Question

Two wires of resistance $R_1$ and $R_2$ at $0^{o}C$ have temperature coefficient of resistance ${\alpha }_1$ and ${\alpha }_2$, respectively. These are joined in series. The effective temperature coefficient of resistance is :-

• A. $\frac{{\alpha }_1+{\alpha }_2}{2}$
• B. $\sqrt{{\alpha }_1{\alpha }_2}$
• C. $\frac{{\alpha }_1{R}_1+{\alpha }_2{R}_2}{R_1+R_2}$
• D. $\frac{\sqrt{R_2{R}_1{\alpha }_1{\alpha }_2}}{\sqrt{R_1^2+R_2^2}}$

Answer 1

The correct option is C $\frac{{\alpha }_1{R}_1+{\alpha }_2{R}_2}{R_1+R_2}$

The thermal coefficient of resistance is $\Rightarrow \alpha =\frac{1}{R}\frac{dR}{dt}$
Equivalent resistance in series= $R_1+R_2=R_{eq.}$
$\frac{d{R}_{eq.}}{dt}=\frac{d{R}_1}{dt}+\frac{d{R}_2}{dt}$
${\alpha }_{eq.}{R}_{eq.}={\alpha }_1{R}_1+{\alpha }_2{R}_2$
${\alpha }_{eq.}=\frac{{\alpha }_1{R}_1+{\alpha }_2{R}_2}{R_1+R_2}$