An air bubble of radius r in water is at a depth h below the...

An air bubble of radius $r$ in water is at a depth $h$ below the water surface at some instant. If $P$ is atmospheric pressure and $d$ and $T$ are the density and surface tension of water, respectively, the pressure inside the bubble will be (A) $P+h d g-\frac{4 T}{r}$ (B) $P+h d g+\frac{2 T}{r}$ (C) $P+h d g-\frac{2 T}{r}$ (D) $P+h d g+\frac{4 T}{r}$

An air bubble of radius $r$ in water is at a depth $h$ below the water surface at some instant. If $P$ is atmospheric pressure and $d$ and $T$ are the density and surface tension of water, respectively, the pressure inside the bubble will be
(A) $P+h d g-\frac{4 T}{r}$
(B) $P+h d g+\frac{2 T}{r}$
(C) $P+h d g-\frac{2 T}{r}$
(D) $P+h d g+\frac{4 T}{r}$

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Key Concepts

Hydrostatic Pressure

Hydrostatic pressure refers to the pressure exerted by a fluid at equilibrium due to the force of gravity. It increases linearly with depth, meaning that any point within a fluid is subject to a pressure that is greater than the pressure at the surface by an amount proportional to the fluid’s density, gravitational acceleration, and the depth beneath the surface.

Surface Tension

Surface tension is the property of a liquid that makes its surface behave like a stretched elastic membrane. It arises from the cohesive forces between liquid molecules and plays a significant role in phenomena involving interfaces, such as the formation and stability of bubbles or droplets.

Laplace Pressure

Laplace pressure is the pressure difference across a curved interface, which is induced by surface tension. For a spherical interface such as a bubble, this pressure difference is given by an expression involving the surface tension and the radius of curvature. In cases with a single interface, this pressure difference is typically expressed as two times the surface tension divided by the radius, reflecting the additional pressure required inside the bubble to balance the external pressure and the constricting effect of surface tension.

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