Question

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

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

Answer 1

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

Rise of a liquid in a capillary tube is given by ,
$h=\frac{2T\cos \theta }{r\rho g}$
or $\cos \theta =hr\rho g/2T$

where $\theta =$ angle of contact ,
$r=$ radius of capillary tube ,
$T=$ surface tension ,
$\rho =$ density of liquid ,
now given that $h$, $T$ and $r$ are constants for all three liquids
so, $$\cos \theta \propto \rho$$
As $\rho$ is increasing $\theta$ will decrease simultaneously and the min. value of $\theta$ is $0$ and maximum is $\frac{\pi }{2}$. So, option c is correct