than in series.

Section 4: EMF and Internal Resistance

A real cell: ε (EMF) with internal resistance r. Terminal voltage V = ε − Ir (discharging) or ε + Ir (charging). Maximum power to external load when R = r: P_max = ε^{2}/(4r). Efficiency at max power = 50%. Short circuit: R = 0, V = 0, I = ε/r.

Section 5: Kirchhoff's Laws

KCL (junction rule): ΣI_in = ΣI_out (conservation of charge). KVL (loop rule): Σ$\Delta V$ = 0 around any closed loop (conservation of energy). Sign convention: with current through resistor → −IR; against → +IR; cell − to + → +ε; cell + to − → −ε. Kirchhoff's laws form the foundation for all circuit analysis, including multi-loop and multi-source circuits.

Section 6: Wheatstone Bridge and Metre Bridge

Wheatstone bridge balance: P/Q = R/S → Ig = 0. At balance, galvanometer arm can be removed. Interchanging battery and galvanometer preserves balance (reciprocity). Metre bridge formula: R/S = l/(100 − l) where l is balance length in cm. If R and S are swapped, new balance length = 100 − l. Most accurate near l = 50 cm (symmetric bridge).

Section 7: Potentiometer

Working principle: V ∝ l for uniform wire with constant current. Potential gradient k = V_wire/L (V/m). At null point (G = 0): ε_test = kl — TRUE EMF measured because no current through test cell. EMF comparison: ε_{1}/ε_{2} = l_{1}/l_{2}. Internal resistance: r = R(l_{1} − l_{2})/l_{2}. Advantage over voltmeter: voltmeter always draws current → measures terminal voltage (< ε); potentiometer draws no current at null → measures true ε. Consistency check: V_wire must be > ε_test for balance to exist.