A soap bubble A of radius 0.03 m and another bubble B of radius 0.04 m are brought together, so that the combined bubble has a common interface of radius r, then the value of r is

Options

A. 0.24 m
B. 0.48 m
C. 0.12 m
D. none of these

Answer

To find the radius \( R \) of the common interface formed when two soap bubbles of different radii are brought together, we can use the formula for the radius of the common interface:

\[ R = \frac{R_1 \times R_2}{R_1 - R_2} \]

where \( R_1 \) and \( R_2 \) are the radii of the two soap bubbles.

### Step 1: Identify the radii of the soap bubbles
Given:
- Radius of bubble A, \( R_1 = 0.03 \, \text{m} \)
- Radius of bubble B, \( R_2 = 0.04 \, \text{m} \)

### Step 2: Substitute the values into the formula
Substituting the values into the formula:

\[ R = \frac{0.03 \times 0.04}{0.03 - 0.04} \]

### Step 3: Calculate the numerator
Calculate the product of the radii:

\[ 0.03 \times 0.04 = 0.0012 \]

### Step 4: Calculate the denominator
Calculate the difference of the radii:

\[ 0.03 - 0.04 = -0.01 \]

### Step 5: Substitute back into the formula
Now substitute the calculated values back into the formula:

\[ R = \frac{0.0012}{-0.01} \]

### Step 6: Calculate the radius \( R \)
Now perform the division:

\[ R = -0.12 \, \text{m} \]

Since the radius cannot be negative, we take the absolute value:

\[ R = 0.12 \, \text{m} \]

### Final Answer
The radius of the common interface \( R \) is \( 0.12 \, \text{m} \).

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