If two soap bubbles of equal radii r coalesce then the radius of curvature of interface between two bubbles will be

Options

A. r
B. 0
C. inifinity
D. 1/2r

Answer

To find the radius of curvature of the interface between two soap bubbles of equal radii \( r \), we can follow these steps:

### Step 1: Understand the Concept of Soap Bubbles
Soap bubbles are thin films of liquid that enclose air. Each bubble has an outer surface and an inner surface. The pressure inside a soap bubble is higher than the pressure outside due to surface tension.

### Step 2: Identify the Radii of the Bubbles
Let the radius of each soap bubble be \( r \). Since both bubbles are of equal size, we have:
- Radius of Bubble 1, \( r_1 = r \)
- Radius of Bubble 2, \( r_2 = r \)

### Step 3: Use the Formula for Pressure Difference
The pressure difference across a soap bubble is given by the formula:
\[ \Delta P = \frac{4T}{r} \]
where \( T \) is the surface tension of the soap solution and \( r \) is the radius of the bubble.

For two bubbles coalescing, we can denote the pressure inside each bubble:
- Pressure inside Bubble 1, \( P_1 = P_0 + \frac{4T}{r} \)
- Pressure inside Bubble 2, \( P_2 = P_0 + \frac{4T}{r} \)

Here, \( P_0 \) is the atmospheric pressure outside the bubbles.

### Step 4: Analyze the Interface
When the two bubbles coalesce, there is an interface between them. The pressure difference across this interface must be considered. The radius of curvature \( R \) of the interface can be determined using the relation for pressure difference across a curved surface:
\[ \Delta P = \frac{2T}{R} \]

### Step 5: Set Up the Equation
Since both bubbles have the same radius and thus the same internal pressure, the pressure difference across the interface can be expressed as:
\[ \Delta P = P_1 - P_2 = 0 \]
However, we need to consider the curvature of the interface. The effective radius of curvature \( R \) for the interface between two bubbles can be derived from the balance of pressures:
\[ \frac{4T}{r} = \frac{2T}{R} \]

### Step 6: Solve for the Radius of Curvature
Rearranging the equation gives:
\[ R = \frac{2r}{2} = r \]

### Conclusion
Thus, the radius of curvature of the interface between the two soap bubbles when they coalesce is equal to the radius of the bubbles:
\[ R = r \]