Question

The effective resistances are $10\Omega ,2.4\Omega$ when two resistors are connected in series and parallel respectively. What are the resistances of individual resistors?

Answer 1

Given $R_S=10\Omega ;R_P=2.4\Omega ;R_1=?;R_2=?$
Solution:
Effective resistance when connected in series and parallel.
$R_S=R_1+R_2=10$ ............$\left(1\right)$
$\frac{1}{R_P}=\frac{1}{R_1}+\frac{1}{R_2}=\frac{R_1+R_2}{R_1{R}_2}$

$\therefore R_P=\frac{R_1{R}_2}{R_1+R_2}=2.4$ ............$\left(2\right)$

From egn $\left(1\right)$ and $\left(2\right)$
$\frac{R_1{R}_2}{10}=2.4$
$\Rightarrow R_1{R}_2=24$
$\therefore R_2=\frac{24}{R_1}$
Substituting $R_2$ in eqn $\left(1\right)$
$R_1+\frac{24}{R_1}=10;R_1^2+24=10R_1$
$R_1^2-10R_1+24=0$
$(R_1-6)(R_1-4)=0$
$\therefore R_1=6\Omega$ or $R_1=4\Omega$
$R_2=\frac{24}{6}$ or $R_2=\frac{24}{4}$
$=4\Omega$. or $=6\Omega$
$R_1=6\Omega ,R_2=4\Omega$.