Two soap bubbles, each of radius r, coaleses in vacuum under isotermal conditions to from a bigger bubble of radius R. Then R is equal to

Options

A. `2^(-1//2)r`
B. `2^(-1//3)r`
C. `2^(1//2)r`
D. 2r

Explanation

To solve the problem of two soap bubbles coalescing into a bigger bubble, we can follow these steps:

### Step 1: Understand the relationship between surface area and volume
When two soap bubbles coalesce, the total surface area of the smaller bubbles is equal to the surface area of the larger bubble formed. The surface area \(A\) of a sphere is given by the formula:
\[ A = 4\pi r^2 \]

### Step 2: Calculate the surface area of the two smaller bubbles
Let the radius of each smaller bubble be \(r\). Therefore, the surface area of one bubble is:
\[ A_1 = 4\pi r^2 \]
For two bubbles, the total surface area \(A_{total}\) is:
\[ A_{total} = 2 \times A_1 = 2 \times 4\pi r^2 = 8\pi r^2 \]

### Step 3: Set the total surface area equal to the surface area of the larger bubble
Let the radius of the larger bubble be \(R\). The surface area of the larger bubble is:
\[ A_{big} = 4\pi R^2 \]
Setting the total surface area of the smaller bubbles equal to the surface area of the larger bubble gives us:
\[ 8\pi r^2 = 4\pi R^2 \]

### Step 4: Simplify the equation
We can cancel \(4\pi\) from both sides:
\[ 2r^2 = R^2 \]

### Step 5: Solve for \(R\)
Taking the square root of both sides, we find:
\[ R = \sqrt{2} r \]
This can also be expressed as:
\[ R = 2^{1/2} r \]

### Final Answer
Thus, the radius \(R\) of the larger bubble formed by the coalescence of two smaller bubbles, each with radius \(r\), is:
\[ R = \sqrt{2} r \]

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