Electrostatics 5 Question 24
26. A parallel plate capacitor $C$ with plates of unit area and separation $d$ is filled with a liquid of dielectric constant $K=2$. The level of liquid is $d / 3$ initially. Suppose the liquid level decreases at a constant speed $v$, the time constant as a function of time $t$ is
$(2008,3 M)$
(a) $\frac{6 \varepsilon _0 R}{5 d+3 v t}$
(b) $\frac{(15 d+9 v t) \varepsilon _0 R}{2 d^{2}-3 d v t-9 v^{2} t^{2}}$
(c) $\frac{6 \varepsilon _0 R}{5 d-3 v t}$
(d) $\frac{(15 d-9 v t) \varepsilon _0 R}{2 d^{2}+3 d v t-9 v^{2} t^{2}}$
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Solution:
- After time $t$, thickeness of liquid will remain $\frac{d}{3}-v t$.
Now, time constant as function of time
$$ \tau _c=C R=\frac{\varepsilon _0(1) \cdot R}{\left(d-\frac{d}{3}+v t\right)+\frac{d / 3-v t}{2}} $$
$$ \text { Applying } C=\frac{\varepsilon _0 A}{d-t+\frac{t}{k}} $$
$$ =\frac{6 \varepsilon _0 R}{5 d+3 v t} $$
$\therefore$ Correct option is (a).
