Current Electricity and Resistivity
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Direct answer
Current I = dQ/dt is the rate of charge flow, carried in metals by electrons drifting with a tiny average velocity v_d = I/(nAe) — typically a fraction of a millimetre per second, even while the electric influence propagates near light speed. Resistance follows R = ρl/A, where resistivity ρ is the material's own property (unit Ω m), independent of shape but strongly dependent on temperature and, at the microscopic level, ρ = m/(ne^2τ) with τ the collision relaxation time. NCERT's benchmark values anchor expectations: copper 1.7 × 10^-8 Ω m, nichrome about 100 × 10^-8 Ω m (hence heater coils), glass 10^10 to 10^14 Ω m. Conductivity σ = 1/ρ, and current density J = I/A links the microscopic picture through J = σE.
What you must remember
- Geometry relation: R = ρl/A — stretching a wire to double its length (volume constant) quadruples R, since l doubles and A halves.
- Drift velocity: v_d = I/(nAe); for a copper wire of 1 mm^2 area carrying 1 A, v_d ≈ 0.07 mm/s with n ≈ 9 × 10^28 per m^3 (an NCERT-worked figure).
- Microscopic Ohm's law: J = σE, with σ = ne^2τ/m; more free electrons or longer collision times mean better conduction.
- Temperature behaviour: metals' ρ rises almost linearly (positive coefficient, α ≈ 0.004 per °C for copper); nichrome's α ≈ 1.7 × 10^-4 per °C — weak, which is why it holds resistance steady in heaters; semiconductors' ρ falls with temperature (negative coefficient).
- Material ladder (NCERT Table 3.1): silver 1.6, copper 1.7, aluminium 2.7, iron 10, nichrome ~100 (all × 10^-8 Ω m); insulators climb to 10^14.
- Colour-code convention: Indian labs read resistors by the mnemonic "B B ROY Great Britain Very Good Wife" — black, brown, red, orange, yellow, green, blue, violet, grey, white for digits 0-9, with the fourth band as tolerance (gold 5%, silver 10%).
- Ohm's law domain: V = IR holds for metallic conductors at constant temperature; it fails for diodes, electrolytes and at extreme temperatures — a standard assertion item.
A worked wire-stretching problem
A copper wire of resistance 10 Ω is stretched uniformly to twice its original length. What is the new resistance? Volume conservation: l → 2l, so A → A/2. Then R_new = ρ(2l)/(A/2) = 4ρl/A = 4 × 10 = 40 Ω. The general rule: R ∝ l^2 at constant volume, so any stretch factor k multiplies resistance by k^2. Now the companion numerical the exam prefers: find the drift velocity in this wire of area 1 mm^2 carrying 3 A. v_d = I/(nAe) = 3/(9 × 10^28 × 10^-6 × 1.6 × 10^-19) = 3/(1.44 × 10^4) ≈ 2.1 × 10^-4 m/s — about a fifth of a millimetre each second. Yet the lamp lights instantly when you flip the switch, because the field establishes itself through the wire at near light speed and electrons everywhere begin drifting at once; the drift is a response to the field, not a relay race of charge carriers. That contrast — 10^-4 m/s drift against instant lighting — is the conceptual heart of this chapter in NEET statement questions.
Where students slip
Stretching problems mislead twice over: candidates double the resistance (forgetting the area halves) or quarter it (forgetting the length doubles); the answer is always k^2, and options carry both errors as bait. In drift-velocity questions, mixing units — area in mm^2 left unconverted, n per cm^3 rather than per m^3 — shifts the answer by six orders of magnitude, so write the unit chain explicitly. The temperature trap runs in the opposite direction for semiconductors: a thermistor's resistance falls as it warms, and "resistance always increases with temperature" is the false assertion the examiner plants. Finally, current is the same through every series element regardless of wire thickness — thinner wire does not "use up" current; it merely drops more of the potential difference, a point NCERT stresses when deriving series resistance.
Frequently asked questions
How does resistance depend on length and area of a conductor?
R = ρl/A: resistance doubles with doubled length and halves with doubled cross-sectional area, with resistivity fixed by the material.
What happens to resistance when a wire is stretched to double its length?
It becomes four times the original, because length doubles and area halves at constant volume (R ∝ l^2).
Why is drift velocity so small compared with the speed of electricity?
Electrons drift at about 10^-4 m/s due to constant collisions, but the electric field propagates through the wire at nearly light speed, starting drift everywhere almost simultaneously.
Why is nichrome preferred for heating elements?
Its resistivity (~100 × 10^-8 Ω m) is high and its temperature coefficient is small (≈ 1.7 × 10^-4 per °C), so it produces ample heat (I^2R) without its resistance drifting much as it glows.
How does resistivity change with temperature in metals versus semiconductors?
In metals, increased lattice vibrations shorten the relaxation time and resistivity rises nearly linearly; in semiconductors, more charge carriers are freed, so resistivity falls.