A charged soap bobbe having surface charge density `sigma` and radius r. If the pressure inside and outsides the soap bubble is the same, then the surface tension of the soap solution is

Options

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

Answer

To solve the problem of finding the surface tension of a charged soap bubble, we will follow these steps:

### Step 1: Understand the Pressure Difference
We start by noting that for a soap bubble, the pressure difference between the inside and outside of the bubble is given by the formula:
\[ \Delta P = P_i - P_o = \frac{4T}{r} \]
where \( P_i \) is the pressure inside the bubble, \( P_o \) is the pressure outside the bubble, \( T \) is the surface tension, and \( r \) is the radius of the bubble.

### Step 2: Consider Electrostatic Pressure
Since the bubble is charged, we need to consider the electrostatic pressure acting on the bubble due to the surface charge density \( \sigma \). The electrostatic pressure \( P_e \) can be expressed as:
\[ P_e = \frac{\sigma^2}{2\epsilon_0} \]
where \( \epsilon_0 \) is the permittivity of free space.

### Step 3: Set Up the Balance Condition
According to the problem, the pressure inside and outside the soap bubble is the same, which means:
\[ P_i - P_o = P_e \]
Substituting the expressions for pressure difference and electrostatic pressure, we get:
\[ \frac{4T}{r} = \frac{\sigma^2}{2\epsilon_0} \]

### Step 4: Solve for Surface Tension \( T \)
Now we can rearrange the equation to solve for the surface tension \( T \):
\[ 4T = \frac{\sigma^2 r}{2\epsilon_0} \]
\[ T = \frac{\sigma^2 r}{8\epsilon_0} \]

### Step 5: Conclusion
Thus, the surface tension of the soap solution is given by:
\[ T = \frac{\sigma^2 r}{8\epsilon_0} \]

### Final Answer
The surface tension of the soap solution is \( \frac{\sigma^2 r}{8\epsilon_0} \).

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