Q.A capillary of diameter d mm is dipped in water such that the water rises to a height of 30 mm. If the radius of the capillary is made $\frac{2}{3}$ of its previous value, then compute the height up to which water will rise in the new capillary?

Capillary Rise: Why Water Climbs a Straw

Imagine dipping a thin glass straw into a glass of water. You expect the water level inside the straw to match the water level outside. But if the straw is narrow enough, the water actually climbs up the straw, standing higher inside than outside. That's capillary rise. If you try the same straw in a glass of mercury, the opposite happens — the mercury level inside the straw sits lower than outside. That's capillary depression.

Why? The answer lies in two forces you can't see: surface tension and adhesion.

The Intuition: A Sticky, Stretchy Skin

Water molecules are attracted to each other (cohesion), which creates a "skin" at the surface — that's surface tension. But water molecules are also attracted to the glass (adhesion). When the glass is clean, adhesion is stronger than cohesion. So at the point where water meets glass, the water molecules are literally pulled upward along the wall.

In a wide container, this upward pull only affects a tiny rim around the edge. But in a narrow tube, the entire surface is close to the wall. The upward pull from the glass is strong enough to drag the whole water column upward against gravity. The water keeps rising until the weight of the lifted column exactly balances the upward pull from surface tension.

For mercury, adhesion to glass is weak — cohesion dominates. So mercury molecules prefer to stick to each other rather than to the glass, and the surface curves downward at the walls. The net effect is a downward pull, so the mercury column is pushed lower than the outside level.

The Geometry: The Contact Angle

The key to describing this quantitatively is the contact angle $\theta$ — the angle the liquid surface makes with the solid wall, measured inside the liquid.

• For water on clean glass, $\theta \approx 0^\circ$ (the water spreads completely, "wetting" the surface).
• For mercury on glass, $\theta \approx 140^\circ$ (the mercury beads up, barely touching the glass).

The curved liquid surface in the tube is called the meniscus. For a wetting liquid ($\theta < 90^\circ$), the meniscus is concave (curved upward). For a non-wetting liquid ($\theta > 90^\circ$), it's convex (curved downward).

The Precise Statement

Consider a narrow cylindrical tube of radius $r$ dipped into a liquid of density $\rho$ and surface tension $T$. The liquid makes a contact angle $\theta$ with the tube wall. The liquid rises (or falls) to a height $h$ given by:

where $g$ is the acceleration due to gravity.

Where Does This Formula Come From?

The upward force comes from surface tension acting along the entire inner circumference of the tube. Surface tension $T$ is a force per unit length, so the total upward force is:

The $\cos\theta$ factor picks out the vertical component of that force — only the part that actually lifts the liquid.

The downward force is the weight of the liquid column. The column is approximately a cylinder of height $h$ and radius $r$, so its volume is $\pi r^2 h$, and its weight is:

At equilibrium, these balance:

Cancel $\pi r$ from both sides:

And solve for $h$:

What the Formula Tells You

• Narrower tube → higher rise. $h$ is inversely proportional to $r$. Halve the radius, double the height. That's why capillary action is only noticeable in thin tubes.
• Stronger surface tension → higher rise. $h$ is proportional to $T$. Liquids with stronger surface tension (like water) climb higher.
• Denser liquid → lower rise. $h$ is inversely proportional to $\rho$. Mercury is very dense, so even if it wet the glass, it wouldn't climb far.
• Contact angle matters critically. If $\theta > 90^\circ$, $\cos\theta$ is negative, so $h$ becomes negative — that's capillary depression (the liquid falls below the outside level). If $\theta = 90^\circ$, $\cos\theta = 0$, and there is no rise or fall at all.

A Quick Example

Water ($T = 0.073\ \text{N/m}$, $\rho = 1000\ \text{kg/m}^3$, $\theta \approx 0^\circ$) in a glass tube of radius $0.5\ \text{mm} = 5\times10^{-4}\ \text{m}$:

So the water rises about 3 cm. That's measurable and easy to see.

Why This Matters

Capillary rise isn't just a textbook curiosity. It's how water moves through soil, how sap rises in plants (through tiny xylem vessels), how ink is drawn into a blotting paper, and how a paper towel soaks up a spill. Every time you see a liquid defy gravity in a narrow space, you're watching surface tension and adhesion at work.

Capillary rise, including Jurin's law formula, is part of the NCERT/CBSE Class 11 Physics mechanical properties of fluids syllabus, and "capillary rise formula and derivation class 11 physics" is a frequently searched numerical topic before exams. It's also a common JEE Main and NEET important-question application of surface tension.