Current Electricity

2.1 Core concepts

• Charges in motion constitute an electric current; in solid conductors the carriers are free electrons against a background of fixed positive ions (NCERT §3.1–3.3, p. 81–83).
• For steady current across an area in time t with net forward charge q, I = q/t; more generally I(t) = lim ΔQ/Δt as Δt→0. SI unit is the ampere; nerve currents are in µA while lightning carries tens of thousands of A (NCERT §3.2, p. 82).
• Without an electric field, electrons move thermally with random directions, so the average velocity is zero and no net current flows (NCERT §3.3, p. 82–83).
• Ohm's law (1828, G.S. Ohm): for many conductors, V ∝ I, written V = IR; R is the resistance of the conductor, SI unit ohm (Ω) (NCERT §3.4, p. 83).
• Geometry of R: doubling length doubles R; halving cross-section doubles R; hence R = ρl/A, where ρ (resistivity) depends only on material (NCERT §3.4, p. 83–84).
• Current density j = I/A (A m⁻²); in vector form E = ρj or equivalently j = σE, where σ = 1/ρ is conductivity (NCERT §3.4, p. 84–85).
• Drift velocity: averaging electron motion under field E with average collision time τ (relaxation time) gives vd = −(eE/m)τ, independent of time (NCERT §3.5, Eq. 3.17, p. 85–86).
• This yields the microscopic form of Ohm's law: j = (ne²τ/m) E and hence σ = ne²τ/m, ρ = m/(ne²τ) (NCERT §3.5, Eqs. 3.21–3.23, p. 86).
• Worked estimate (Example 3.1): in a Cu wire (A = 1.0 × 10⁻⁷ m², I = 1.5 A, n = 8.5 × 10²⁸ m⁻³), drift speed vd ≈ 1.1 mm s⁻¹ — about 10⁻⁵ times the thermal speed and 10⁻¹¹ times the speed of EM signal propagation (NCERT §3.5, Example 3.1, p. 86–87).
• Mobility μ = |vd|/E = eτ/m, SI unit m² V⁻¹ s⁻¹ (NCERT §3.5.1, Eqs. 3.24–3.25, p. 88–89).
• Limitations of Ohm's law: (a) V not proportional to I, (b) V–I relation depends on sign of V (diode), (c) V–I relation not single-valued (GaAs) (NCERT §3.6, p. 89).
• Materials by resistivity: metals 10⁻⁸–10⁻⁶ Ω m; insulators ~10¹⁸ times that of metals; semiconductors in between, with ρ that decreases with rising temperature (NCERT §3.7, p. 89–90).
• Temperature dependence for metals over a limited range: ρT = ρ0 [1 + α(T − T0)]; α (temperature coefficient of resistivity, dimension K⁻¹) is positive for metals; alloys like nichrome, manganin, constantan have very weak α and are used in standard resistors (NCERT §3.8, Eq. 3.26, p. 90).
• Microscopic explanation via ρ = m/(ne²τ): in metals n ≈ const, τ falls with T so ρ rises; in semiconductors/insulators n rises strongly with T and dominates, so ρ falls (NCERT §3.8, p. 91).
• Electrical energy and power: in time Δt charge IΔt drops through V, dissipating ΔW = IVΔt, so P = IV = I²R = V²/R (ohmic loss) (NCERT §3.9, Eqs. 3.31–3.33, p. 92–93).
• High-voltage transmission: with cable resistance Rc, wasted power Pc = P²Rc/V², so raising V reduces transmission loss (NCERT §3.9, Eq. 3.35, p. 93).
• A cell has emf ε = V₊ + V₋ — the open-circuit potential difference between terminals — and internal resistance r; when current I flows, terminal voltage V = ε − Ir, and I = ε/(R + r); maximum current Imax = ε/r (NCERT §3.10, Eqs. 3.36–3.40, p. 93–95).
• Cells in series (same orientation): εeq = ε1 + ε2, req = r1 + r2; if a cell is reversed its emf enters with a negative sign (NCERT §3.11, Eqs. 3.45–3.47, p. 95–96).
• Cells in parallel: 1/req = 1/r1 + 1/r2 and εeq/req = ε1/r1 + ε2/r2; extends to n cells (NCERT §3.11, Eqs. 3.56–3.59, p. 96–97).
• Kirchhoff's rules: (a) Junction rule — at any junction Σ I_in = Σ I_out (conservation of charge); (b) Loop rule — algebraic sum of changes of potential around any closed loop is zero (conservation of energy) (NCERT §3.12, p. 97–98).
• Wheatstone bridge: four resistors R1, R2, R3, R4 with battery across one diagonal and galvanometer across the other; balance (Ig = 0) gives R2/R1 = R4/R3, equivalently R1/R2 = R3/R4; a practical realisation is the metre bridge, which uses this balance to find an unknown resistance (NCERT §3.13, Eqs. 3.62–3.64, p. 100–101).

