A uniform charged shell is reassembled in the form of a sphere of same radius but charge uniformly distributed through out of its volume. Find the ratio of initial potential energy to work required for it.

Options

A. 2
B. 3
C. 5
D. 6

Answer

To find the ratio of the initial potential energy of a uniformly charged shell to the work required to reassemble it into a uniformly charged sphere of the same radius, we can follow these steps:

### Step 1: Understand the Initial Configuration
The initial configuration is a uniformly charged shell with charge \( Q \) and radius \( R \). The potential energy \( U_{\text{shell}} \) of a uniformly charged shell can be calculated using the formula:
\[ U_{\text{shell}} = \frac{3}{5} \frac{Q^2}{R} \]

### Step 2: Calculate the Potential Energy of the Final Configuration
The final configuration is a uniformly charged sphere with the same charge \( Q \) and radius \( R \). The potential energy \( U_{\text{sphere}} \) of a uniformly charged sphere can be calculated using the formula:
\[ U_{\text{sphere}} = \frac{3}{5} \frac{Q^2}{R} \]
This is the same as the potential energy of the shell since both configurations have the same charge and radius.

### Step 3: Work Done in Reassembling
When the shell is reassembled into a sphere, the work done \( W \) is equal to the change in potential energy. However, since both configurations have the same potential energy, the work done is:
\[ W = U_{\text{sphere}} - U_{\text{shell}} = 0 \]

### Step 4: Calculate the Ratio of Initial Potential Energy to Work Done
Now we can find the ratio of the initial potential energy to the work done:
\[ \text{Ratio} = \frac{U_{\text{shell}}}{W} \]
Since \( W = 0 \), this ratio is mathematically undefined. However, we can interpret this as the potential energy being finite while the work done is zero.

### Conclusion
Thus, the ratio of the initial potential energy to the work required for reassembly is undefined due to the work done being zero.