A parallel plate capacitor half filled with a dielectric of dielectric constant 4 has a capacitance of 10 μF. The capacitance without the dielectric is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 6.25 μF
Let \( C_0 = \varepsilon_0A/d \) be the capacitance without the dielectric. Half filling means a dielectric slab of thickness \( d/2 \) with an air gap of \( d/2 \), which is equivalent to two capacitors in series.
\( \displaystyle C_1 = \frac{K\varepsilon_0A}{d/2} = 2KC_0,\qquad C_2 = \frac{\varepsilon_0A}{d/2} = 2C_0 \)
\( \displaystyle \frac{1}{C} = \frac{1}{2KC_0}+\frac{1}{2C_0} = \frac{1+K}{2KC_0} \Rightarrow C = \frac{2K}{1+K}\,C_0 \)
With \( K = 4 \), \( C = \frac{8}{5}C_0 = 1.6\,C_0 \).
\( \displaystyle C_0 = \frac{10\,\mu\mathrm{F}}{1.6} = 6.25\,\mu\mathrm{F} \)