JEE Mains Physics · Mechanical Properties of Fluids

Excess Pressure and Capillary Rise

A curved liquid surface has a higher pressure on its concave side, 2T/r for one surface and 4T/r for a soap bubble, and the same curvature lifts a liquid up a narrow tube to a height h = 2T cos θ/ρgr.

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Why this matters

Twenty-four PYQs, six of them numeric, and two from 2026. Eleven are excess pressure: an air bubble at a depth, ratios of soap bubbles, a liquid column that balances one, and two bubbles in contact or one inside another; six use surface tension to hold up a liquid, in straight, tilted and U-shaped tubes and for a drop floating half-immersed; seven are statement questions on contact angle, the shape of the meniscus, and what heat and soap do to the rise.

Concept 1 of 3: Excess pressure inside drops and bubbles

A curved surface under tension squeezes what is inside it, like a stretched balloon. So the pressure inside is higher than outside, and the smaller the radius, the larger the difference. A drop, or an air bubble inside a liquid, has one surface:

2T/r

. A soap bubble in air has two:

4T/r

Definition

One surface (a drop; an air bubble in a liquid): $\Delta P = \dfrac{2T}{r}$ . Two surfaces (a soap bubble in air): $\Delta P = \dfrac{4T}{r}$ .
Air bubble at depth h in a liquid: $P_{\text{in}} - P_0 = \rho g h + \dfrac{2T}{r}$ .
$\Delta P \propto 1/r$ : if A's excess pressure is $1/k$ of B's, then $r_A = k\,r_B$ and $V_A = k^{3}V_B$ .
A liquid column balancing a soap bubble: $\rho g h = \dfrac{4T}{r}$ .
Two soap bubbles in contact: the common surface has radius $\dfrac{r_1 r_2}{r_2 - r_1}$ and bulges into the larger bubble.
A bubble inside a bubble: the excess pressures add, $\dfrac{4T}{r_1} + \dfrac{4T}{r_2}$ for the inner one.
Two bubbles merging at constant temperature, the atmosphere neglected: $R^{2} = R_1^{2} + R_2^{2}$ .

Excess pressure

\Delta P = \frac{2T}{r}\ \text{(one surface)}, \qquad \Delta P = \frac{4T}{r}\ \text{(soap bubble)}

Worked example

An air bubble of radius 0.5 mm is 15 cm below the surface of water. Surface tension is 0.072 N/m. By how much does the pressure inside the bubble exceed atmospheric pressure? (

\rho = 1000\ \text{kg/m}^{3}

g = 10\ \text{m/s}^{2}

)

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2025 · 23 January 2025 · Q96Moderate

Example 1 · Mechanical Properties of Fluids · Excess Pressure and Capillary Rise

An air bubble of radius 1.0 mm is observed at a depth of 20 cm below the free surface of a liquid having surface tension

0.095\text{ }J/m^{2}

and density

10^{3}\text{ }kg/m^{3}

. The difference between pressure inside the bubble and atmospheric pressure

\_\_\_\_N/m^{2}

. (Take

g = 10\text{ }m/s^{2}

)

An air bubble in a liquid has ONE surface

Only a soap bubble in air has a film with two faces. An air bubble under water is bounded by water on one side only, so its excess pressure is 2T/r, not 4T/r.

Add the depth term

At depth h, the liquid just outside the bubble is already at P₀ + ρgh. The pressure inside exceeds atmospheric by ρgh + 2T/r.

The common surface uses the difference of the radii

It is r₁r₂/(r₂ − r₁). The sum r₁r₂/(r₁ + r₂) looks like a parallel-resistor formula and is a common wrong option.

Drill 10 more on excess pressure inside drops and bubbles

Concept 2 of 3: Capillary rise

In a narrow glass tube water clings to the glass and its surface curves up. Just under a curved surface the pressure is lower, so the liquid climbs until the weight of the column makes up the difference. A narrower tube curves the surface more and lifts the liquid higher.

Definition

$h = \dfrac{2T\cos\theta}{\rho g r}$ . For water on clean glass $\theta \approx 0$ , so $h = \dfrac{2T}{\rho g r}$ .
$h \propto \dfrac{T}{\rho r}$ . Doubling both T and ρ leaves h unchanged. Small changes: $\dfrac{\Delta h}{h} = \dfrac{\Delta T}{T} - \dfrac{\Delta\rho}{\rho} - \dfrac{\Delta r}{r}$ .
Tilted tube: the vertical height stays h. The length of liquid along the tube is $\dfrac{h}{\cos\alpha}$ , with α the tilt from the vertical.
U-tube with limbs of radii $r_1 < r_2$ : the narrow limb stands higher by $\Delta h = \dfrac{2T}{\rho g}\left(\dfrac{1}{r_1} - \dfrac{1}{r_2}\right)$ .
Hot water rises less than cold, because surface tension falls as the temperature rises.
Surface tension as a force along a line of contact: $F = T \times$ length. A drop of radius R floating half-immersed is held up by $2\pi R T$ as well as by the upthrust.

