A soap bubble of radius `r_(1)` is placed on another soap bubble of radius `r_(2)(r_(1) lt r_(2))`. The radius `R` of the soapy film separating the two bubbles is :

Options

A. `r_(1) + r_(2)`
B. `sqrt(r_(1)^(2) + r_(2)^(2))`
C. `(r_(1)^(3) + r_(2)^(3))`
D. `(r_(2)r_(1))/(r_(2) - r_(1))`

Explanation

To find the radius \( R \) of the soapy film separating two soap bubbles of radii \( r_1 \) and \( r_2 \) (where \( r_1 < r_2 \)), we can use the concept of pressure differences due to surface tension. Here’s a step-by-step solution:

### Step 1: Understand the Pressure Inside the Bubbles
The pressure inside a soap bubble is given by the formula:
\[ P = P_0 + \frac{4\sigma}{r} \]
where \( P_0 \) is the atmospheric pressure, \( \sigma \) is the surface tension, and \( r \) is the radius of the bubble.

### Step 2: Write the Pressure Equations
For the smaller bubble (radius \( r_1 \)):
\[ P_1 = P_0 + \frac{4\sigma}{r_1} \]
For the larger bubble (radius \( r_2 \)):
\[ P_2 = P_0 + \frac{4\sigma}{r_2} \]

### Step 3: Set Up the Pressure Difference
Since the pressure inside the smaller bubble is greater than that inside the larger bubble, we can express the pressure difference when moving from the first bubble to the second bubble:
\[ P_1 - P_2 = \frac{4\sigma}{r_1} - \frac{4\sigma}{r_2} \]

### Step 4: Relate the Pressure Difference to the Soap Film
The pressure difference across the soap film is also given by:
\[ P_1 - P_2 = \frac{4\sigma}{R} \]
where \( R \) is the radius of the soap film.

### Step 5: Equate the Two Expressions
Now we can set the two expressions for the pressure difference equal to each other:
\[ \frac{4\sigma}{R} = \frac{4\sigma}{r_1} - \frac{4\sigma}{r_2} \]

### Step 6: Simplify the Equation
We can cancel \( 4\sigma \) from both sides (assuming \( \sigma \neq 0 \)):
\[ \frac{1}{R} = \frac{1}{r_1} - \frac{1}{r_2} \]

### Step 7: Find a Common Denominator
To combine the fractions on the right side, we find a common denominator:
\[ \frac{1}{R} = \frac{r_2 - r_1}{r_1 r_2} \]

### Step 8: Invert to Find \( R \)
Taking the reciprocal gives us:
\[ R = \frac{r_1 r_2}{r_2 - r_1} \]

### Final Answer
Thus, the radius \( R \) of the soapy film separating the two bubbles is:
\[ R = \frac{r_1 r_2}{r_2 - r_1} \]