Physics · Mechanical Properties Of Fluids · NEET
Water sticks to glass more than it sticks to itself (adhesion beats cohesion). So the water surface inside the tube becomes concave (curves upward). Surface tension acts along this curved surface and pulls the water column up. The water keeps rising until the weight of the lifted column exactly balances this upward surface tension pull. That balance point gives the final height h.
Step 1: The surface tension force acts around the circle where water meets the tube. Length of this circle = 2πr. The vertical part of this force = T × (2πr) × cosθ, because the force acts at angle θ to the wall. Step 2: This upward force lifts a water column of height h. Weight of that column = (volume) × ρ × g = (πr²h) × ρ × g. Step 3: At equilibrium, upward pull = weight: T(2πr)cosθ = πr²hρg. Step 4: Cancel π and one r, then solve for h: h = 2T cosθ / (ρgr).
Thinner tube gives a higher rise. In the formula h = 2T cosθ / (ρgr), the radius r is in the denominator. So h is inversely proportional to r (h ∝ 1/r). Halve the radius and the height doubles. This is why the word 'capillary' comes from the Latin for hair - a hair-thin tube gives a very large rise.
For mercury on glass, cohesion (mercury sticking to itself) beats adhesion. The angle of contact θ is obtuse (more than 90°), so cosθ is negative. Putting a negative cosθ in h = 2T cosθ / (ρgr) makes h negative, which means the liquid level drops below the outside level instead of rising. The mercury surface is also convex (curves down), not concave.
It acts along the line of contact where the liquid meniscus touches the inner wall of the tube - a circle of length 2πr (the circumference). The force per unit length is T, and it pulls along the liquid surface at the angle of contact θ to the wall. Only the vertical component, T cosθ per unit length, does the lifting. Multiply by the circle length 2πr to get the total upward force.
Three liquids of densities ρ1, ρ2 and ρ3 (with ρ1 > ρ2 > ρ3), having the same value of surface tension T, rise to the same height in three identical capillaries. The angles of contact θ1, θ2 and θ3 obey:
Try the real previous-year questions from this chapter — each with the answer and a full solution.
Surface tension T is measured in newtons per metre (N/m), because it is force per unit length. You can also read it as energy per unit area, joules per square metre (J/m²), which gives the same unit.
It is the angle between the tube wall and the liquid surface, measured inside the liquid at the point they touch. For water on clean glass θ is small (near 0°), so cosθ is close to 1 and the rise is large. For mercury θ is obtuse, so cosθ is negative and the level falls.
No. The rise height h depends only on T, θ, ρ, g, and r. It does not depend on how far the tube is pushed into the liquid. If the tube is shorter than the calculated h, the water just reaches the top and the meniscus flattens instead of overflowing.
The concave meniscus means the pressure just under the surface is lower than the pressure just above it by 2T/R (R is the meniscus radius). This lower pressure is what the atmosphere pushes against, forcing water up until the column weight ρgh balances the pressure difference - which again gives h = 2T cosθ / (ρgr).
It usually decreases. Heating a liquid lowers its surface tension T, and since h is directly proportional to T, a smaller T means a smaller rise (density change is minor by comparison).
