Question

An infinite number of identical capacitors each of capacitance $1\mu \text{F}$ are connected as shown in the adjoining figure. Then, equivalent capacitance between $\text{A}$ and $\text{B}$ is

• A. Infinite
• B. $2\mu \text{F}$
• C. $4\mu \text{F}$
• D. $8\mu \text{F}$

Answer 1

The correct option is B $2\mu \text{F}$

From the given circuit, we can replace the capacitors in series in each branch with an equivalent capacitor.
Again, these equivalent capacitors will be in parallel to each other.
Now let's find the equivalent capacitance of these capacitors, which are connected in parallel.

$\therefore C_{eq}=1+\frac{1}{2}+\frac{1}{4}+---+\frac{1}{\infty }$

This is a GP series, in which

$a\left(\text{first term}\right)=1;r\left(\text{common ratio}\right)=\frac{1}{2}$

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

$\Rightarrow S_{\infty }=C_{eq}=\frac{1}{1/2}=2\mu \text{F}$

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