Question

Two conductors have the same resistance at $0^{o}C$ but their temperature coefficients of resistance are ${\alpha }_1$ and ${\alpha }_2$. The respective temperature coefficients of their series and parallel combinations are nearly

• A. $\frac{{\alpha }_1+{\alpha }_2}{2},{\alpha }_1+{\alpha }_2$
• B. ${\alpha }_1+{\alpha }_2,\frac{{\alpha }_1+{\alpha }_2}{2}$
• C. ${\alpha }_1+{\alpha }_2,\frac{{\alpha }_1{\alpha }_2}{{\alpha }_1+{\alpha }_2}$
• D. $\frac{{\alpha }_1+{\alpha }_2}{2},\frac{{\alpha }_1+{\alpha }_2}{2}$

Answer 1

The correct option is D $\frac{{\alpha }_1+{\alpha }_2}{2},\frac{{\alpha }_1+{\alpha }_2}{2}$

Let $R_0$ be the initial resistance of both conductors
.$\cdot$. At temperature 6 their resistance will be,
$R_1=R_0(1+{\alpha }_1\Theta )$ and $R_2=R_0(1+{\alpha }_2\Theta )$
for, series combination, $R_s=R_1+R_2$
$R_{s0}(1+{\alpha }_s\Theta )=R_0(1+{\alpha }_1\Theta )+R_0(1+{\alpha }_2\Theta )$
where $R_{s0}=R_0+R_0=2R_0$
.$\cdot$.
$2R_0(1+{\alpha }_s\Theta )=2R_0+R_0\Theta ({\alpha }_1+{\alpha }_2)$
or ${\alpha }_s=\frac{{\alpha }_1+{\alpha }_2}{2}$
for parallel combination, $R_p=\frac{R_1{R}_2}{R_1+R_2}$
$R_{p0}(1+{\alpha }_p\Theta )=\frac{R_0(1+{\alpha }_1\Theta )R_0(1+{\alpha }_2\Theta )}{R_0(1+{\alpha }_1\Theta )+R_0(1+{\alpha }_2\Theta )}$
where, $R_{p0}=\frac{R_0{R}_0}{R_0+R_0}=\frac{R_0}{2}$
$\cdot$.
$\frac{R_0}{2}(1+{\alpha }_p\Theta )=\frac{R_0^2(1+{\alpha }_1\Theta +{\alpha }_2\Theta +{\alpha }_1{\alpha }_2\Theta )}{R_0(2+{\alpha }_1\Theta +{\alpha }_2\Theta )}$
as ${\alpha }_1$ and ${\alpha }_2$ are small quantities
${\alpha }_1{\alpha }_2$ is negligible
or ${\alpha }_p=\frac{{\alpha }_1+{\alpha }_2}{2+({\alpha }_1+{\alpha }_2)\Theta }=\frac{{\alpha }_1+{\alpha }_2}{2}[1-({\alpha }_1+{\alpha }_2\left)\Theta \right]$ as $({\alpha }_1+{\alpha }_2{)}^2$ is negligible
.$\cdot$.${\alpha }_p=\frac{{\alpha }_1+{\alpha }_2}{2}$