Question

$N$ conducting plares are placed face to face as shown in figure. Distance between any two plates is $d$. Area of the plates is $A,\frac{A}{2},\frac{A}{4},\frac{A}{8}...\frac{A}{2(n-1)}$. The equivalent capacitance of the system is _____.

Answer 1

Remember the summation

${\sum }_{k=0}^{n-1}{2}^k=\frac{1-2^n}{1-2}=2^n-1$

All the caoacitors are in series. So their reciprocals will get added up. So$⟶$

$\frac{1}{C}=\frac{d\left({\sum }_{k=0}^{n-1}{2}^k\right)}{{\epsilon }_{0}A}=\frac{d(2^n-1)}{{\epsilon }_{0}A}$

$\therefore C=\frac{{\epsilon }_{0}A}{(2^n-1)d}$