Q 12-02-100JEE MainJEE Main 2023 (6 Apr, Shift 1)Medium
A parallel plate capacitor with plate area $A$ and plate separation $d$ is filled with a dielectric material of dielectric constant $K=4$. The thickness of the dielectric material is $x$, where $x<d$. Let $C_1$ and $C_2$ be the capacitance of the system for $x=\dfrac13d$ and $x=\dfrac{2d}{3}$, respectively. If $C_1=2\ \mu\text{F}$, the value of $C_2$ is ______ $\mu\text{F}$.
Numerical value type. Enter your answer.
Answer: 3
$C=\dfrac{\epsilon_0A}{d-x+\frac xK}$.
$x=\dfrac d3$: $d-\dfrac d3+\dfrac d{12}=\dfrac34d$, so $C_1=\dfrac43\dfrac{\epsilon_0A}{d}=2\Rightarrow\dfrac{\epsilon_0A}{d}=1.5\ \mu\text{F}$.
$x=\dfrac{2d}{3}$: $d-\dfrac{2d}3+\dfrac d6=\dfrac d2$, so $C_2=2\times1.5=3\ \mu\text{F}$.
Solution by Sreeraj P, M.Sc Physics