Question

Find the effective capacitance across the points $\text{X}$ and $\text{Y}$, of the infinite ladder shown below.

• A. $1\mu \text{F}$
• B. $2\mu \text{F}$
• C. $3\mu \text{F}$
• D. $4\mu \text{F}$

Answer 1

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

Let the equivalent capacitance across $\text{X}$ and $\text{Y}$ be $C$.

Since the ladder is infinite, so the equivalent capacitance across the points $\text{A}$ and $\text{B}$ (right part) will also be equal to $C$ (approximately).


So, the circuit can be redrawn as:


From the above circuit, we can see that $C$ and $2\mu \text{F}$ are in series. So, their equivalent will be

$C^{\prime }=\frac{C\times 2}{C+2}=\frac{2C}{C+2}$


Both capacitors are in parallel, so equivalent capacitance between the points $\text{X}$ and $\text{Y}$ is

$C_{eq}=1+\frac{2C}{C+2}$

$\Rightarrow C=1+\frac{2C}{C+2}$

$C^2-C-2=0$

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

$\therefore C=2orC=-1$

As $C$ can not have a negative value.

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

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

Why this question? Tip: In infinite ladder, first recognize the repeating ladder. And then assume that the capacitance of the whole circuit is equal to the capacitance of the circuit excluding one repeating ladder because of infinite structure.