A parallel plate capacitor with plate area A and plate separation… - Vidyadip

MCQ

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 $........... \mu F$

A
$4$
B
$5$
C
$2$
✓
$3$

✓

Answer

Correct option: D.

$3$

d
$\text { For } x =\frac{ d }{3}$

$C _1=\frac{\epsilon_0 A }{\left(\frac{ d / 3}{ k }+\frac{2 d }{3}\right)}=\frac{\epsilon_0 A }{\frac{ d }{12}+\frac{2 d }{3}}$

$=\frac{\epsilon_0 A }{ d } \times\left(\frac{12}{9}\right)$

$C _1=\frac{4}{3} \frac{\epsilon_0 A }{ d }=2 \mu F$

$\text { for } x =\frac{2 d }{3}$

$C _2=\frac{\epsilon_0 A }{\left(\frac{2 d / 3}{ k }+\frac{ d }{3}\right)}=\frac{\epsilon_0 A }{ d } \times 2$

$\Rightarrow \frac{6}{4} \times 2=3 \mu F$

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