Two soap bubbles are blown. In first soap bubble excess pressure is 4 times of the second soap bubble. The ratio of the radii of the first and second soap bubble is

Options

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

Explanation

To solve the problem of finding the ratio of the radii of two soap bubbles given that the excess pressure in the first bubble is four times that in the second bubble, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Excess Pressure in Soap Bubbles**:
The excess pressure (\( P \)) inside a soap bubble is given by the formula:
\[ P = \frac{4T}{R} \]
where \( T \) is the surface tension and \( R \) is the radius of the bubble. Since there are two surfaces in a soap bubble, the factor of 4 is used.

2. **Set Up the Relationship Between the Two Bubbles**:
Let \( P_1 \) be the excess pressure in the first bubble and \( P_2 \) be the excess pressure in the second bubble. According to the problem:
\[ P_1 = 4P_2 \]

3. **Express the Excess Pressures in Terms of Radii**:
Using the formula for excess pressure, we can express \( P_1 \) and \( P_2 \) in terms of their respective radii \( R_1 \) and \( R_2 \):
\[ P_1 = \frac{4T}{R_1} \]
\[ P_2 = \frac{4T}{R_2} \]

4. **Substitute the Relationship into the Equation**:
From the relationship \( P_1 = 4P_2 \), we can substitute the expressions for \( P_1 \) and \( P_2 \):
\[ \frac{4T}{R_1} = 4 \left(\frac{4T}{R_2}\right) \]

5. **Simplify the Equation**:
Cancel \( 4T \) from both sides (assuming \( T \neq 0 \)):
\[ \frac{1}{R_1} = \frac{4}{R_2} \]

6. **Cross Multiply to Find the Ratio of Radii**:
Cross multiplying gives:
\[ R_2 = 4R_1 \]

7. **Find the Ratio of the Radii**:
To find the ratio \( \frac{R_1}{R_2} \):
\[ \frac{R_1}{R_2} = \frac{R_1}{4R_1} = \frac{1}{4} \]

### Final Answer:
The ratio of the radii of the first and second soap bubble is:
\[ \frac{R_1}{R_2} = \frac{1}{4} \]