Question

The equivalent capacity of the infinite network shown in the figure (across AB) is :

(Capacity of each capacitor is 1 $\mu$F)

• A. $\infty$
• B. $1\mu F$
• C. $\left(\frac{\sqrt{3}-1}{2}\right)\mu F$
• D. $\left(\frac{\sqrt{3}+1}{2}\right)\mu F$

Answer 1

The correct option is C $\left(\frac{\sqrt{3}-1}{2}\right)\mu F$

If we remove one set from infinite it is still equal to infinite series, assume the equivalent capacitance is $C_{eff}$, capacitance across OE can be replaced by $C_{eff}$.
now $1\mu F$ and $C_{eff}$ are in parallel, their equivalent capacitance is $C_{eff}+1$.
Now 2 capacitors of $1\mu F$ and $C_{eff}+1$ are in series.
$\frac{1}{C_{eff}}=\frac{1}{1}+\frac{1}{1}+\frac{1}{1+C_{eff}}$
$\frac{1}{C_{eff}}=2+\frac{1}{1+C_{eff}}$
$\frac{1}{C_{eff}}=\frac{2+2C_{eff}+1}{1+C_{eff}}$
$1+C_{eff}=3C_{eff}+2C_{eff}^2$
$2C_{eff}^2+2C_{eff}-1=0$
$\Rightarrow C_{eff}=\frac{\sqrt{3}-1}{2}$