Question

An Aluminium $(\alpha =4\times {10}^{-3}{K}^{-1})$ resistance R${}_1$ and a carbon $(\alpha =0.5\times {10}^{-3}{K}^{-1})$ resistance R${}_2$ are connected in series to have a resultant resistance of 36$\Omega$ at all temperatures. The values of

R${}_1$ and R${}_2$ in $\Omega$ respectively are :

• A. 32, 4
• B. 16, 20
• C. 4, 32
• D. 20, 16

Answer 1

The correct option is C 4, 32

Here, resistance of aluminium is $R_1$ and resistance of carbon is $R_2$ and
$R_1+R_2=36$ ........(1)

$R_{1^{\prime }}=R_1[1+{\alpha }_1\Delta T]$

$R_{2^{\prime }}=R_2[1+{\alpha }_2\Delta T]$

$R_{1^{\prime }}+R_{2^{\prime }}$ is independent of T

$\Rightarrow R_1{\alpha }_1+R_2{\alpha }_2=0$

Also, $\frac{R_1}{R_2}=\frac{{\alpha }_2}{{\alpha }_1}$

or, $\frac{R_1}{R_2}=\frac{0.5\times {10}^{-3}}{04\times {10}^{-3}}$

or, $\frac{R_1}{R_2}=\frac{1}{8}$

$\therefore R_2=8R_1$

using this value in equation (1)
$R_1+8R_1=36$
$9R_1=36$
$\therefore R_1=4\Omega$ and $R_2=32\Omega$