Question

A very thin metal sheet is inserted halfway between the parallel plates of an air-gap capacitor. The sheet is thin compared to the distance between the plates, and it does not touch either plate when fully inserted. The system had capacitance, $C$, before the plate is inserted.
What is the equivalent capacitance of the system after the sheet is fully inserted?

• A. $\frac{1}{4}C$
• B. $\frac{1}{2}C$
• C. $C$
• D. $2C$
• E. $4C$

Answer 1

Answer: (C)

$C$

Initially (before metal sheet inserted) the capacitance of a parallel plate capacitor is $C=\frac{A{ϵ}_0}{d}$ where A be the area of plates and d be the separation between parallel plates.
When a metal sheet inserted fully halfway between the parallel plates, the capacitance will be divided into two capacitors $C_1,C_2$ and they are in series.
Thus, $C_1=\frac{A{ϵ}_0}{\left(d/2\right)}=2C$ and $C_2=\frac{A{ϵ}_0}{\left(d/2\right)}=2C$
The equivalent capacitance , $C_{12}=\frac{C_1{C}_2}{C_1+C_2}=\frac{2C\times 2C}{2C+2C}=C$