A parallel plate capacitor with plates of unit area and separation is filled with a liquid of dielectric constant . The level of liquid is initially. Suppose the liquid level decreases at a constant speed , the time constant as a function of time is

Question

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]

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(1)

Time constant as a function of time, $d$

Capacitance, $C = \frac{ε_{0} A}{d - T + \frac{T}{k}}$

and, $T (thickness of liquid after time t) = \frac{d}{3} - v t$

$∴$ $T_{C} = C R = \frac{ε_{0} A R}{(d - \frac{d}{3} + v t) + \frac{(\frac{d}{3} - v t)}{k}}$

Given: $A = 1$ and $K = 2$

$∴$ $T_{C} = \frac{ε_{0} \times 1 \times R}{(d - \frac{d}{3} + v t) + \frac{(\frac{d}{3} - v t)}{2}}$

or, $T_{C} = \frac{6 ε_{0} R}{5 d + 3 v t}$

Solution

Q.
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]

1 $\frac{6 ε_{0} R}{5 d + 3 v t}$

2 $\frac{(15 d + 9 v t) ε_{0} R}{2 d^{2} - 3 d v t - 9 v^{2} t^{2}}$

3 $\frac{6 ε_{0} R}{5 d - 3 v t}$

4 $\frac{(15 d - 9 v t) ε_{0} R}{2 d^{2} - 3 d v t - 9 v^{2} t^{2}}$

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Solution

Q. 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]

1 6ε0R5d+3vt

2 (15d+9vt)ε0R2d2-3dvt-9v2t2

3 6ε0R5d-3vt

4 (15d-9vt)ε0R2d2-3dvt-9v2t2