Capillary rise

h = \frac{2T\cos\theta}{\rho g r}

Worked example

Water (surface tension 0.07 N/m, contact angle 0,

\rho = 1000\ \text{kg/m}^{3}

) stands in a glass capillary of radius 0.2 mm. How high does it rise? How long is the column if the tube is tilted at 60° to the vertical? (

g = 10\ \text{m/s}^{2}

)

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The same idea in a real exam question:

JEE Mains · 2021 · Paper 18 · Q5Moderate

Example 2 · Mechanical Properties of Fluids · Excess Pressure and Capillary Rise

Two narrow bores of diameter

5.0\text{ }mm

and

8.0\text{ }mm

are joined together to form a U-shaped tube open at both ends. If this U-tube contains water, what is the difference in the level of two limbs of the tube. [Take surface tension of water

T = 7.3 \times10^{- 2}Nm^{- 1}

, angle of contact

= 0,\text{ }g = 10{\text{ }ms}^{- 2}

and density of water

= 1.0 \times10^{3}\text{ }kg{\text{ }m}^{- 3}

]

Bores are often given as diameters

Halve each diameter before using 1/r₁ − 1/r₂. Using diameters halves the answer.

A tilted tube keeps the vertical height

The liquid still rises to the same vertical height h; only the length along the tube grows, to h/cos α.

Drill 5 more on capillary rise

Concept 3 of 3: Contact angle, meniscus shape and surface-tension facts

The contact angle is the angle, inside the liquid, between the solid and the liquid surface where they meet. It is set by a contest: cohesion holds the liquid together, adhesion pulls it onto the solid. So it belongs to the pair of materials, not to the liquid alone, and its cosine decides whether the liquid rises or falls in a tube.

Definition

Acute angle: adhesion wins, the meniscus is concave and the liquid rises.
Obtuse angle: cohesion wins, the meniscus is convex and the liquid falls.
Exactly 90°: $\cos\theta = 0$ , the surface is flat and the liquid neither rises nor falls.
Surface tension comes from the extra energy of the molecules AT THE SURFACE compared with those inside.
Soap and detergent LOWER the surface tension of water.
Heating lowers surface tension, so the capillary rise is smaller in hot water. Heating also lowers a liquid's viscosity, so hot water flows faster.

Case	Contact angle	Meniscus and capillary
Water on clean glass	About 0°, acute	Concave; the water rises
Mercury on glass	About 140°, obtuse	Convex; the mercury falls below the outside level
Water on grease or wax	Obtuse	Water forms beads and does not wet the surface, so washing with water alone cannot remove a grease stain
Soapy water on grease	Made acute by the detergent	The water spreads and wets the grease; detergent lowers T
Adhesion and cohesion in balance	90°	Flat surface; no rise and no fall A liquid that neither rises nor falls has a contact angle of 90°, not 0°.

h = 2T\cos\theta/\rho g r

, the sign of

\cos\theta

gives the direction: positive for an acute angle (rise), zero at 90°, negative for an obtuse angle (fall).

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The same idea in a real exam question:

JEE Mains · 2024 · 4 April 2024 · Q100Moderate

Example 3 · Mechanical Properties of Fluids · Excess Pressure and Capillary Rise

Given below are two statements: Statement I: The contact angle between a solid and a liquid is a property of the material of the solid and liquid as well. Statement II: The rise of a liquid in a capillary tube does not depend on the inner radius of the tube. In the light of the above statements, choose the correct answer from the options given below:

The contact angle belongs to the pair

The same water makes about 0° with clean glass and an obtuse angle with wax. A statement that it depends on the liquid alone, or on the solid alone, is false.

Soap water has LOWER surface tension

Detergents are added to water because they lower its surface tension. A statement that soap water has the higher surface tension is false.

Gases are less viscous than liquids

Statement pairs here often slip in a viscosity fact. The viscosity of a gas is far smaller than that of a liquid, so a statement saying the opposite is false.

Drill 6 more on contact angle, meniscus shape and surface-tension facts

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Excess pressure inside drops and bubbles
Excess pressure
$\Delta P = \frac{2T}{r}\ \text{(one surface)}, \qquad \Delta P = \frac{4T}{r}\ \text{(soap bubble)}$
Capillary rise
Capillary rise
$h = \frac{2T\cos\theta}{\rho g r}$

Reference tables (1)

Contact angle, meniscus shape and surface-tension facts5 rows

Case	Contact angle	Meniscus and capillary
Water on clean glass	About 0°, acute	Concave; the water rises
Mercury on glass	About 140°, obtuse	Convex; the mercury falls below the outside level
Water on grease or wax	Obtuse	Water forms beads and does not wet the surface, so washing with water alone cannot remove a grease stain
Soapy water on grease	Made acute by the detergent	The water spreads and wets the grease; detergent lowers T
Adhesion and cohesion in balance	90°	Flat surface; no rise and no fall A liquid that neither rises nor falls has a contact angle of 90°, not 0°.

h = 2T\cos\theta/\rho g r

, the sign of

\cos\theta

gives the direction: positive for an acute angle (rise), zero at 90°, negative for an obtuse angle (fall).

Watch out for (8)

An air bubble in a liquid has ONE surface
→ Excess pressure inside drops and bubbles
Add the depth term
→ Excess pressure inside drops and bubbles
The common surface uses the difference of the radii
→ Excess pressure inside drops and bubbles
Bores are often given as diameters
→ Capillary rise
A tilted tube keeps the vertical height
→ Capillary rise
The contact angle belongs to the pair
→ Contact angle, meniscus shape and surface-tension facts
Soap water has LOWER surface tension
→ Contact angle, meniscus shape and surface-tension facts
Gases are less viscous than liquids
→ Contact angle, meniscus shape and surface-tension facts

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