A capacitor of capacitance 45 pF45\text{ pF}45 pF is made from two parallel metal plates separated by an air gap.
The capacitor is charged so that it stores a charge of 5.4×10−10 C5.4 \times 10^{-10}\text{ C}5.4×10−10 C; it is then isolated.
A sheet of dielectric material of dielectric constant 5.05.05.0 is inserted between the plates so that it completely fills the space between them. The dielectric does not discharge the capacitor and does not change the separation of the plates.
Calculate the difference between the initial energy stored by the capacitor and the energy stored when the dielectric has been fully inserted.
Explain how the polar molecules in the dielectric material cause the capacitance of the capacitor to increase when the dielectric is inserted.
In another design, a different parallel-plate capacitor with circular plates of diameter DDD and air gap separation ddd needs to be replaced because it is too large.
The same maximum capacitance is required using circular plates that have one-third (1/31/31/3) of the diameter of the original plates (D′=13DD' = \frac{1}{3}DD′=31D).
Explain, with numerical detail, two ways in which this can be achieved by changing the plate separation or by introducing a dielectric material (while maintaining the new circular plates).
