Question

Two identical parallel plate capacitors A and B are connected to battery of V volts with the switch S closed. The switch is now opened and the free space between the plates of the capacitors is filled with a dielectric of dielectric constant K. Find the ratio of the total electrostatic energy stored in both capacitors before and after the introduction of the dielectric.

Answer 1

Energy stored before introduction of dielectric,
$U_{iA}=U{U}_{iB}=U{U}_i=U+U=2U$
After introduction of dielectric,
$U_{fA}=KU$
$U_{fB}=\frac{U}{K}$
$U_f=KU+\frac{U}{K}=U(K+\frac{1}{K})$
Ratio =$\frac{U_i}{U_f}=\frac{2U}{U(K+\frac{1}{K})}=\frac{2K}{K^2+1}$