Two separate air bubbles (radii `0.002 cm` and `0.004`) formed of the same liquid (surface tension `0.07 N//m`) come together to form a double bubble. Find the radius and the sense of curvature of the internal film surface common to both the bubbles.

Explanation

To solve the problem of finding the radius and the sense of curvature of the internal film surface common to two air bubbles, we will follow these steps:

### Step 1: Understand the Problem
We have two air bubbles with radii:
- \( r_A = 0.002 \, \text{cm} = 0.00002 \, \text{m} \)
- \( r_B = 0.004 \, \text{cm} = 0.00004 \, \text{m} \)

Both bubbles are formed in the same liquid with a surface tension \( T = 0.07 \, \text{N/m} \).

### Step 2: Calculate the Excess Pressure in Each Bubble
The excess pressure \( P \) inside a bubble is given by the formula:
\[ P = P_0 + \frac{4T}{r} \]
Where \( P_0 \) is the atmospheric pressure (which will cancel out later).

For bubble A:
\[ P_A = P_0 + \frac{4T}{r_A} = P_0 + \frac{4 \times 0.07}{0.00002} \]

For bubble B:
\[ P_B = P_0 + \frac{4T}{r_B} = P_0 + \frac{4 \times 0.07}{0.00004} \]

### Step 3: Calculate the Excess Pressures
Calculating \( P_A \):
\[ P_A = P_0 + \frac{0.28}{0.00002} = P_0 + 14000 \, \text{Pa} \]

Calculating \( P_B \):
\[ P_B = P_0 + \frac{0.28}{0.00004} = P_0 + 7000 \, \text{Pa} \]

### Step 4: Determine the Pressure Difference
Since \( P_A > P_B \), the pressure difference is:
\[ \Delta P = P_A - P_B = (P_0 + 14000) - (P_0 + 7000) = 7000 \, \text{Pa} \]

### Step 5: Calculate the Radius of the Common Film Surface
The excess pressure in the film surface common to both bubbles can be expressed as:
\[ P = \frac{4T}{r} \]
Setting this equal to the pressure difference:
\[ \frac{4T}{r} = 7000 \]
Substituting \( T = 0.07 \):
\[ \frac{4 \times 0.07}{r} = 7000 \]
Solving for \( r \):
\[ r = \frac{4 \times 0.07}{7000} = \frac{0.28}{7000} \approx 0.00004 \, \text{m} = 0.004 \, \text{cm} \]

### Step 6: Determine the Sense of Curvature
The sense of curvature is determined by the relationship between the pressures in the bubbles. Since \( P_A > P_B \), the internal film surface will be concave towards the smaller bubble (bubble A).

### Final Answer
- The radius of the internal film surface is \( r = 0.004 \, \text{cm} \).
- The sense of curvature is concave towards the smaller bubble.