Q113 / 150 · MHT CET 2026 (16 April Shift 2) · Physics
A capacitor has capacity C when it's parallel plates are separated by air medium of thickness 'd'. A slab of material of dielectric constant K having area equal to that of plates but thickness $\dfrac{d}{2}$ is inserted between the plates. Capacitance of the capacitor in the presence of slab will be
- A$\text{KC}$
- B$2\text{KC}$
- C$\dfrac{\text{KC}}{\text{K}+1}$
- Correct answerD$\dfrac{2\text{KC}}{\text{K}+1}$
Solution
Initial capacitance of the parallel plate capacitor with air is given by
$C = \dfrac{\epsilon_0 A}{d}$
When a dielectric slab of thickness $t = \dfrac{d}{2}$ and dielectric constant $K$ is inserted, the system can be considered as two capacitors in series.
Capacitance of the air-filled part of thickness $\dfrac{d}{2}$ is
$C_1 = \dfrac{\epsilon_0 A}{\dfrac{d}{2}} = \dfrac{2\epsilon_0 A}{d} = 2C$
Capacitance of the dielectric-filled part of thickness $\dfrac{d}{2}$ is
$C_2 = \dfrac{K\epsilon_0 A}{\dfrac{d}{2}} = \dfrac{2K\epsilon_0 A}{d} = 2KC$
The equivalent capacitance $C'$ of these two capacitors in series is
$C' = \dfrac{C_1 C_2}{C_1 + C_2}$
Substituting the values of $C_1$ and $C_2$:
$C' = \dfrac{(2C)(2KC)}{2C + 2KC}$
$C' = \dfrac{4KC^2}{2C(K + 1)}$
$C' = \dfrac{2KC}{K + 1}$
Answer: $\dfrac{2\text{KC}}{\text{K}+1}$
$C = \dfrac{\epsilon_0 A}{d}$
When a dielectric slab of thickness $t = \dfrac{d}{2}$ and dielectric constant $K$ is inserted, the system can be considered as two capacitors in series.
Capacitance of the air-filled part of thickness $\dfrac{d}{2}$ is
$C_1 = \dfrac{\epsilon_0 A}{\dfrac{d}{2}} = \dfrac{2\epsilon_0 A}{d} = 2C$
Capacitance of the dielectric-filled part of thickness $\dfrac{d}{2}$ is
$C_2 = \dfrac{K\epsilon_0 A}{\dfrac{d}{2}} = \dfrac{2K\epsilon_0 A}{d} = 2KC$
The equivalent capacitance $C'$ of these two capacitors in series is
$C' = \dfrac{C_1 C_2}{C_1 + C_2}$
Substituting the values of $C_1$ and $C_2$:
$C' = \dfrac{(2C)(2KC)}{2C + 2KC}$
$C' = \dfrac{4KC^2}{2C(K + 1)}$
$C' = \dfrac{2KC}{K + 1}$
Answer: $\dfrac{2\text{KC}}{\text{K}+1}$
Hence, the correct answer is option: 'D'.