A soap bubble (surface tension`=T`) is charged to maximum surface density of charge`= sigma`. When it is just going to burst. Its radius `R` is given by:

Options

A. `R=(sigma^(2))/(8epsilon_(0)T)`
B. `R=8epsilon_(0)T/(sigma^(2))`
C. `R=(sigma)/(sqrt(8(epsilon_(0)T))`
D. `R=(sqrt(8epsilon_(0)T))/(sigma)`

Explanation

To find the radius \( R \) of a soap bubble that is just about to burst when it is charged to a maximum surface charge density \( \sigma \), we can follow these steps:

### Step 1: Understand the pressures involved
When a soap bubble is charged, two types of pressures act on it:
1. **Pressure due to surface tension** \( P_s \)
2. **Pressure due to electric charge** \( P_e \)

### Step 2: Write the expression for pressure due to surface tension
The pressure inside the soap bubble due to surface tension is given by the formula:
\[ P_s = \frac{4T}{R} \]
where \( T \) is the surface tension and \( R \) is the radius of the bubble.

### Step 3: Write the expression for pressure due to electric charge
The pressure inside the soap bubble due to the electric charge 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 4: Set the pressures equal to each other
When the soap bubble is just about to burst, the pressure due to surface tension equals the pressure due to the electric charge:
\[ P_e = P_s \]
Substituting the expressions from Steps 2 and 3, we get:
\[ \frac{\sigma^2}{2\epsilon_0} = \frac{4T}{R} \]

### Step 5: Solve for the radius \( R \)
Rearranging the equation to solve for \( R \):
\[ R = \frac{4T \cdot 2\epsilon_0}{\sigma^2} \]
This simplifies to:
\[ R = \frac{8T}{\sigma^2} \]

### Conclusion
Thus, the radius \( R \) of the soap bubble when it is just going to burst is given by:
\[ R = \frac{8T}{\sigma^2} \]