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The resistance of a conductor is 10 Ω at 500 C and 15 Ω at 1000 C. Calculate its resistance at 00 C.

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The resistance of a conductor is 10 Ω at 500 C and 15 Ω at 100 0 C. Calculate its resistance at 0 0 C.
Solution:
\(\theta_1\) = 50 \(^\circ\) C
resistance at 50 \(^\circ\) C, \(R_1\) = 10 \(\Omega\)
\(\theta_2\) = 100 \(^\circ\) C
resistance at 100 \(^\circ\) C, \(R_2\) = 15 \(\Omega\)
\(\theta\) = 0\(^\circ\)
resistance at 0\(^\circ\) C, \(R\)=?
We have,
\[\begin{align*} \alpha&=\frac{R_2-R_1}{R_1 \Delta \theta}\\ &=\frac{15-10}{10\times 50}\\ &=\frac{5}{10 \times 50}\\ &=\frac{1}{100}\hspace{0.1cm} K^{-1}\\ \end{align*}\] Now, \[\begin{align*} R&=R_1(1+\alpha \Delta \theta)\\ &=10(1+\frac{1}{100} \times (0-50))\\ &=10(1-\frac{50}{100})\\ &=10\times \frac{1}{2}\\ \therefore R&=5 \hspace{0.1cm} \Omega \end{align*}\] Hence the resistance at 0\(^\circ\) C is 5 \(\Omega\).

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