An experimental electrostatic transducer uses a parallel-plate capacitor with an air gap and a capacitance of 120 pF120\text{ pF}120 pF.
The capacitor is charged so that it stores a charge of 9.6×10−10 C9.6 \times 10^{-10}\text{ C}9.6×10−10 C; it is then electrically isolated.
A sheet of synthetic polymer of dielectric constant (relative permittivity) 6.06.06.0 is inserted between the plates so that it completely fills the space between them. The polymer 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 square plates of side length LLL and air gap separation ddd needs to be replaced because it is too large.
The same maximum capacitance is required using square plates that have one-quarter (1/41/41/4) of the side length of the original plates (L′=14LL' = \frac{1}{4}LL′=41L).
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 square plates).
