Question

A constant current $i$ is passed through a resistor. Taking the temperature coefficient of resistance into account, indicate which of the plots shown in the figure best represents the rate of production of thermal energy in the resistor

• A. b
• B. a
• C. c
• D. d

Answer 1

The correct option is D d

Step 1: Derive rate of production of thermal energy

We know that thermal energy produced,

$U=i^{2}Rt\to \left(1\right)$

Where $i$ is current, $R$ is resistance and $t$ is time

Also for $T^{\circ }C$ increase in temperature

$R=Ro(1+\propto T)\to \left(2\right)$

From equation (2) and (1), we get

$\Rightarrow U{i}^2{R}_0(1+\propto T)t$

Differentiating w.r.t. ${}^{\prime }{t}^{\prime }$ taking temperature constant,

$\Rightarrow \frac{dU}{dt}=i^2{R}_0(1+\propto T)$

Step 2: Plot variation of thermal energy with temperature

At $T=0^{\circ }C,$

$\Rightarrow \frac{dU}{dt}=i^2{R}_0$

Also,

$\frac{du}{dt}\propto T$

Hence, the plot between $\frac{dU}{dt}$ and $T$ is a straight line with intercept on $y-$axis as $i^2{R}_0$.

This is best represented by curve $d$.

Hence, option (d) is correct.