Capillary Height Calculator (Jurin's Law)
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/3/2026
Jurin's law gives the capillary rise height of a liquid in a thin tube: h = 2×σ×cos(θ) ÷ (ρ×g×r). For water in a 1 mm diameter tube (perfect wetting), the water rises about 3 cm above the outside level — the classic textbook example of capillarity.
Explanation
Jurin's law describes an everyday, easily observable phenomenon: when a very thin tube is dipped into a liquid, the liquid inside doesn't stop at the same level as the liquid around it, but rises (or drops) by a height that depends on the balance between two opposing forces — the liquid's surface tension, which pulls along the tube's walls, and the weight of the raised column of liquid, which opposes it. The outcome of this balance depends crucially on the contact angle θ between the liquid and the tube wall: when the liquid "wets" the wall well (a contact angle close to 0°, like water on clean glass), surface tension wins out and the liquid RISES in the tube, higher still the thinner the tube is — a relationship inversely proportional to the radius, so a tube ten times thinner produces a rise ten times higher. Conversely, a non-wetting liquid (a contact angle greater than 90°, like mercury on glass) is pushed away from the wall rather than drawn to it, and the level inside the tube DROPS below the outside level — this is exactly why old mercury barometers and thermometers had to use tubes wide enough for this effect to stay negligible in the reading. Unlike Pascal's law, which describes how pressure is transmitted through a fluid regardless of tube size, capillarity becomes significant precisely when the tube is very thin — it's the ratio between the surface-tension force (proportional to the tube's perimeter, so to the radius) and the weight of the raised liquid (proportional to volume, so to the radius squared) that explains why the capillary effect intensifies as the tube narrows. This phenomenon also explains how water rises through the thin capillary vessels of plants, or how ink is absorbed by blotting paper or a candle wick.
Example: water in a 1 mm diameter glass tube
Inputs
Surface tension: 0.0728 N/m (water). Contact angle: 0° (perfect wetting). Density: 1000 kg/m³. Tube radius: 0.0005 m (0.5 mm).
Calculation
h = (2 × 0.0728 × cos(0°)) ÷ (1000 × 9.81 × 0.0005) = 0.1456 ÷ 4.905 ≈ 0.02969 m.
Result
The water rises about 2.97 cm above the outside level in this thin tube — the classic example used to illustrate capillarity in physics.
Frequently asked questions
Why is the capillary height negative for some liquids?
A negative height means the liquid DROPS in the tube instead of rising, a phenomenon called capillary depression. This happens when the contact angle exceeds 90° (a non-wetting liquid, cos(θ) becomes negative), as is the case for mercury on glass: surface tension then pushes the liquid away from the tube wall rather than drawing it up, which lowers the level inside below the outside level.
Why is the capillary effect negligible in a large container?
Because the surface-tension force acts along the tube's PERIMETER (proportional to the radius), while the weight of liquid to be lifted depends on the VOLUME of the column (proportional to the radius squared). As the radius grows, the weight therefore increases much faster than the surface-tension force available to lift it, which quickly makes capillary rise imperceptible in a glass or a basin — the same scaling argument that also explains why viscous effects dominate at very small scales in flows at low Reynolds number, even though the same underlying physics is technically still at play.
Does this formula apply to a tube with a non-circular cross-section?
This form of Jurin's law assumes a cylindrical tube with a circular cross-section, where the radius r simply captures both the wetted perimeter and the cross-sectional area. For a tube with a different cross-section (rectangular, between two parallel flat plates), the relationship between the surface-tension force and the weight of the column takes a different geometric form, and a formula adapted to that specific geometry must be used instead.