Capillary rise

Water climbs a narrow tube against gravity; capilla is Latin for hair, and the thinner the tube the higher the climb.

Water meets glass at an acute angle, so the water surface in a capillary of radius a is concave. The surface is part of a sphere of radius r = a sec θ, giving a pressure difference 2S/r = (2S/a) cos θ across it.

The water just below the meniscus is therefore below atmospheric pressure. Points A (inside the tube, level with the outer surface) and B (on the outer surface) must be at equal pressure, which lifts a column of height h.

hρg = 2S cos θ / a, so h = 2S cos θ/(ρga). The rise comes from surface tension and is larger for a narrower tube, typically a few cm for fine capillaries.

Water in a tube of radius a = 0.05 cm with S = 0.073 N m⁻¹ (θ taken as 0): h = 2 × 0.073/(10³ × 9.8 × 5 × 10⁻⁴) = 2.98 × 10⁻² m = 2.98 cm.

If the meniscus is convex, as for mercury in glass, cos θ is negative and the liquid stands lower inside the capillary than outside: capillary depression.