The pressure inside an air bubble of radius 2 cm formed 20 cm below an open water surface is (Given surface tension of water = `70 xx 10^(-3) Nm^(-1)`)

Options

A. `0.3 xx 10^(5) Nm^(-2)`
B. `0.2 xx 10^(5) Nm^(-2)`
C. `2.03 xx 10^(5) Nm^(-2)`
D. `1.03 xx 10^(5) Nm^(-2)`

Answer

To find the pressure inside an air bubble formed 20 cm below the surface of water, we will follow these steps:

### Step 1: Identify the given values
- Radius of the air bubble, \( R = 2 \, \text{cm} = 0.02 \, \text{m} \)
- Depth of the bubble below the water surface, \( h = 20 \, \text{cm} = 0.20 \, \text{m} \)
- Surface tension of water, \( T = 70 \times 10^{-3} \, \text{N/m} \)
- Density of water, \( \rho = 1000 \, \text{kg/m}^3 \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)
- Atmospheric pressure, \( P_0 = 1.01 \times 10^5 \, \text{Pa} \)

### Step 2: Calculate the pressure due to the height of the water column
The pressure due to the height of the water column can be calculated using the formula:
\[ P_h = \rho g h \]
Substituting the values:
\[ P_h = 1000 \, \text{kg/m}^3 \times 10 \, \text{m/s}^2 \times 0.20 \, \text{m} = 2000 \, \text{Pa} = 0.02 \times 10^5 \, \text{Pa} \]

### Step 3: Calculate the excess pressure inside the bubble
The excess pressure inside an air bubble is given by the formula:
\[ P_e = \frac{2T}{R} \]
Substituting the values:
\[ P_e = \frac{2 \times 70 \times 10^{-3} \, \text{N/m}}{0.02 \, \text{m}} = \frac{140 \times 10^{-3}}{0.02} = 7 \times 10^3 \, \text{Pa} = 0.007 \times 10^5 \, \text{Pa} \]

### Step 4: Calculate the total pressure inside the bubble
The total pressure inside the bubble can be calculated by adding the atmospheric pressure, the pressure due to the water column, and the excess pressure:
\[ P_{\text{total}} = P_0 + P_h + P_e \]
Substituting the values:
\[ P_{\text{total}} = 1.01 \times 10^5 \, \text{Pa} + 0.02 \times 10^5 \, \text{Pa} + 0.007 \times 10^5 \, \text{Pa} \]
\[ P_{\text{total}} = 1.01 \times 10^5 + 0.02 \times 10^5 + 0.007 \times 10^5 \]
\[ P_{\text{total}} = 1.03 \times 10^5 \, \text{Pa} \]

### Final Answer
The pressure inside the air bubble is \( 1.03 \times 10^5 \, \text{Pa} \).

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