A parallel-plate capacitor containing a polar dielectric with a dielectric constant $k$, connected to a battery with an e.m.f. $E$, is moved from room temperature ($25\text{ }^\circ\text{C}$) into a freezer. What happens to the electrostatic potential energy of the capacitor? (Assume thermal expansion of the plates is neglected and the battery's e.m.f. remains constant.)
Answer
At room temperature (25 °C), thermal energy causes molecules to vibrate and collide violently and randomly. This thermal chaos fights against the electric field from the battery, constantly knocking the molecular dipoles out of alignment.
When capacitor is placed in a cool freezer, the thermal kinetic energy of the molecules drops significantly. With less thermal agitation disrupting them, the electric field becomes much more effective at lining up those molecular dipoles in an orderly fashion.
Because more dipoles successfully align with the field, the material's internal polarization increases. A higher degree of polarization means the material can store more bound charge, which directly translates to a higher dielectric constant ($k$).
The capacitance of a parallel plate capacitor filled with a dielectric is given by: $$C = \frac{k \varepsilon_0 A}{d}$$
Because thermal expansion is to be neglected, the surface area of the plates ($A$) and the distance between them ($d$) remain constant.
Consequently, because $k$ increases while $A$ and $d$ remain constant, the overall capacitance ($C$) increases.
Since the voltage ($V$) is constant and the capacitance ($C$) has increased due to the higher dielectric constant at lower temperatures, the electrostatic potential energy ($U$) increases.
The battery forces the voltage across the capacitor to remain constant and equal to its e.m.f. ($V = E$).
The electrostatic potential energy ($U$) stored in a capacitor connected to a battery is given by:$$U = \frac{1}{2} C V^2$$
While the thermal energy of the material's molecules decreases in the freezer, the electrical energy stored in the electric field increases because the battery injects more charge into the system to accommodate the higher capacitance.
The animation below shows dielectric thermal agitation calming down as temperature reduces.
