Question

An infinite number of identical capacitors each of capacitance $4\mu \text{F}$ are connected as in the adjoining figure. If a battery of $10\text{V}$ is connected across $\text{AB}$, what is the charge flown through the battery?

• A. $40\mu \text{C}$
• B. $80\mu \text{C}$
• C. $20\mu \text{C}$
• D. $160\mu \text{C}$

Answer 1

The correct option is B $80\mu \text{C}$



Capacitors in series in each branch can be replaced by equivalent capacitor,

$\therefore$ The given circuit can be simplified as follows,



All these capacitors are in parallelel across $\text{AB}$

$C_{eq}=4+2+1+\frac{1}{2}+....\infty$ .........(1)

As we can see from eq (1),

The capacitance in the each parallel branches is multiplied by a common factor $\frac{1}{2}$ i.e., It is like a geometric progression.

Using G.P for sum of infinite series,

$$S_{\infty }=\frac{a}{1-r}$$

$a=4$ (first term)

$r=\frac{1}{2}$ (common ratio)

$S_{\infty }=\frac{4}{1-\frac{1}{2}}=8\mu \text{F}$

Using $$Q=CV$$

$Q=80\mu \text{C}$

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