Jurin Capillarity

Jurin's law: capillary rise

Jurin's law tells you the equilibrium height a liquid rises (or falls) inside a thin tube because of capillarity: h = (2γ·cos θ)/(ρ·g·r), where γ is surface tension (N/m), θ is the contact angle, ρ is density (kg/m³), g = 9.81 m/s² and r is the tube radius (m). On clean glass, water has θ ≈ 0°, so it wets the surface completely. Take a tube with a 0.1 mm radius and you get h ≈ 14.8 cm. Mercury behaves the opposite way: on glass its contact angle is about 140° and cos θ < 0, so the column actually drops below the reservoir level. Capillarity in the xylem on its own lifts sap roughly 1 m. Tall trees reach tens of meters because leaf evapotranspiration generates negative pressure, the cohesion–tension theory at work. And when a towel pulls water up through its fibers, that's capillarity too.

Applications

You'll find it behind paper chromatography, where pigments separate as the solvent climbs the strip. It governs water movement through soils, which matters for irrigation and agronomy. Civil engineers deal with it as rising damp in walls ("umidade ascendente"). It also shows up in micropipetting and capillary blood collection, in the path water takes through a plant from root to xylem to leaves, in the ink feed of a fountain pen, and in how porous media behave when you analyze filters and ceramics.

FAQ

Why does h scale as 1/r? Surface tension pulls along the tube perimeter, which grows with r, while the weight of the lifted column grows with r². Divide the force by the weight and one r cancels out, leaving h ∝ 1/r. That's why a narrower tube lifts water higher.

Can capillarity really push water up a tree? Not on its own. A 100 m sequoia would need a tube narrower than 0.15 µm in radius for that to work. What actually does the job is evapotranspiration: as the leaves lose water, they pull the continuous column upward, held together by the cohesion between water molecules.

How do I stop rising damp in walls? You have to break the capillary path. That means an impermeable layer at the base of the wall, such as a chemical barrier or a PVC blanket, or a ventilated air gap. Painting over the wet patch won't do it. The water just keeps climbing through the masonry.