Question

Derive the formula for the capacitance of a parallel plate capacitor partially filled with a dielectric medium between the plates.

Hint:

When a parallel plate capacitor is partially filled with a dielectric, treat the system as two capacitors in parallel: one with the dielectric and one without it. This allows you to calculate the total capacitance by summing the capacitances of the two regions.

Answer 1

Capacitance of a Parallel Plate Capacitor: A parallel plate capacitor consists of two conducting plates separated by a distance \(d\), with area \(A\) of each plate. The capacitance of such a capacitor is given by the formula: \[ C = \frac{\varepsilon_0 A}{d}, \] where: - \( C \) is the capacitance,
- \( \varepsilon_0 \) is the permittivity of free space,
- \( A \) is the area of one of the plates,
- \( d \) is the separation between the plates.
Effect of Dielectric Medium: When a dielectric medium with dielectric constant \( \kappa \) (also known as relative permittivity) partially fills the space between the plates, the capacitance is modified. The dielectric constant \( \kappa \) increases the capacitance by a factor of \( \kappa \). If the dielectric only fills part of the space between the plates, the system can be treated as two capacitors in parallel: 1. One part of the capacitor (with area \( A_1 \)) is filled with the dielectric. 2. The other part (with area \( A_2 \)) is filled with air or some other dielectric with permittivity \( \varepsilon_0 \).
Capacitance of the Region with Dielectric: The capacitance for the part of the capacitor with dielectric is: \[ C_1 = \frac{\kappa \varepsilon_0 A_1}{d_1}, \] where: - \( A_1 \) is the area of the plates covered by the dielectric,
- \( d_1 \) is the distance between the plates covered by the dielectric.
Capacitance of the Region with Air: The capacitance for the part of the capacitor with air (or vacuum) is: \[ C_2 = \frac{\varepsilon_0 A_2}{d_2}, \] where:
- \( A_2 \) is the area of the plates not covered by the dielectric,
- \( d_2 \) is the distance between the plates not covered by the dielectric.
Total Capacitance: Since the areas \( A_1 \) and \( A_2 \) add up to the total area \( A \), and the distances \( d_1 \) and \( d_2 \) add up to the total distance \( d \), the total capacitance is the sum of the capacitances of the two regions: \[ C_{\text{total}} = C_1 + C_2 = \frac{\kappa \varepsilon_0 A_1}{d_1} + \frac{\varepsilon_0 A_2}{d_2}. \] Formula for Capacitance: For the case where the dielectric partially fills the space between the plates (i.e., \( d_1 + d_2 = d \)), the final formula for the capacitance becomes: \[ C = \frac{\varepsilon_0 A}{d} \left( \kappa \frac{d_1}{d} + \frac{d_2}{d} \right). \] This is the general formula for the capacitance of a parallel plate capacitor partially filled with a dielectric medium.