Two soap bubbles, one of radius `50 mm` and the other of radius `80 mm`, are brought in contact so that they have a common interface. The radius of the curvature of the common interface is

Options

A. `0.003 m`
B. `0.133m`
C. `1.2m`
D. `8.9m`

Explanation

To find the radius of curvature of the common interface between two soap bubbles, we can use the formula derived from the principles of fluid mechanics. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have two soap bubbles with radii \( R_1 = 50 \, \text{mm} \) and \( R_2 = 80 \, \text{mm} \) that are in contact with each other. We need to find the radius of curvature \( R \) of the common interface.

### Step 2: Use the Formula for Radius of Curvature
The formula for the radius of curvature \( R \) of the common interface between two soap bubbles is given by:

\[ \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} \]

This formula arises from the balance of pressures inside the bubbles.

### Step 3: Substitute the Values
Substituting the values of \( R_1 \) and \( R_2 \) into the formula:

\[ \frac{1}{R} = \frac{1}{50} + \frac{1}{80} \]

### Step 4: Find a Common Denominator
To add the fractions, we need a common denominator. The least common multiple of 50 and 80 is 400. Thus, we rewrite the fractions:

\[ \frac{1}{50} = \frac{8}{400}, \quad \frac{1}{80} = \frac{5}{400} \]

### Step 5: Add the Fractions
Now we can add the fractions:

\[ \frac{1}{R} = \frac{8}{400} + \frac{5}{400} = \frac{13}{400} \]

### Step 6: Invert to Find \( R \)
To find \( R \), we take the reciprocal of both sides:

\[ R = \frac{400}{13} \, \text{mm} \]

### Step 7: Calculate the Value
Now, we can calculate the value:

\[ R \approx 30.77 \, \text{mm} \]

### Conclusion
The radius of curvature of the common interface is approximately \( 30.77 \, \text{mm} \).

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