The temperature coefficient of resistance of a wire is \( 0.00125\,\mathrm{K^{-1}} \), taking 300 K as the reference temperature. The resistance of the wire at 300 K is \( 1\,\Omega \). At what temperature will its resistance be \( 2\,\Omega \)?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 1100 K
Over a limited range of temperature, the resistivity of a metal varies as \( \rho_T=\rho_0\,[1+\alpha\,(T-T_0)] \). Here \( \rho_0 \) is its value at the reference temperature \( T_0 \), and \( \alpha \) is the temperature coefficient. For a wire of fixed length and cross-section, \( R\propto\rho \). So the same relation holds for the resistance, with \( R_0=1\,\Omega \) at \( T_0=300\,\mathrm{K} \).
\( R_T=R_0\,[1+\alpha\,(T-T_0)] \)
\( 2\,\Omega=1\,\Omega\times\left[1+0.00125\,\mathrm{K^{-1}}\,(T-300\,\mathrm{K})\right] \)
\( 0.00125\,\mathrm{K^{-1}}\,(T-300\,\mathrm{K})=1 \)
\( \begin{aligned} T-300\,\mathrm{K} &= \frac{1}{0.00125\,\mathrm{K^{-1}}} \\ &= 800\,\mathrm{K} \end{aligned} \)
\( \Rightarrow\; T=1100\,\mathrm{K} \)
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,
\( \rho_T=\rho_0\,[1+\alpha\,(T-T_0)] \)\( \qquad (3.26) \)
where ρT 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.”