What is meant by capillarity? Derive …

Capillarity is a physical phenomenon in which liquids flow without the help of gravity.  Liquids even rise to a height against gravity, through narrow tubes.

Capillary action is due to the phenomenon of Surface tension of liquid as well as adhesive forces between liquid molecules and molecules of the narrow tube. Surface tension is due to cohesive attraction among liquid molecules.

Derivation: When a thin (open or closed at the top) tube is inserted into a liquid in a container, the liquid inside the tube rises to a height h above the liquid surface outside.  Let the diameter of the tube be D.  The density of liquid be ρ.  The surface tension of the liquid be S.
Weight of liquid column acting downwards = m g
        W = ρ (πD²/4) h g     --(1)
The surface on the top liquid inside the capillary tube has a trough (cup) like shape. Assume the angle of contact with the walls be Ф.  Surface tension is the contact force per unit length along the circumference of top surface. This force pulls the liquid vertically upwards.
    Force upwards = S * πD * CosФ  ----- (2)
=>    h =  4 S CosФ / (ρ D g)