Question

In the given network of potential difference between $p$ and $q$ is $2V$ and $C_2=3C_1$. The potential difference between $a$ & $b$ in V is $10\times X$. The value of $X$ is

Answer 1

The on the capacitor $C_2$ in between p and q is $Q_{pq}=C_2\times 2=2C_2$
As the capacitors $C_1$ and $C_2$ in between b and q are in series so charge on $C_1$ is equal to $Q_{pq}$.
Potential across $C_1$ is $V_{bp}=\frac{Q_{pq}}{C_1}=\frac{2C_2}{C_1}=\frac{2\left(3C_1\right)}{C_1}=6V$
Thus, $V_{bq}=V_{bp}+V_{pq}=6+2=8V$
The equivalent capacitance $C_{bq}=C_2+\frac{C_1{C}_2}{C_1{C}_2}=3C_1+\frac{C_1\left(3C_1\right)}{C_1+3C_1}=\frac{15}{4}{C}_1$
and $Q_{bq}=C_{bq}{V}_{bq}=\frac{15}{4}{C}_1\left(8\right)=30C_1$
As $C_{bq}$ and $C_1$ are in series so charge on $C_1$ is equal to $Q_{bq}$
Thus $V_{ab}=\frac{Q_{ab}}{C_1}=\frac{30C_1}{C_1}=30V$