The resistivity
1. increases with temperature for semiconductors
2. increases with the applied electric field for metal.
3. varies inversely with the diameter of metallic conductor.
4. depends inversely on the number of free electrons per unit volume for metallic conductor.
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 4 only
From the drift model, \( \rho = m/(ne^{2}\tau) \): resistivity is a property of the material, set by the carrier density n and the relaxation time \( \tau \) (both depend on temperature), not by the size of the sample.
(1) increases with temperature for semiconductors — Incorrect. In a semiconductor the number of carriers rises rapidly with temperature, so its resistivity decreases.
(2) increases with the applied electric field for metal. — Incorrect. A metal obeys Ohm’s law: at a given temperature its resistivity does not depend on the applied field.
(3) varies inversely with the diameter of metallic conductor. — Incorrect. The resistance \( R = \rho l/A \) depends on the diameter; the resistivity does not.
(4) depends inversely on the number of free electrons per unit volume for metallic conductor. — Correct. \( \rho = m/(ne^{2}\tau) \), so \( \rho \propto 1/n \).
NCERT (2026-27):
Ohm’s Law – “… where the constant of proportionality \( \rho \) depends on the material of the conductor but not on its dimensions. \( \rho \) is called resistivity.”
Temperature Dependence of Resistivity – “… Unlike metals, the resistivities of semiconductors decrease with increasing temperatures.”
Temperature Dependence of Resistivity – “… \( \rho \) thus depends inversely both on the number n of free electrons per unit volume and on the average time \( \tau \) between collisions.”