Question

A $10 \,V$ battery is connected to three capacitors, $C_{1}=2 \,\mu F$, $C_{2}=3 \,\mu F$ and $C_{3}=5\, \mu F$, as shown in figure. The charges on the capacitors $C_{1}, C_{2}$ and $C_{3}$ respectively are,

• A. $2 \,\mu C , 3\, \mu C , 5\, \mu C$
• B. $5\, \mu C , 10\, \mu C , 15\, \mu C$
• C. $10\, \mu C , 15\, \mu C , 25\, \mu C$
• D. $4\, \mu C, 6\, \mu C, 10 \,\mu C$

Answer 1

Answer: (C)

The capacitors $C_{1}$ and $C_{2}$ are connected in parallel, the equivalent capacitance $C^{\prime}$ is given by
$C' =C_{1}+C_{2}=2\, \mu F +3\, \mu F =5\, \mu F$
The corresponding circuit diagram is shown in figure.

The capacitors $C'$ and $C_{3}$ are connected in series and equivalent capacitance $C''$ is given by
$\frac{1}{C''}=\frac{1}{C'}+\frac{1}{C_{3}}=\frac{1}{5 \,\mu F }+\frac{1}{5 \,\mu F } $;
$ C''=2.5\, \mu F$
The corresponding equivalent diagram is shown in figure.

Total charge of the given circuit is
$Q=C'' V=2.5 \times 10=25\, \mu C$
$Q=Q_{1}+Q_{2}$ or $25=$
$C_{1} V' +C_{2} V'$
$25=2 V'+3 V'$ (where $V'$ is the potential difference across $A B$ ) or $V'=5\, V$

$\therefore $ The charge on capacitor $C_{1}$ is
$Q_{1}=C_{1} V'=(2 \mu F)(5 V )=10 \,\mu C$
The charge on capacitor $C_{2}$ is
$Q_{2}=C_{2} V'=(3 \,\mu F )(5 V )=15\, \mu C$
The charge on capacitor $C_{3}$ is $Q=25\, \mu C$.