Question

Two metal plates each of area $A$ form a parallel plate capacitor with air in between the plates. The distance between the plates is $d$. A metal plate of thickness $\frac{d}{2}$ and of same area $A$ is inserted between the plates to form two capacitors of capacitances $C_1$ and $C_2$ as shown in the figure. The effective capacitance of the two capacitors is C' and the capacitance of the capacitor initially is C, then $\frac{C^{\prime }}{C}$ is :

• A. $1$
• B. $6$
• C. $2$
• D. $4$

Answer 1

Answer: (C)

Let the gap for capacitor 1 be x and that for capacitor 2 be y. Then,

$C_1=\frac{{ϵ}_{0}A}{x}$ and $C_2=\frac{{ϵ}_{0}A}{y}$

$x+y=d/2$

Since the two are in series,

$C^{\prime }=\frac{C_1{C}_2}{C_1+C_2}={ϵ}_{0}A\frac{\frac{1}{xy}}{\frac{1}{x}+\frac{1}{y}}$

$=\frac{{ϵ}_{0}A}{x+y}=\frac{2{ϵ}_{0}A}{d}=2C$