Question

Two capacitors of capacitance $1\mu \text{F}$ and $2\mu \text{F}$ are charged to potential difference $20\text{V}$ and $15\text{V}$ as shown in figure. If now terminal $\text{B}$ and $\text{C}$ are connected together while terminal $\text{A}$ is connected with positive terminal of battery and terminal $\text{D}$ of capacitor connected with negative terminal of battery with an emf $30\text{V}$.


The final charges on $1\mu \text{F}$ and $2\mu \text{F}$ capacitors respectively will be;

• A. $\frac{20}{3}\mu \text{C},\frac{80}{3}\mu \text{C}$
• B. $\frac{50}{3}\mu \text{C},\frac{80}{3}\mu \text{C}$
• C. $\frac{120}{7}\mu \text{C},\frac{80}{7}\mu \text{C}$
• D. $\frac{50}{6}\mu \text{C},\frac{40}{3}\mu \text{C}$

Answer 1

The correct option is B $\frac{50}{3}\mu \text{C},\frac{80}{3}\mu \text{C}$

Before connection, charge on each capacitor:

$q_1=C_1{V}_1=1\times 20=20\mu \text{C}$
$q_2=C_2{V}_2=2\times 15=30\mu \text{C}$


Let the battery supplies charge $q$ in circuit and the charges on capacitor will be as shown in figure. So after connection distribution of charge will be


Applying $KVL$ in loop

$-\left(\frac{20+q}{C_1}\right)-\left(\frac{30+q}{C_2}\right)+30=0$

[taking in clockwise direction]

$\Rightarrow -\frac{(20+q)}{1}-\frac{(30+q)}{2}+30=0$

$\Rightarrow 2(20+q)+(30+q)=60$

$\therefore q=\frac{-10}{3}\mu \text{C}$

Therefore, changes in capacitors are,

$q_1=20+q=20-\frac{10}{3}=\frac{50}{3}\mu \text{C}$

Similarly, charge on $2\mu \text{F}$

$q_2=30+q$

$\Rightarrow q_2=30-\frac{10}{3}=\frac{80}{3}\mu \text{C}$

Hence, option $\left(b\right)$ is correct.

Why this question? Tip:In questions involving two capacitors and a battery connected in parallel with them, either we can apply $KVL$ in loop or$\Sigma \Delta V_{capacitor}=\Delta V_{Battery}$ i.e., $V_1+V_2=V_{Battery}$ Caution: While traversing in a closed loop, always follow one convention. If we consider crossing from +ve plate of capacitor to -ve plate, it can be given negative sign if potential drop is considered negative throughout.