Dip a strip of paper towel into a puddle and watch the wet line creep upward against gravity. Stand a thin glass tube in a dish of water and the water inside rises noticeably above the surface outside. Nobody pushed it. No pump, no pressure source — just the liquid itself, climbing on its own. The same effect carries sap up the stem of a plant, draws ink from a fountain pen onto paper, and lets a tree wick moisture from soil through roots finer than a hair.
This article explains what surface tension is, why it produces capillary rise, how to calculate the rise height for a given liquid and tube, and where the calculation quietly goes wrong.
Why this calculation matters
Capillary action is not a laboratory curiosity. It governs how fuel moves through a wick, how moisture migrates through concrete and soil, how solder flows into a joint, and how a heat pipe returns its working fluid to the evaporator without any moving parts. Microfluidic chips route droplets through channels narrower than a human hair using nothing but surface forces. Get the surface-tension balance wrong and the device simply does not move liquid where it needs to go.
It also matters in the other direction. In a packed column, a soil sample, or a porous filter, capillary forces can trap liquid you are trying to drain, or pull liquid into places you want to keep dry. Knowing the rise height tells you how tall a wick must be, how fine a pore can be before water bridges it, and whether a coating will self-level or bead up. The calculation is short, but it sits underneath a lot of real engineering decisions.
The core formula
Surface tension, written gamma, is the energy cost of creating new liquid surface, measured in newtons per metre. Molecules inside a liquid are pulled equally in all directions; molecules at the surface are pulled inward, so the surface behaves like a stretched elastic skin that resists being enlarged.
When a liquid meets a solid wall, a second quantity enters: the contact angle theta, the angle the liquid surface makes with the wall. A small contact angle means the liquid wets the solid and is pulled along it; a large angle means the liquid is repelled. Water on clean glass has a contact angle near zero; mercury on glass has an angle well above 90 degrees, which is why mercury is pushed down a tube rather than up.
In a narrow circular tube, the wetted rim pulls the liquid column upward until that pull is balanced by the weight of the raised liquid. Setting the two equal gives the capillary rise height:
h = 2 * gamma * cos(theta) / (rho * g * r)
Here h is the rise height, gamma the surface tension, theta the contact angle, rho the liquid density, g the acceleration of gravity, and r the inner radius of the tube. The result, often called Jurin's law, carries one striking message: rise height is inversely proportional to tube radius. Halve the radius and the liquid climbs twice as high.
A companion quantity is the Laplace pressure, the pressure jump across the curved liquid surface:
delta_p = 2 * gamma / R_curvature
where R_curvature is the radius of the curved meniscus. The tighter the curve, the larger the pressure difference — the same physics that makes a small bubble harder to inflate than a large one.
A worked example
Take water rising in a fine glass capillary tube. Use surface tension gamma = 0.0728 N/m, and assume the water wets the clean glass so the contact angle theta is close to zero, which makes cos(theta) = 1. The tube has an inner radius r = 0.5 mm = 5e-4 m. Water density is rho = 1000 kg/m^3, and g = 9.81 m/s^2.
Step 1 — assemble the numerator.
2 * gamma * cos(theta) = 2 * 0.0728 * 1 = 0.1456
Step 2 — assemble the denominator.
rho * g * r = 1000 * 9.81 * 5e-4 = 4.905
Step 3 — divide to get the rise height.
h = 0.1456 / 4.905 = 0.0297 m
So the water climbs about 30 mm — three centimetres — above the level in the surrounding dish, with no external help at all. That is a substantial height for a tube only one millimetre across.
The radius dependence is worth feeling directly. Shrink the tube to r = 0.1 mm and the rise jumps to roughly 150 mm. Open it up to r = 5 mm and the rise drops to about 3 mm. The narrower the tube, the higher the liquid rises — which is exactly why capillary effects dominate in fine pores and fabrics and become almost invisible in wide pipes.
Common mistakes
Using diameter where the formula wants radius. The equation takes the tube radius r, not the diameter. Slipping in the diameter halves the predicted rise. When in doubt, check the units and sanity-check against the expected order of magnitude.
Forgetting the contact angle entirely. The cos(theta) term is not optional. For water on clean glass it is close to one and easy to overlook, but for a partially wetting surface theta might be 60 degrees, cutting the rise by half. For a non-wetting pair the cosine goes negative and the liquid is depressed, not raised.
Assuming surface tension is a fixed constant. Gamma drops as temperature rises, and even a trace of surfactant or contamination can lower it sharply. A value measured for pure water at 20 C will not describe soapy water or hot water.
Applying Jurin's law to wide tubes. The simple formula assumes the meniscus is a near-perfect spherical cap, which holds only when the tube is narrow compared with the capillary length. In a wide tube the meniscus flattens and the equation overpredicts the rise.
Mixing unit systems for surface tension. Surface tension is often quoted in mN/m or dyn/cm. Water's 72.8 mN/m is 0.0728 N/m. Dropping the conversion throws the answer off by a factor of a thousand.
Try the interactive NovaSolver calculator
Working one case by hand is instructive, but a feel for how rise height trades off against tube size and wetting comes faster when the numbers update live. The Surface Tension & Capillary Rise Simulator on NovaSolver lets you pick a liquid preset or enter custom surface tension, density, contact angle and tube radius, then returns the capillary rise height, the Laplace pressure difference and the Bond number, alongside a diagram that redraws the meniscus and droplet contact-angle shape as you adjust the inputs.
Related calculators
- Archimedes' Buoyancy Calculator — for the other classic surface-and-fluid balance, the upward force on a submerged or floating body.
- Hydrostatic Pressure Calculator — to see how pressure builds with depth, the weight term that capillary rise has to overcome.
- Fluid Pressure Calculator — for converting between pressure, force and area when sizing wicks, seals and small passages.
You can browse the rest in the fluid dynamics tools hub.
Closing note
Surface tension and capillary action turn a liquid's own skin into a pump. The governing relation is compact — rise height equals twice the surface tension times the cosine of the contact angle, divided by density, gravity and tube radius — but it explains a remarkable range of behaviour, from sap in a stem to solder in a joint. Keep three things straight: use the radius, never forget the contact angle, and watch your units on gamma. Do that, and the climb of a liquid up a thin tube becomes a number you can predict instead of a trick you merely admire.
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