The ratio of radii of two bubbles is `2 : 1`. What is the ratio of excess pressures inside them ?

Options

A. `1 : 2`
B. `1 : 4`
C. `2 : 1`
D. `4 : 1`

Explanation

To solve the problem, we need to find the ratio of excess pressures inside two bubbles given that the ratio of their radii is 2:1.

### Step-by-Step Solution:

1. **Understand the Formula for Excess Pressure**:
The excess pressure inside a soap bubble is given by the formula:
\[ P = \frac{4T}{r} \]
where \( P \) is the excess pressure, \( T \) is the surface tension, and \( r \) is the radius of the bubble.

2. **Define the Variables**:
Let:
- \( r_1 \) be the radius of the first bubble.
- \( r_2 \) be the radius of the second bubble.
- Given that the ratio of the radii is \( r_1 : r_2 = 2 : 1 \), we can express this as:
\[ r_1 = 2r \quad \text{and} \quad r_2 = r \]
for some radius \( r \).

3. **Calculate the Excess Pressure for Each Bubble**:
- For bubble 1 (with radius \( r_1 \)):
\[ P_1 = \frac{4T}{r_1} = \frac{4T}{2r} = \frac{2T}{r} \]
- For bubble 2 (with radius \( r_2 \)):
\[ P_2 = \frac{4T}{r_2} = \frac{4T}{r} \]

4. **Find the Ratio of Excess Pressures**:
Now, we can find the ratio of the excess pressures:
\[ \frac{P_1}{P_2} = \frac{\frac{2T}{r}}{\frac{4T}{r}} = \frac{2T}{4T} = \frac{2}{4} = \frac{1}{2} \]

5. **Conclusion**:
The ratio of excess pressures inside the two bubbles is:
\[ P_1 : P_2 = 1 : 2 \]

### Final Answer:
The ratio of excess pressures inside the two bubbles is \( 1 : 2 \).

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