Question

Rows of capacitors containing $1,2,4,8,....\infty$ capacitors, each of capacitance $2\mu \text{F}$, are connected in parallel as shown in figure. The potential difference across $\text{AB}=10\text{volts}$, then:

• A. Equivalent capacitance across $\text{AB}$ is $2\mu \text{F}$.
• B. Charge on the capacitor in the first row is less than on any other capacitor
• C. Charge on the capacitor in the first row is more than on any other capacitor
• D. Energy of all the capacitors is $50\text{J}$

Answer 1

The correct option is C Charge on the capacitor in the first row is more than on any other capacitor

Given : $C=2\mu \text{F}$

$C_{eq}=C+\frac{C}{2}+\frac{C}{4}+\frac{C}{8}+\frac{C}{16}+.....$

$\Rightarrow C_{eq}=C\left(\frac{1}{1-\frac{1}{2}}\right)=2\left(\frac{1}{\frac{1}{2}}\right)=4\mu \text{F}$

Charge on first row capacitor is $q_1=2\times 10\mu C=20\mu \text{C}$

Charge on second row capacitor is $q_2=1\times 10\mu C=10\mu \text{C}$

Charge on third row capacitor is $q_3=\frac{1}{2}\times 10\mu C=5\mu \text{C}$

Therefore charge on the capacitor in the first row is more than on any other system of capacitors connected in rows.

Energy stored in all capacitor is $=\frac{1}{2}{C}_{eq}.V^2=\frac{1}{2}\times 4\times {10}^{-6}\times (10{)}^2=0.2\text{mJ}$