Question

Define capacitance. Derive capacitance of a parallel plate capacitor having dielectric slab between its plates.

Hint:

The dielectric material increases the capacitance by reducing the electric field between the plates, allowing more charge to be stored.

Answer 1

Step 1: Definition of Capacitance.
Capacitance is defined as the ability of a capacitor to store charge per unit potential difference between its plates. It is given by: \[ C = \dfrac{Q}{V} \] where:
- \( C \) is the capacitance, - \( Q \) is the charge on one plate of the capacitor, - \( V \) is the potential difference between the plates.
Step 2: Parallel Plate Capacitor without Dielectric.
For a parallel plate capacitor without a dielectric, the capacitance is given by: \[ C_0 = \dfrac{\epsilon_0 A}{d} \] where: - \( \epsilon_0 \) is the permittivity of free space, - \( A \) is the area of each plate, - \( d \) is the separation between the plates. Step 3: Parallel Plate Capacitor with Dielectric.
When a dielectric material is inserted between the plates of the capacitor, the capacitance increases by a factor of the dielectric constant \( K \) (also called the relative permittivity). The new capacitance is: \[ C = K \cdot C_0 = K \cdot \dfrac{\epsilon_0 A}{d} \] where \( K \) is the dielectric constant of the material. The dielectric reduces the effective electric field between the plates, thereby allowing more charge to be stored for the same applied voltage. Step 4: Conclusion.
The capacitance of a parallel plate capacitor with a dielectric slab between its plates is: \[ C = \dfrac{K \epsilon_0 A}{d} \]