Question

Three liquids of densities ${\rho }_1,{\rho }_{2}and{\rho }_3$ $\left(with{\rho }_1>{\rho }_2>{\rho }_3\right),$ having the same value of surface tension T, rise to the same height in three identical capillaries.The angle of contact ${\theta }_1{\theta }_{2}and{\theta }_3$ obey-

• A. $\frac{\pi }{2}<{\theta }_1<{\theta }_2<{\theta }_3<\pi$
• B. $\pi >{\theta }_1>{\theta }_2>{\theta }_3>\frac{\pi }{2}$
• C. $\frac{\pi }{2}>{\theta }_1>{\theta }_2>{\theta }_3⩾0$
• D. $0⩽{\theta }_1<{\theta }_2<{\theta }_3<\frac{\pi }{2}$

Answer 1

The correct option is D $0⩽{\theta }_1<{\theta }_2<{\theta }_3<\frac{\pi }{2}$

Rise of a liquid in a capillary tube is given by ,

$h=\frac{2Tcos\theta }{r\rho g}$

$⟹\cos \theta =hr\rho g/2T$

now given that $h,T$ and $r$ are constants for all three liquids

so $\cos \theta \propto \rho$

${\rho }_1>{\rho }_2>{\rho }_3$

so $\cos {\theta }_1>\cos {\theta }_2>\cos {\theta }_3$

$⟹{\theta }_1<{\theta }_2<{\theta }_3$

now as the liquid is rising in all three capillaries therefore angles of contact will be acute so

$0<{\theta }_1<{\theta }_2<{\theta }_3<\frac{\pi }{2}$