The pressure inside a small air bubble of radius `0.1`mm situated just below the surface of water will be equal to [Take surface tension of water `70xx10^-3Nm^-1` and atmospheric pressure`=1.013xx10^5Nm^-2`]

Options

A. `2.954xx10^3Pa`
B. `1.027xx10^3Pa`
C. `1.027xx10^5Pa`
D. `2.054xx10^5Pa`

Explanation

To find the pressure inside a small air bubble of radius \(0.1 \, \text{mm}\) situated just below the surface of water, we can use the formula for the pressure difference across the interface of a bubble. The relevant parameters are the surface tension of water and the atmospheric pressure.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Radius of the bubble, \( R = 0.1 \, \text{mm} = 0.1 \times 10^{-3} \, \text{m} \)
- Surface tension of water, \( S = 70 \times 10^{-3} \, \text{N/m} \)
- Atmospheric pressure, \( P_{\text{atm}} = 1.013 \times 10^{5} \, \text{N/m}^2 \)

2. **Understand the Pressure Difference Formula:**
- For a bubble just below the surface of a liquid, the pressure difference \( \Delta P \) is given by:
\[ \Delta P = \frac{2S}{R} \]
- Here, the factor is \(2\) because the bubble is in contact with the liquid only on one side (the water surface).

3. **Calculate the Pressure Difference:**
- Substitute the values into the formula:
\[ \Delta P = \frac{2 \times (70 \times 10^{-3})}{0.1 \times 10^{-3}} \]
- Calculate:
\[ \Delta P = \frac{140 \times 10^{-3}}{0.1 \times 10^{-3}} = 1400 \, \text{Pa} \]

4. **Calculate the Pressure Inside the Bubble:**
- The pressure inside the bubble \( P_{\text{inside}} \) is given by:
\[ P_{\text{inside}} = P_{\text{atm}} + \Delta P \]
- Substitute the values:
\[ P_{\text{inside}} = 1.013 \times 10^{5} + 1400 \]
- Convert \(1400 \, \text{Pa}\) to \(N/m^2\) for consistency:
\[ P_{\text{inside}} = 1.013 \times 10^{5} + 0.0014 \times 10^{5} = 1.013 + 0.0014 \times 10^{5} \]
- Add the two pressures:
\[ P_{\text{inside}} = 1.0144 \times 10^{5} \, \text{N/m}^2 \]

5. **Final Result:**
- Therefore, the pressure inside the bubble is approximately:
\[ P_{\text{inside}} \approx 1.0144 \times 10^{5} \, \text{N/m}^2 \]