Find the depth at which an air bubble of radius 0.7 mm will remain in equilibrium in water . Given surface tension of water = `7.0 xx 10^(-2)` N/m.

Answer

To find the depth at which an air bubble of radius 0.7 mm will remain in equilibrium in water, we can use the relationship between pressure, surface tension, and the radius of the bubble.

### Step-by-Step Solution:

1. **Understand the Forces Acting on the Bubble**:
- The bubble experiences an upward buoyant force due to the pressure of the water and a downward force due to the surface tension at the interface of the bubble.

2. **Calculate the Pressure Difference due to Surface Tension**:
- The pressure difference (ΔP) across the surface of a bubble can be given by the formula:
\[ \Delta P = \frac{4 \sigma}{r} \]
where \( \sigma \) is the surface tension and \( r \) is the radius of the bubble.

3. **Substitute the Given Values**:
- Given:
- Radius \( r = 0.7 \, \text{mm} = 0.7 \times 10^{-3} \, \text{m} \)
- Surface tension \( \sigma = 7.0 \times 10^{-2} \, \text{N/m} \)
- Substitute these values into the pressure difference formula:
\[ \Delta P = \frac{4 \times 7.0 \times 10^{-2}}{0.7 \times 10^{-3}} \]

4. **Calculate ΔP**:
- Performing the calculation:
\[ \Delta P = \frac{0.28}{0.7 \times 10^{-3}} = 400 \, \text{Pa} \]

5. **Relate Pressure Difference to Depth**:
- The pressure at a depth \( h \) in a fluid is given by:
\[ P = \rho g h \]
where \( \rho \) is the density of water (approximately \( 1000 \, \text{kg/m}^3 \)) and \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)).
- Set the pressure difference equal to the hydrostatic pressure:
\[ \Delta P = \rho g h \]

6. **Solve for Depth \( h \)**:
- Rearranging the equation gives:
\[ h = \frac{\Delta P}{\rho g} \]
- Substitute the values:
\[ h = \frac{400}{1000 \times 9.81} \]

7. **Calculate \( h \)**:
- Performing the calculation:
\[ h = \frac{400}{9810} \approx 0.0407 \, \text{m} \approx 0.041 \, \text{m} \text{ or } 4.1 \, \text{cm} \]

### Final Answer:
The depth at which the air bubble will remain in equilibrium in water is approximately **4.1 cm**.

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