The excess pressure inside a soap bubble of radius R is (S is the surface tension)

Options

A. `2S/R`
B. `4S/R`
C. `S/R`
D. `S/2R`

Answer

To find the excess pressure inside a soap bubble of radius \( R \) with surface tension \( S \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Structure of a Soap Bubble**:
- A soap bubble consists of two surfaces: an outer surface and an inner surface. Each surface contributes to the pressure difference inside the bubble.

2. **Pressure Increase Across the First Surface**:
- When we cross the first surface of the soap bubble, the pressure increases due to surface tension. The increase in pressure (\( \Delta P_1 \)) across the first surface can be given by the formula:
\[ \Delta P_1 = \frac{2S}{R} \]
- Here, \( S \) is the surface tension and \( R \) is the radius of the bubble.

3. **Pressure Increase Across the Second Surface**:
- Similarly, when we cross the second surface, there is another increase in pressure (\( \Delta P_2 \)):
\[ \Delta P_2 = \frac{2S}{R} \]

4. **Total Pressure Increase Inside the Bubble**:
- The total increase in pressure inside the bubble (\( \Delta P \)) is the sum of the increases from both surfaces:
\[ \Delta P = \Delta P_1 + \Delta P_2 = \frac{2S}{R} + \frac{2S}{R} = \frac{4S}{R} \]

5. **Excess Pressure Definition**:
- The excess pressure inside the soap bubble is defined as the difference between the pressure inside the bubble and the pressure outside the bubble. Therefore, the excess pressure \( P_{excess} \) is:
\[ P_{excess} = \frac{4S}{R} \]

### Final Answer:
Thus, the excess pressure inside a soap bubble of radius \( R \) with surface tension \( S \) is given by:
\[ P_{excess} = \frac{4S}{R} \]