Question

A resistance $R$ of the thermal coefficient of resistivity = $\alpha$ is connected in parallel with a resistance = $3R$, having thermal coefficient of resistivity = $2\alpha$. The value of ${\alpha }_{eff}$ is given as ${\alpha }_{eff}=\frac{x}{4}\alpha$. Find $x.$

Answer 1

The equivalent resistance at $0^{\circ }C$ is

$R_0=\frac{R_{10}{R}_{20}}{R_{10}+R_{20}}$ ......(1)

The equivalent resistance at $t^{\circ }C$ is

$R_t=\frac{R_{1t}{R}_{2t}}{R_{1t}+R_{2t}}$......(2)

But, $R_{1t}=R_{10}(1+\alpha t)$ ....(3)

$R_{2t}=R_{20}(1+2\alpha t)$.......(4)

And $R=R0(1+{\alpha }_{eff}t)$......(5)

From all above equations

${\alpha }_{eff}=\frac{5}{4}\alpha$

Hence $x=5$