Question

Two identical parallel plate capacitors A and B connected to a battery of V volts w/ the switch S closed. The switch is now opened and the free space between the plates of the capacitors is filled w/ 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

The electrostatic energy stored in a capacitor with capacitance $C_0$, with a voltage difference V applied to it, and without dielectric, is given by
$U_0 = \frac{1}{2} C_0 V^2$
Now let's assume we fill the space between the two plates of the capacitor with a dielectric with constant k. The new capacitance of the capacitor is
$C_k = k C_0$
So, the energy stored now is
$U_k = \frac{1}{2}C_k V^2= \frac{1}{2}kC_0 V^2$

Therefore, the ratio between the energies stored in the capacitor before and after the introduction of the dielectric is
$\frac{U_k}{U_0}= \frac{ \frac{1}{2}kC_0 V^2 }{ \frac{1}{2} C_0 V^2}= k$