The radii of two soap bubbles are `r_(1) and r_(2) (r_(2) lt r_(1))`. They meet to produce a double bubble. The radius of their common interface is

Options

A. `(r_(1)r_(2))/(r_(1) + r_(2))`
B. `(r_(1) + r_(2))/ (2)`
C. `(r_(1)r_(2))/(r_(1)-r_(2))`
D. `r_(2)-r_(1)`

Explanation

To find the radius of the common interface of two soap bubbles with radii \( r_1 \) and \( r_2 \) (where \( r_2 < r_1 \)), we can follow these steps:

### Step 1: Understand the Pressure Difference in Soap Bubbles
The excess pressure inside a soap bubble is given by the formula:
\[ \Delta P = P_{\text{inside}} - P_{\text{outside}} = \frac{4T}{R} \]
where \( T \) is the surface tension of the soap solution and \( R \) is the radius of the bubble.

### Step 2: Set Up the Equations for Each Bubble
For bubble 1 with radius \( r_1 \):
\[ P_1 - P_0 = \frac{4T}{r_1} \quad \text{(1)} \]
For bubble 2 with radius \( r_2 \):
\[ P_2 - P_0 = \frac{4T}{r_2} \quad \text{(2)} \]
Here, \( P_0 \) is the external pressure.

### Step 3: Set Up the Equation for the Common Interface
For the common interface of the double bubble, the excess pressure difference is:
\[ P_2 - P_1 = \frac{4T}{r} \]
where \( r \) is the radius of the common interface.

### Step 4: Substitute the Pressures from Equations (1) and (2)
Substituting \( P_1 \) and \( P_2 \) from equations (1) and (2):
\[ \left( P_0 + \frac{4T}{r_2} \right) - \left( P_0 + \frac{4T}{r_1} \right) = \frac{4T}{r} \]
This simplifies to:
\[ \frac{4T}{r_2} - \frac{4T}{r_1} = \frac{4T}{r} \]

### Step 5: Simplify the Equation
Dividing through by \( 4T \) (assuming \( T \neq 0 \)):
\[ \frac{1}{r_2} - \frac{1}{r_1} = \frac{1}{r} \]

### Step 6: Rearranging the Equation
Rearranging gives us:
\[ \frac{1}{r} = \frac{1}{r_1} - \frac{1}{r_2} \]

### Step 7: Finding the Radius of the Common Interface
Taking the reciprocal of both sides, we find:
\[ r = \frac{r_1 r_2}{r_1 - r_2} \]

### Final Answer
Thus, the radius of the common interface of the double bubble is:
\[ r = \frac{r_1 r_2}{r_1 - r_2} \]

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