What is capillarity? Derive an expression for the ascent of liquid in a capillary.

Explanation

The rise or fall of a liquid in a narrow tube is called capillarity.
Let us consider a capillary tube which is held vertically in a beaker containing water, the water rises in the capillary tube to a height h due to surface tension.

The surface tension force `F_(T)` acts along the tangent at the point of contact downwards and its reaction force upwards. Surface tension T, is resolved into two components.
(i) Horizontal component T `sintheta` and
(ii) Vertical component T `costheta` acting upwards, all along the whole circumference of the meniscus.
Tota upward force `=(Tcostheta)(2pir)=2pirTcostheta`
Where `theta` is the angle of contact, r is the radius of the tube. Let `rho` be the density of water and h be the height to which the liquid rises inside the tube. Then,
`{:(("the volume of"),("liquid column"),("in the tube,V")):}={:(("volume of the"),("liquid column of"),("radius r height h")):}+{:(("volume of the liquid of radius"),("r and height h-volume of"),("the hemisphere of radius of r")):}`
`V=pir^(2)h+(pir^(2)xxr-(2)/(3)pir^(3))`
`rArrV=pir^(2)h+(1)/(3)pir^(3)`
The upward force supports the weight of the liquid column above the free surface, therefore,
`2pirTcostheta=pir^(2)(h+(1)/(3)r)rhog`
`rArr" "T=((h+(1)/(3)r)rhog)/(2costheta)`
If the capillary is a very fine tube of radius (i.e., radius is very small) then `(r)/(3)` can be neglected when it is compared to the height h. Therefore,
`T=(rrhogh)/(2costheta)`