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Q.

The apparent depth of water in cylindrical water tank of diameter 2R cm is reducing at the rate of x cm/minute when water drained out a constant rate. The amount of water drained in c.c. per minute is ( $n_{1} =$ refractive index of air, $n_{2} =$ refractive index of water)

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a

$x π R^{2} n_{1} / n_{2}$

b

$x π R^{2} n_{2} / n_{1}$

c

$2 π R n_{1} / n_{2}$

d

$π R^{2} x$

answer is B.

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Detailed Solution

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Apparent depth $h' = \frac{h}{μ} = \frac{n_{1}}{n_{2}} h$ $\Rightarrow \frac{d h^{1}}{d t} = \frac{n_{1}}{n_{2}} \frac{d h}{d t} \Rightarrow x = \frac{n_{1}}{n_{2}} \frac{d h}{d t} \Rightarrow \frac{d h}{d t} = \frac{x n_{2}}{n_{1}}$
Rate of volume flux
$\frac{d v}{d t} = π R^{2} \frac{d h}{d t} = π R^{2} (x \frac{n_{2}}{n_{1}})$

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