Surface tension of a soap solution is T. There is a soap bubble of radius r. Calculate the amount of charge that must be spread uniformly on its surface so that its radius becomes 2r. Atmospheric pressure is `P_0`. Assume that air temperature inside the bubble remains constant.

Answer

To solve the problem of calculating the amount of charge that must be spread uniformly on the surface of a soap bubble so that its radius doubles from \( r \) to \( 2r \), we will follow these steps:

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

### Step 2: Calculate the Initial Pressure
For the initial radius \( r \):
\[ P_1 = P_0 + \frac{4T}{r} \]

### Step 3: Calculate the Pressure After Radius Doubles
When the radius doubles to \( 2r \), the pressure inside the bubble becomes:
\[ P_2 = P_0 + \frac{4T}{2r} = P_0 + \frac{2T}{r} \]

### Step 4: Find the Change in Pressure
The change in pressure \( \Delta P \) as the bubble expands from radius \( r \) to \( 2r \) is:
\[ \Delta P = P_2 - P_1 = \left(P_0 + \frac{2T}{r}\right) - \left(P_0 + \frac{4T}{r}\right) = \frac{2T}{r} - \frac{4T}{r} = -\frac{2T}{r} \]

### Step 5: Relate Change in Pressure to Electrostatic Pressure
The electrostatic pressure \( P_e \) due to the charge on the bubble is given by:
\[ P_e = \frac{\sigma^2}{2\epsilon_0} \]
where \( \sigma \) is the surface charge density and \( \epsilon_0 \) is the permittivity of free space.

### Step 6: Set Up the Equilibrium Condition
At equilibrium, the total pressure inside the bubble must equal the sum of the surface tension pressure and the electrostatic pressure:
\[ \frac{4T}{2r} = \frac{\sigma^2}{2\epsilon_0} - \frac{2T}{r} \]

### Step 7: Solve for Surface Charge Density
Rearranging gives:
\[ \frac{\sigma^2}{2\epsilon_0} = \frac{4T}{2r} + \frac{2T}{r} = \frac{4T + 4T}{2r} = \frac{8T}{2r} = \frac{4T}{r} \]
Thus,
\[ \sigma^2 = \frac{8T\epsilon_0}{r} \]

### Step 8: Calculate Total Charge on the Bubble
The total charge \( Q \) on the surface of the bubble can be calculated using:
\[ Q = \sigma \times \text{Area of the bubble} = \sigma \times 4\pi (2r)^2 = \sigma \times 16\pi r^2 \]
Substituting \( \sigma \):
\[ Q = \sqrt{\frac{8T\epsilon_0}{r}} \times 16\pi r^2 = 16\pi r \sqrt{8T\epsilon_0} \]

### Final Answer
Thus, the amount of charge that must be spread uniformly on the surface of the soap bubble so that its radius becomes \( 2r \) is:
\[ Q = 16\pi r \sqrt{8T\epsilon_0} \]