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.
The electrostatic energy stored in a capacitor with capacitance [tex]C_0[/tex], with a voltage difference V applied to it, and without dielectric, is given by [tex]U_0 = \frac{1}{2} C_0 V^2 [/tex] 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 [tex]C_k = k C_0[/tex] So, the energy stored now is [tex]U_k = \frac{1}{2}C_k V^2= \frac{1}{2}kC_0 V^2 [/tex]
Therefore, the ratio between the energies stored in the capacitor before and after the introduction of the dielectric is [tex] \frac{U_k}{U_0}= \frac{ \frac{1}{2}kC_0 V^2 }{ \frac{1}{2} C_0 V^2}= k[/tex]
