The apparent depth of water in cylindrical water tank of diameter `2R cm` is reducing at the rate of `x cm // min ` when water is being drained out at 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 )

Options

A. `xpiR^(2)n_(1)//n_(2)`
B. `xpiR^(2)n_(2)//n_(1)`
C. `2piRn_(1)//n_(2)`
D. `piR^(2)x`

Answer

B

Explanation

Apparent depth `h^(')=(h)/(._(air)mu_(liquid))`
`implies(dh^('))/(dt)=(1)/(._(a)mu_(w))=(1)/(._(a)mu_(w))(dh)/(dt)`
`impliesx=(1)/(._(a)mu_(w))(dh)/(dt)implies(dh)/(dt) =._(a)mu_(w)x`
Now volume of water `V=piR^(2)h`
`implies(dV)/(dt)=piR^(2)(dh)/(dt)=piR^(2). ._(a)mu_(w)x`
`= ._(a)mu_(w)piR^(2)x=(mu_(w))/(mu_(a))pi R^(2)x=((n_(2))/(n_(1)))piR^(2)x`