The resistance of a wire at \(300 \mathrm{~K}\) is found to be \(0.3 \Omega\). If the temperature coefficient of resistance of wire is \(1.5 \times 10^{-3} \mathrm{~K}^{-1}\), the temperature at which the resistance becomes \(0.6 \Omega\) is

\(\text {Given, }\mathrm{R}_{300} =0.3 \Omega, \mathrm{R}_{\mathrm{t}}=0.6 \Omega, \)
\( \mathrm{T} =300 \mathrm{~K}=27^{\circ} \mathrm{C}\)
Temperature coefficient of resistance,
\(\alpha=1.5 \times 10^{-3} \mathrm{~K}^{-1}\)
\(\therefore \mathrm{R}_{300}=\mathrm{R}_{0}(1+\alpha \times 27)\)
\(0.3=\mathrm{R}_{0}\left(1+1.5 \times 10^{-3} \times 27\right) \quad \text{...(i)}\)
Again, \(\mathrm{R}_{\mathrm{t}}=\mathrm{R}_{0}(1+\alpha \mathrm{t})\)
\(0.6=\mathrm{R}_{0}\left(1+1.5 \times 10^{-3} \times \mathrm{t}\right) \quad \text{...(ii)}\)
Dividing Eq. (ii) by Eq. (i), we get \(\frac{0.6}{0.3}=\frac{1+1.5 \times 10^{-3} \mathrm{t}}{1+1.5 \times 10^{-3} \times 27}\)
\(\Rightarrow 2\left(1+1.5 \times 10^{-3} \times 27\right)=1+1.5 \times 10^{-3} \mathrm{t}\)
\(\Rightarrow 2+81 \times 10^{-3}=1+1.5 \times 10^{-3} \mathrm{t}\)
\(\Rightarrow 2+0.081=1+1.5 \times 10^{-3} \mathrm{t}\)
\(\Rightarrow \mathrm{t}=\frac{1.081}{1.5 \times 10^{-3}}=720^{\circ} \mathrm{C}=993 \mathrm{~K}\)