The variation of resistivity of a metallic conductor with temperature…

The correct answer is Option (1) → \( \rho_T = \rho_0[1 + \alpha(T - T_0)] \)

For a metal, the resistivity increases almost linearly with temperature over a limited range. The temperature coefficient of resistivity \( \alpha \) is the fractional increase in resistivity per degree rise, measured from the reference value \( \rho_0 \) at \( T_0 \).

    \( \displaystyle \alpha = \frac{\rho_T - \rho_0}{\rho_0\,(T - T_0)} \;\Rightarrow\; \rho_T = \rho_0[1 + \alpha(T - T_0)] \)

Option (1) \( \rho_T = \rho_0[1 + \alpha(T - T_0)] \) — Correct. This is the defining relation: the increase \( \rho_T - \rho_0 \) is \( \rho_0\,\alpha \) times the temperature rise.

Option (2) \( \rho_0 = \rho_T[1 + \alpha(T_0 - T)] \) — Incorrect. It treats \( \rho_T \) as the reference value. Inverting the correct relation gives \( \rho_0 = \rho_T/[1 + \alpha(T - T_0)] \), which matches this form only approximately, for small \( \alpha(T - T_0) \).

Option (3) \( \rho_T = \rho_0[1 + \alpha(1 - T_0)] \) — Incorrect. \( 1 - T_0 \) subtracts a temperature from a pure number, so the expression is not dimensionally consistent and does not contain \( T \) at all.

Option (4) \( \rho_T = [\rho_0 + \alpha(T - T_0)] \) — Incorrect. \( \alpha \) has the unit \( \mathrm{K^{-1}} \), so \( \alpha(T - T_0) \) is a pure number and cannot be added to a resistivity.

NCERT (2026-27):
Temperature Dependence of Resistivity – “… Over a limited range of temperatures, that is not too large, the resistivity of a metallic conductor is approximately given by,

    \( \displaystyle \rho_T = \rho_0\,[1 + \alpha\,(T - T_0)] \qquad (3.26) \)

where ρ_T is the resistivity at a temperature T and ρ₀ is the same at a reference temperature T₀. α is called the temperature co-efficient of resistivity, and from Eq. (3.26), the dimension of α is (Temperature)^–1.”