A sperical ballon od radius R charged uniformly on its surface with surface density `sigma`. Find work done against electric forces in expanding it upto radius 2R.

Explanation

To solve the problem of finding the work done against electric forces in expanding a spherical balloon from radius \( R \) to \( 2R \) with a uniform surface charge density \( \sigma \), we can follow these steps:

### Step-by-Step Solution:

1. **Determine the Charge on the Balloon:**
The total charge \( Q \) on the surface of the balloon can be calculated using the surface charge density \( \sigma \) and the surface area of the sphere.
\[ Q = \sigma \cdot A = \sigma \cdot 4\pi R^2 \]

2. **Calculate the Initial Potential Energy \( U_i \):**
The potential energy \( U \) of a uniformly charged sphere of radius \( R \) is given by:
\[ U_i = \frac{1}{4\pi \epsilon_0} \cdot \frac{Q^2}{2R} \]
Substituting \( Q \):
\[ U_i = \frac{1}{4\pi \epsilon_0} \cdot \frac{(\sigma \cdot 4\pi R^2)^2}{2R} = \frac{1}{4\pi \epsilon_0} \cdot \frac{16\pi^2 \sigma^2 R^4}{2R} = \frac{8\pi \sigma^2 R^3}{\epsilon_0} \]

3. **Calculate the Final Potential Energy \( U_f \):**
When the radius is expanded to \( 2R \), the new potential energy \( U_f \) is:
\[ U_f = \frac{1}{4\pi \epsilon_0} \cdot \frac{Q^2}{2(2R)} = \frac{1}{4\pi \epsilon_0} \cdot \frac{(\sigma \cdot 4\pi (2R)^2)^2}{4R} \]
Simplifying this:
\[ U_f = \frac{1}{4\pi \epsilon_0} \cdot \frac{16\pi^2 \sigma^2 (4R^2)}{4R} = \frac{1}{4\pi \epsilon_0} \cdot \frac{64\pi^2 \sigma^2 R^4}{4R} = \frac{16\pi \sigma^2 R^3}{\epsilon_0} \]

4. **Calculate the Change in Potential Energy \( \Delta U \):**
The work done \( W \) against electric forces is equal to the change in potential energy:
\[ W = \Delta U = U_f - U_i \]
Substituting the values of \( U_f \) and \( U_i \):
\[ W = \frac{16\pi \sigma^2 R^3}{\epsilon_0} - \frac{8\pi \sigma^2 R^3}{\epsilon_0} = \frac{8\pi \sigma^2 R^3}{\epsilon_0} \]

5. **Final Result:**
The work done against electric forces in expanding the balloon from radius \( R \) to \( 2R \) is:
\[ W = \frac{8\pi \sigma^2 R^3}{\epsilon_0} \]