A soap bubble of radius `'r'` and surface tension `'T'` is given a potential of `'V'` volt. If the new radiys `'R'` of the bubble is related to its initial radius by equation.
`P_(0)[R^(3) - r^(3)] + [lambda T [R^(2) - r^(2)] - epsilon_(0) V^(2)R//2 = 0`, where `P_(0)` is the atmospheric pressure. Then find `lambda`

Explanation

To find the value of \(\lambda\) in the equation given for the soap bubble, we can follow these steps:

### Step 1: Understand the Problem
We have a soap bubble with an initial radius \( r \) and surface tension \( T \) that is given a potential \( V \). The relationship between the initial radius and the new radius \( R \) is given by the equation:

\[ P_0 [R^3 - r^3] + [\lambda T [R^2 - r^2] - \epsilon_0 V^2 R / 2] = 0 \]

where \( P_0 \) is the atmospheric pressure.

### Step 2: Analyze the Internal Pressure
For a soap bubble, the internal pressure before expansion can be expressed as:

\[ P_1 = P_0 + \frac{4T}{r} \]

And after expansion, the internal pressure is:

\[ P_2 = P_0 + \frac{4T}{R} + P_e \]

where \( P_e \) is the pressure due to the charge on the bubble.

### Step 3: Determine the Electric Pressure
The electric pressure \( P_e \) can be expressed in terms of the electric field \( E \). The electric field \( E \) due to a potential \( V \) at a distance \( R \) is given by:

\[ E = \frac{V}{R} \]

Thus, the electric pressure can be calculated as:

\[ P_e = \frac{1}{2} \epsilon_0 E^2 = \frac{1}{2} \epsilon_0 \left(\frac{V}{R}\right)^2 = \frac{\epsilon_0 V^2}{2R^2} \]

### Step 4: Set Up the Equation
Now, we can equate the pressures before and after the expansion:

\[ P_0 + \frac{4T}{r} = P_0 + \frac{4T}{R} + \frac{\epsilon_0 V^2}{2R^2} \]

### Step 5: Simplify the Equation
By cancelling \( P_0 \) from both sides, we get:

\[ \frac{4T}{r} = \frac{4T}{R} + \frac{\epsilon_0 V^2}{2R^2} \]

### Step 6: Rearranging Terms
Rearranging gives:

\[ \frac{4T}{r} - \frac{4T}{R} = \frac{\epsilon_0 V^2}{2R^2} \]

### Step 7: Multiply by \( R^2 \)
To eliminate the fractions, multiply through by \( R^2 \):

\[ 4T R^2 \left(\frac{1}{r} - \frac{1}{R}\right) = \frac{\epsilon_0 V^2}{2} \]

### Step 8: Factor Out Terms
This can be rewritten as:

\[ 4T R^2 \left(\frac{R - r}{rR}\right) = \frac{\epsilon_0 V^2}{2} \]

### Step 9: Relate to the Given Equation
Now, we can relate this to the original equation given in the problem. The left-hand side corresponds to the term involving \(\lambda\) in the equation:

\[ \lambda T (R^2 - r^2) \]

### Step 10: Identify \(\lambda\)
By comparing coefficients, we can identify \(\lambda\):

\[ \lambda = \frac{8}{r} \]

### Final Answer
Thus, the value of \(\lambda\) is:

\[ \lambda = \frac{8}{r} \]