Question

The series of two batteries, both of the same emf $10\text{V}$, but different internal resitance of $20\Omega$ and $5\Omega$, is connected to the parallel combination of two resistors $30\Omega$ and $R\Omega$. The voltage difference across the battery of internal resistance $20\Omega$ is zero, the value of $R$ (in $\Omega$) is

Answer 1

The resistance of $30\Omega$ is in parallel with $R$, their effective resistance
$\frac{1}{R^{\prime }}=\frac{1}{30}+\frac{1}{R}$
$R^{{}^{\prime }}=\frac{30R}{30+R}...\left(i\right)$
Also, $V=IR$
$\Rightarrow 10=\frac{20\times 20}{R^{\prime }+25}$
$\Rightarrow R^{\prime }+25=40\Omega$
$\Rightarrow R^{\prime }=15\Omega$
Using (i)
$\Rightarrow R^{\prime }=15=\frac{30R}{30+R}$
$\Rightarrow 30+R=2R$
$\Rightarrow R=30\Omega$