Capillarity
Simple Explanation
Capillarity is the rise (or fall) of a liquid inside a narrow tube, caused by the balance between surface tension pulling the liquid up the tube walls and gravity pulling it back down: h=2γcosθ/(ρgr).
Why Do We Need It?
Capillary action moves water up through soil into plant roots and stems, draws ink through a fountain pen's tip, and lets a paper towel soak up a spill — all from the same balance of forces.
See It
A narrow vertical tube dipped into a wide liquid reservoir, with the liquid level inside the tube risen higher than the reservoir surface, and the height difference h marked
Formula
Capillary Rise (or Fall)
h = 2γcosθ / (ρgr)
How high (or low) a liquid rises inside a narrow tube, due to the balance between surface tension pulling the liquid up the tube walls and gravity pulling it back down.
- h
- — height the liquid rises (or falls) in the tube, in metres (m)
- γ
- — surface tension coefficient of the liquid, in newtons per metre (N/m)
- θ
- — contact angle between the liquid and the tube wall, in degrees
- ρ
- — density of the liquid, in kilograms per cubic metre (kg/m³)
- r
- — radius of the tube, in metres (m)
When to use it: Whenever the height a liquid rises or falls inside a thin (capillary) tube needs to be predicted.
Worked Example
Find the height of capillary rise
Water (γ=0.072 N/m, ρ=1000 kg/m³, θ≈0°) rises in a glass capillary tube of radius 0.0005 m. Find the height of the rise.
Why Does This Work?
Surface tension pulls the liquid up along the tube's inner wall; as the liquid column rises, its increasing weight pulls back down — the liquid keeps rising until the upward surface-tension force exactly balances the downward weight of the risen column, which is exactly the condition this formula expresses.
Real-Life Example
Water moving up through a paper towel
Dipping the corner of a paper towel into a spill causes the liquid to visibly climb upward through the fibers.
The tiny gaps between the paper towel's fibers act like a network of capillary tubes, drawing the liquid upward by exactly the same surface-tension-versus-gravity balance.
Practice
Water (γ=0.072 N/m, ρ=1000 kg/m³, θ≈0°) rises in a capillary tube of radius 0.001 m. Find the height of the rise.
HardCommon mistake
Forgetting that a SMALLER tube radius causes a GREATER capillary rise (since r is in the denominator) — it is easy to assume a wider tube would let more liquid rise higher, but the opposite is true.
Quick Review
- h = 2γcosθ/(ρgr).
- A narrower tube produces a greater rise.
- Caused by the balance between surface tension and the weight of the risen liquid.