Question

The equivalent capacitance of the infinite ladder shown in the figure between points $\text{A}$ & $\text{B}$ will be

• A. $2\mu \text{F}$
• B. $1\mu \text{F}$
• C. $4\mu \text{F}$
• D. $\frac{1}{2}\mu \text{F}$

Answer 1

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


The above diagram is an example of infinite capacitor network.
Now assuming capacitance of above network across $\text{C}$ and $\text{D}$ is $C(=C_{CD})$. So network will become

Now the equivalent capacitance of above diagram is given by

$C_{eq}=1+\frac{2\times C}{2+C}=\frac{2+3C}{2+C}......\left(1\right)$

and this is an infinite capacitor network, so

$C_{AB}=C_{eq}=C$

So, eq. $\left(1\right)$ becomes

$C=\frac{2+3C}{2+C}$

$\Rightarrow C^2+2C=3C+2$

$\Rightarrow C^2-C-2=0$

$\Rightarrow (C+1)(C-2)=0$

Since, capacitance will be positive

$\therefore C=2\mu \text{F}$

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