Question

Resistance of a resistor at temperature $t$ ${}^o\text{C}$ is $R_t=R_0(1+\alpha t+\beta t^2)$, where $R_0$ is the resistance at $0{}^o\text{C}$. The temperature coefficient of resistance at temperature $t{}^o\text{C}$ is

• A. $\frac{(1+\alpha t+\beta t^2)}{\alpha +2\beta t}$
• B. $(\alpha +2\beta t)$
• C. $\frac{\alpha +2\beta t}{(1+\alpha t+\beta t^2)}$
• D. $\frac{\alpha +2\beta t}{2(1+\alpha t+\beta t^2)}$

Answer 1

The correct option is C $\frac{\alpha +2\beta t}{(1+\alpha t+\beta t^2)}$

Given,
$R_t=R_0(1+\alpha t+\beta t^2)$

The temperature coefficient of a conductor $\alpha$ is given by,$$\alpha =\frac{1}{R_t}\frac{d{R}_t}{dt}$$$\Rightarrow \alpha =\frac{1}{R_t}{R}_0(\alpha +2\beta t)$

$\Rightarrow \alpha =\frac{R_0(\alpha +2\beta t)}{R_0(1+\alpha t+\beta t^2)}$

$\Rightarrow \alpha =\frac{\alpha +2\beta t}{1+\alpha t+\beta t^2}$

Hence, option (c) is correct answer.