A small air bubble of radius 'r' is at a depth 'h' below the water surface (density of water`= rho)`. Surface tension of water is T, atmospheric pressure is `p_(0)`. Find pressure inside the air bubble for the condition `r lt lt h`

Explanation

To find the pressure inside the air bubble at a depth 'h' below the water surface, we will follow these steps:

### Step-by-Step Solution:

1. **Understand the Pressure at Depth**:
The pressure at a depth 'h' in a fluid is given by the hydrostatic pressure formula:
\[ P_h = P_0 + \rho g h \]
where:
- \( P_h \) is the pressure at depth 'h',
- \( P_0 \) is the atmospheric pressure at the water surface,
- \( \rho \) is the density of water,
- \( g \) is the acceleration due to gravity,
- \( h \) is the depth below the water surface.

2. **Consider the Effect of Surface Tension**:
The air bubble is subjected to surface tension, which affects the pressure inside the bubble. For a spherical bubble, the pressure difference between the inside and outside of the bubble due to surface tension is given by:
\[ \Delta P = \frac{2T}{r} \]
where:
- \( T \) is the surface tension of water,
- \( r \) is the radius of the bubble.

3. **Relate Inside Pressure to Outside Pressure**:
The pressure inside the bubble \( P \) can be expressed as:
\[ P = P_h + \Delta P \]
Substituting the expressions for \( P_h \) and \( \Delta P \):
\[ P = (P_0 + \rho g h) + \frac{2T}{r} \]

4. **Final Expression for Pressure Inside the Bubble**:
Combining all the terms, we have:
\[ P = P_0 + \rho g h + \frac{2T}{r} \]

### Final Answer:
The pressure inside the air bubble at a depth 'h' is given by:
\[ P = P_0 + \rho g h + \frac{2T}{r} \]