Question

Find the pressure on an air bubble of radius \( r \) at a height \( h \) from the surface of the water.

• A. \( \frac{4T}{r} \)
• B. \( \frac{2T}{r} \)
• C. \( \frac{T}{r} \)
• D. \( \frac{8T}{r} \)

Hint:

The pressure on an air bubble depends on its size and the height at which it is located, with surface tension playing a significant role in the pressure inside the bubble.

Answer 1

The Correct Option is B

The pressure on an air bubble is influenced by both the pressure due to the depth of the water and the surface tension of the water at the surface of the bubble. The total pressure on the air bubble is given by the sum of the atmospheric pressure, hydrostatic pressure, and the pressure due to surface tension. The pressure due to surface tension is: \[ P = \frac{4T}{r} \] Where: - \( T \) is the surface tension, - \( r \) is the radius of the bubble. In addition to this, the hydrostatic pressure at height \( h \) from the surface of the water is given by: \[ P_{\text{hydrostatic}} = \rho g h \] Where: - \( \rho \) is the density of water, - \( g \) is the acceleration due to gravity. Therefore, the total pressure on the bubble is the sum of the surface tension contribution and the hydrostatic pressure, which results in the expression: \[ P_{\text{total}} = \frac{2T}{r} \]