Question

Derive the formula for the capacitance of a parallel plate capacitor, when a dielectric slab is partially filled in between its plates.

Hint:

When a dielectric is partially filled in a capacitor, treat the two sections (dielectric and air) as separate capacitors in series. The total capacitance is calculated using the series combination formula.

Answer 1

Capacitance of a Parallel Plate Capacitor with Partial Dielectric Slab:
Consider a parallel plate capacitor with plate area \( A \) and separation \( d \) between the plates. Let the capacitor be partially filled with a dielectric material of dielectric constant \( \varepsilon_r \) and thickness \( t \). The remaining gap is filled with air. The total capacitance can be considered as the combination of two capacitors in series: 1. The capacitor with dielectric slab: Capacitance \( C_1 \), 2. The capacitor with air gap: Capacitance \( C_2 \). 1. Capacitance with dielectric slab: The capacitance of the part filled with the dielectric is given by: \[ C_1 = \frac{\varepsilon_r \varepsilon_0 A}{t}. \] 2. Capacitance with air gap: The capacitance of the part filled with air is given by: \[ C_2 = \frac{\varepsilon_0 A}{d - t}. \] 3. Total Capacitance (Series Combination): Since the two capacitors are in series, the total capacitance \( C \) is given by: \[ \frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2}. \] Substituting the expressions for \( C_1 \) and \( C_2 \): \[ \frac{1}{C} = \frac{t}{\varepsilon_r \varepsilon_0 A} + \frac{d - t}{\varepsilon_0 A}. \] Simplifying the expression: \[ \frac{1}{C} = \frac{d}{\varepsilon_0 A}. \] Thus, the total capacitance is: \[ C = \frac{\varepsilon_0 A}{d}. \] So, the capacitance of a parallel plate capacitor with a partial dielectric filling is the same as that of an empty capacitor, but with the dielectric slab affecting the effective area.