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 $0.095 \mathrm{~J} / \mathrm{m}^{2}$ and density $10^{3} \mathrm{~kg} / \mathrm{m}^{3}$. The difference between pressure inside the bubble and atmospheric pressure $\_\_\_\_$ $\mathrm{N} / \mathrm{m}^{2}$. (Take $g=10 \mathrm{~m} / \mathrm{s}^{2}$)

JEE Main · 2025 · 23 Jan 2025 · Shift 2 · Fluid Mechanics

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 J / m^{2}

and density

1 0^{3} kg / m^{3}

. The difference between pressure inside the bubble and atmospheric pressure

____

N / m^{2}

.
(Take

g = 10 m / s^{2}

)

Answer: 2190

An air bubble inside a liquid has a single curved liquid surface around it, so the excess pressure due to surface tension is

\frac{2 T}{R}

, exactly as for a liquid drop. Being at depth

h

also adds the hydrostatic pressure

ρ g h

that the bubble must exceed:

Δ P = P_{in} - P_{a t m} = ρ g h + \frac{2 T}{R}

With

ρ = 1 0^{3} kg/m^{3}

g = 10 m/s^{2}

h = 0.2 m

T = 0.095 N/m

and

R = 1 0^{- 3} m

ρ g h = 1 0^{3} \times 10 \times 0.2 = 2000 N/m^{2}

\frac{2 T}{R} = \frac{2 \times 0.095}{1 0 ^{- 3}} = 190 N/m^{2}

Δ P = 2000 + 190 = 2190 N/m^{2}

JEE Main · 2025 · 23 Jan 2025 · Shift 2 · Integer answer