Surface Tension & CapillarityJEE Main

Capillary rise and excess pressure — interactive Physics simulation for IIT-JEE.

Concept

Surface tension γ is energy per unit area (or force per unit length) of a liquid surface. In a narrow tube the curved meniscus sustains a pressure difference that pulls the liquid up if it wets the wall (θ < 90°) or pushes it down if it doesn't (mercury). Curved surfaces always hold an excess pressure on their concave side.

Key formula

h = \frac{2\gamma\cos\theta}{\rho g r}, \qquad \Delta P_{drop} = \frac{2\gamma}{r}, \qquad \Delta P_{soap\,bubble} = \frac{4\gamma}{r}

Derivation

The meniscus is a spherical cap of radius $R = r/\cos\theta$ . Across it, $\Delta P = 2\gamma/R$ . This pressure deficit lifts a column until hydrostatic balance: $\rho g h = 2\gamma\cos\theta / r$ .

A soap bubble has TWO surfaces (inner + outer), doubling the droplet's excess pressure to $4\gamma/r$ .

Scenarios to explore

Surface Tension & Capillarity — Capillary rise, contact angle & excess pressure in bubbles.

Real-world applications

Water transport in plant xylem and paper towels.
Detergents lower γ to help water spread & wet fabrics.
Insects walking on water; needle floating despite ρ > ρ_water.

JEE exam tips

Small bubble = higher pressure: joined bubbles inflate the BIG one.
Tube shorter than h? Liquid doesn't overflow — the meniscus flattens to reduce R.
Work to blow a soap bubble of radius r: W = 8πr²γ (two surfaces).

Common mistakes

Using 2γ/r for a soap bubble — it has two surfaces: 4γ/r.
Dropping cos θ, or using it with the wrong sign for non-wetting liquids.
Halving instead of doubling rise when the radius halves (h ∝ 1/r).

Exam traps to avoid

Excess pressure uses the meniscus radius R, not always the tube radius r (they differ by cos θ).
In a freely falling lift, capillary rise → the liquid climbs to the tube's full length (g_eff = 0).