Question

Find the capacitance of the system shown in figure.

Answer 1

The capacitance without dielectric is $C_0=\frac{A{ϵ}_0}{d}$

The capacitance with dielectric is $C=\frac{AK{ϵ}_0}{d}=K{C}_0$

Here the capacitances 1,3 are in series and 2, 4 are in series and then they are in parallel.

So, $C_{13}=\frac{C_1{C}_3}{C_1+C_3}=\frac{C_0\left(3C_0\right)}{C_0+3C_0}=\left(3/4\right)C_0$ and $C_{24}=\frac{\left(2C_0\right)\left(4C_0\right)}{2C_0+4C_0}=\left(4/3\right)C_0$

Total capacitance of the system is $C_t=C_{13}+C_{24}=\left(3/4\right)C_0+\left(4/3\right)C_0=\left(25/12\right)C_0$