Question

Obtain an expression, for the risc of a liquid in a capillary tube.

Answer 1

we have to obtain an expression for the rise of a liquid in a capillary tube.

solution : Let's consider a liquid in such a way that its angle of contact with the capillary tube is acute so we have concave surface as shown in 2nd figure. Let S is the surface tension of the liquid , a is the radius of capillary tube , r is the radius of concave surface of the liquid and $\rho$ is the density of the liquid.

now, pressure difference between two sides of the top surface , $P_a-P_0=\frac{2S}{r}$
here, $r=asec\theta$ see figure,

so, $P_a-P_0=\frac{2S}{asec\theta}$

or, $P_a-P_0=\frac{2S}{a}cos\theta$...(1)

now, if we consider point A outside and point B inside then, they must be at the same pressure.
$P_0+\rho hg=P_a$

or, $P_a-P_0=\rho hg$.....(2)

from equations (1) and (2),

$\rho hg=\frac{2S}{a}cos\theta$

thus , height of liquid rises is given by
$h=\frac{2Scos\theta}{\rho ga}$