The radii of two soap bubbles are `r_(i)` and `r_(2)`. In isothermal conditions, two meet together in vacuum. Then the radius kof the resultant bubble is given by

Options

A. `R=(r_1+r_2)//2`
B. `R=r_1(r_1r_2+r_2)`
C. `R^2=r_1^2+r_2^2`
D. `R=r_1+r_2`

Answer

To solve the problem of finding the radius of the resultant bubble when two soap bubbles with radii \( r_1 \) and \( r_2 \) meet in vacuum under isothermal conditions, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have two soap bubbles with radii \( r_1 \) and \( r_2 \).
- They meet in a vacuum, and we need to find the radius \( R \) of the resultant bubble formed.

2. **Surface Energy of Soap Bubbles**:
- The surface energy \( E \) of a soap bubble is given by the formula:
\[ E = 4 \pi r^2 \sigma \]
where \( \sigma \) is the surface tension of the soap solution.

3. **Applying Isothermal Conditions**:
- Under isothermal conditions, the total surface energy remains constant. Therefore, the sum of the surface energies of the two original bubbles equals the surface energy of the resultant bubble:
\[ E_1 + E_2 = E_R \]
- This can be expressed as:
\[ 4 \pi r_1^2 \sigma + 4 \pi r_2^2 \sigma = 4 \pi R^2 \sigma \]

4. **Simplifying the Equation**:
- We can cancel \( 4 \pi \sigma \) from all terms (assuming \( \sigma \) is constant and non-zero):
\[ r_1^2 + r_2^2 = R^2 \]

5. **Finding the Radius of the Resultant Bubble**:
- Rearranging the equation gives us:
\[ R^2 = r_1^2 + r_2^2 \]
- Taking the square root of both sides, we find:
\[ R = \sqrt{r_1^2 + r_2^2} \]

### Final Result:
The radius \( R \) of the resultant bubble is given by:
\[ R = \sqrt{r_1^2 + r_2^2} \]