CBSE 2025, Set 1

Question

A parallel plate capacitor has plate area $A$ and plate separation $d$. Half of the space between the plates is filled with a material of dielectric constant $K$ in two ways as shown in the figure. ![Two configurations of dielectric in capacitor](https://www.edvaya.com/images/cbse/physics/2025/set1/q23-capacitor-dielectric-two-ways.webp) *(a) Dielectric fills half the gap vertically (side by side — parallel combination)*
*(b) Dielectric fills half the gap horizontally (one on top of other — series combination)* Find the values of the capacitance of the capacitors in the two cases.

Edvaya · Accepted Answer

The correct answer is Subjective. **Case (a): Dielectric fills half the plate area (side by side — parallel combination)** The capacitor acts as two capacitors in **parallel**: - $C_1$ = capacitor with dielectric K, area $A/2$, separation $d$: $$C_1 = \frac{K\varepsilon_0 (A/2)}{d} = \frac{K\varepsilon_0 A}{2d}$$ - $C_2$ = capacitor without dielectric, area $A/2$, separation $d$: $$C_2 = \frac{\varepsilon_0 (A/2)}{d} = \frac{\varepsilon_0 A}{2d}$$ Total capacitance (parallel): $$\boxed{C_a = C_1 + C_2 = \frac{\varepsilon_0 A}{2d}(K + 1)}$$ --- **Case (b): Dielectric fills half the separation (one on top of other — series combination)** The capacitor acts as two capacitors in **series**: - $C_1$ = capacitor with dielectric K, full area $A$, separation $d/2$: $$C_1 = \frac{K\varepsilon_0 A}{d/2} = \frac{2K\varepsilon_0 A}{d}$$ - $C_2$ = capacitor without dielectric, full area $A$, separation $d/2$: $$C_2 = \frac{\varepsilon_0 A}{d/2} = \frac{2\varepsilon_0 A}{d}$$ Total capacitance (series): $$\frac{1}{C_b} = \frac{1}{C_1} + \frac{1}{C_2} = \frac{d}{2K\varepsilon_0 A} + \frac{d}{2\varepsilon_0 A} = \frac{d(K+1)}{2K\varepsilon_0 A}$$ $$\boxed{C_b = \frac{2K\varepsilon_0 A}{(K+1)d}}$$ > **Comparison:** For $K > 1$: $C_a > C_b$. The parallel configuration always gives higher capacitance than the series configuration for the same dielectric material.

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Physics > Electrostatics And Current Electricity > Electrostatic Potential And Capacitance

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