2.2 Definitions to memorise

2.3 Diagrams / processes to remember

• Fig. 3.1 (p. 83): metallic cylinder with +Q and −Q on end discs — electrons drift to neutralise charges, motivating the need for a cell to maintain steady current.
• Fig. 3.2 (p. 83–84): rectangular slab of length l and area A illustrating R = ρl/A.
• Fig. 3.3 (p. 85): zig-zag electron path A → B between collisions, with a slight drift B → B′ opposite to E.
• Fig. 3.4 (p. 86): cylinder of unit area and length vd containing charge carriers — used to derive I = neAvd.
• Fig. 3.5 (p. 89): V vs I for a good conductor — solid curve deviates from dashed Ohm's-law line at large V.
• Fig. 3.6 (p. 89): characteristic curve of a diode (asymmetric forward/reverse) — Ohm's-law failure type (b).
• Fig. 3.7 (p. 89): V vs I for GaAs showing non-unique V for the same I — Ohm's-law failure type (c).
• Fig. 3.8 (p. 90): ρ of copper vs T — linear in the usual range, deviates at very low T.
• Fig. 3.9 (p. 90): ρ of nichrome — very weak dependence on T.
• Fig. 3.10 (p. 90): ρ of a typical semiconductor — decreases with T.
• Fig. 3.11 (p. 93): cell driving a resistor R; heat dissipated in R comes from chemical energy of the electrolyte.
• Fig. 3.12 (p. 94): electrolytic cell with positive (P) and negative (N) terminals; emf is V₊ + V₋.
• Fig. 3.13 (p. 95): two cells in series — yields εeq = ε1 + ε2, req = r1 + r2.
• Fig. 3.14 (p. 96): two cells in parallel — yields 1/req = 1/r1 + 1/r2.
• Fig. 3.15 (p. 98): junction a where I3 = I1 + I2; loop rules for closed loops ahdcba and ahdefga.
• Fig. 3.16 (p. 98) and Fig. 3.17 (p. 99): cubical network of 12 equal resistors (Example 3.5, Req = 5R/6) and a non-symmetric mesh (Example 3.6) — classic Kirchhoff applications.
• Fig. 3.18 (p. 101): Wheatstone bridge — battery on AC, galvanometer on BD, balance R2/R1 = R4/R3.

2.4 Common confusions / NTA trap points

• Current is a scalar, not a vector — even though we draw arrows; currents do not add by the parallelogram law. The arrow only indicates direction along the wire (Points to Ponder 1, p. 104).
• "V = IR is Ohm's law" is wrong as stated: V = IR is the definition of R and applies to any conducting device. Ohm's law is the stronger claim that R is independent of V (i.e., the V–I plot is a straight line through the origin) (Points to Ponder 2, p. 105).
• Drift speed vs signal speed vs thermal speed: drift speed in a metal is ~mm s⁻¹, thermal speed of ions ~10² m s⁻¹, while the electric field itself propagates near the speed of light. Distractors swap these in MCQs (NCERT Example 3.1 & 3.2, p. 87–88).
• Sign of α: positive for metals (ρ increases with T) but negative for semiconductors and insulators (ρ decreases with T). NTA likes to flip this.
• Terminal voltage: when the cell is discharging through R, V = ε − Ir (V < ε); when it is being charged from outside, the formula reverses sign (V = ε + Ir for the terminal voltage of the battery being charged). Many students always write ε − Ir.
• Wheatstone bridge balance condition is sometimes mis-written. The NCERT writes it as R1/R2 = R3/R4 (Summary, p. 104) and equivalently R2/R1 = R4/R3 from the loop derivation (Eq. 3.64a, p. 101) — both forms are correct provided the arms are paired consistently (adjacent arms on the same side of the galvanometer).