Question

Four identical plates $1,2,3,$ and $4$ are placed parallel to each other at equal distance as shown in the figure. Plates $1$ and $4$ are joined together and the space between $2$ and $3$ is filled with a dielectric of dielectric constant $k=2$. The capacitance of the system between $1$ and $3$ & $2$ and $4$ are $C_1$ and $C_2$ respectively. The ratio $C_1/C_2$ is :

• A. $\frac{5}{3}$
• B. $1$
• C. $\frac{3}{5}$
• D. $\frac{5}{7}$

Answer 1

The correct option is B $1$

The capacitance of parallel plate capacitor is $C=\frac{A{ϵ}_0}{d}$
The capacitance of parallel plate capacitor with dielectric is $C^{\prime }=\frac{Ak{ϵ}_0}{d}=kC=2C$
Here 1 and 2 plate and 3 and 4 plate will make the capacitor of same capacitance $C$ and 2 and 3 will make the capacitance of $C^{\prime }$.
Here the capacitance in between 1 and 3 is $C_1=\frac{C{C}^{\prime }}{C+C^{\prime }}=\frac{C.2C}{C+2C}=2C/3$
and the capacitance in between 2 and 4 is $C_2=\frac{C{C}^{\prime }}{C+C^{\prime }}=\frac{C.2C}{C+2C}=2C/3$
$\therefore \frac{C_1}{C_2}=1$