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.

Answer

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

### Step 1: Calculate the initial potential energy of the hollow charged shell
The potential energy \( U_i \) of a uniformly charged hollow shell with total charge \( Q \) and radius \( R \) is given by the formula:

\[ U_i = \frac{kQ^2}{2R} \]

where \( k \) is Coulomb's constant.

### Step 2: Calculate the final potential energy of the solid sphere
When the charge is uniformly distributed throughout the volume of the sphere, the potential energy \( U_f \) of the solid sphere is given by:

\[ U_f = \frac{3kQ^2}{5R} \]

### Step 3: Calculate the work done in reassembling the shell into a solid sphere
The work done \( W \) in reassembling the shell into a solid sphere is equal to the change in potential energy:

\[ W = U_f - U_i \]

Substituting the values from Steps 1 and 2:

\[ W = \frac{3kQ^2}{5R} - \frac{kQ^2}{2R} \]

### Step 4: Simplify the expression for work done
To simplify \( W \), we need a common denominator, which is \( 10R \):

\[ W = \left(\frac{3kQ^2}{5R} \cdot \frac{2}{2}\right) - \left(\frac{kQ^2}{2R} \cdot \frac{5}{5}\right) \]

This gives:

\[ W = \frac{6kQ^2}{10R} - \frac{5kQ^2}{10R} = \frac{kQ^2}{10R} \]

### Step 5: Find the ratio of initial potential energy to work done
Now, we can find the ratio \( \frac{U_i}{W} \):

\[ \frac{U_i}{W} = \frac{\frac{kQ^2}{2R}}{\frac{kQ^2}{10R}} \]

This simplifies to:

\[ \frac{U_i}{W} = \frac{10}{2} = 5 \]

### Final Answer
Thus, the ratio of the initial potential energy to the work required for reassembly is:

\[ \text{Ratio} = 5 \]

---