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 $x =\frac{1}{3} d$ and $x =\frac{2 d }{3}$, respectively. If $C _1=2 \mu F$ the value of $C _2$ is $........... \mu F$
JEE MAIN 2023, Diffcult
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$\text { For } x =\frac{ d }{3}$
$C _1=\frac{\epsilon_0 A }{\left(\frac{ d / 3}{ k }+\frac{2 d }{3}\right)}=\frac{\epsilon_0 A }{\frac{ d }{12}+\frac{2 d }{3}}$
$=\frac{\epsilon_0 A }{ d } \times\left(\frac{12}{9}\right)$
$C _1=\frac{4}{3} \frac{\epsilon_0 A }{ d }=2 \mu F$
$\text { for } x =\frac{2 d }{3}$
$C _2=\frac{\epsilon_0 A }{\left(\frac{2 d / 3}{ k }+\frac{ d }{3}\right)}=\frac{\epsilon_0 A }{ d } \times 2$
$\Rightarrow \frac{6}{4} \times 2=3 \mu F$
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