Question

The resistance of wire at ${20}^{\circ }C$ is $20\Omega$, and at ${300}^{\circ }C$ resistance of wire is $50\Omega$. The temperature at which its resistance will be $40\Omega$ is

• A. ${100}^{\circ }C$
• B. ${107}^{\circ }C$
• C. ${207}^{\circ }C$
• D. ${200}^{\circ }C$

Answer 1

The correct option is C ${207}^{\circ }C$

$R=R_0\{1+\alpha \Delta \theta \}$
So,
$20=R_0\{1+\alpha .20\}$
$50=R_0\{1+\alpha .300\}$
$\frac{2}{5}=\frac{1+20\alpha }{1+300\alpha }\Rightarrow 2+600\alpha =5+100\alpha$
$500\alpha =3$
$\alpha =6\times {10}^{-3}$
Hence,
$R=R_0\{1+\alpha \Delta \theta \}$
$\Rightarrow 50=R_0\{1+6\times {10}^{-3}\times 300\}$
$=R_0=\frac{50}{2.8}$

$\therefore R=\frac{50}{2.8}\{1+6\times {10}^{-3}\times \Delta \theta \}$
$\Rightarrow 40=\frac{50}{2.8}\{1+6\times {10}^{-3}\times \Delta \theta \}$
$\therefore \Delta \theta ={206.67}^{0}C$ which is close to ${207}^{0}C$.