A positive charge .Q. is fixed at a point, A negatively charged of mass .m. and charge .q. is revolving in a circular path of radius radius `r_(1)` with .Q. as the centre. The change in potential energy to change the radius of the circular path from `r_(1) and r_(2)` in joule is

Options

A. zero
B. `(1)/(4piino)Qq[(1)/(r_(1))-(1)/(r_(2))]`
C. `(1)/(piino)Qq[(1)/(r_(1))-(1)/(r_(2))]`
D. `(Qq)/(4piino)[(1)/(r_(2))-(1)/(r_(1))]`

Answer

To solve the problem, we need to calculate the change in potential energy when the radius of the circular path of a negatively charged particle changes from \( r_1 \) to \( r_2 \) while revolving around a fixed positive charge \( Q \).

### Step-by-Step Solution:

1. **Understanding the System**:
- We have a positive charge \( Q \) fixed at a point.
- A negatively charged particle with charge \( q \) and mass \( m \) is revolving in a circular path of radius \( r_1 \) around the charge \( Q \).

2. **Potential Energy Calculation**:
- The potential energy \( U \) of a charge \( q \) in the electric field of another charge \( Q \) at a distance \( r \) is given by:
\[ U = -\frac{k Q q}{r} \]
- Here, \( k \) is Coulomb's constant, \( k = \frac{1}{4 \pi \epsilon_0} \).

3. **Potential Energy at Radius \( r_1 \)**:
- The potential energy \( U_1 \) when the radius is \( r_1 \):
\[ U_1 = -\frac{k Q q}{r_1} \]

4. **Potential Energy at Radius \( r_2 \)**:
- The potential energy \( U_2 \) when the radius is \( r_2 \):
\[ U_2 = -\frac{k Q q}{r_2} \]

5. **Change in Potential Energy**:
- The change in potential energy \( \Delta U \) when the radius changes from \( r_1 \) to \( r_2 \) is given by:
\[ \Delta U = U_2 - U_1 \]
- Substituting the expressions for \( U_1 \) and \( U_2 \):
\[ \Delta U = \left(-\frac{k Q q}{r_2}\right) - \left(-\frac{k Q q}{r_1}\right) \]
- Simplifying this gives:
\[ \Delta U = -\frac{k Q q}{r_2} + \frac{k Q q}{r_1} = k Q q \left(\frac{1}{r_1} - \frac{1}{r_2}\right) \]

6. **Final Expression**:
- Therefore, the change in potential energy when the radius changes from \( r_1 \) to \( r_2 \) is:
\[ \Delta U = k Q q \left(\frac{1}{r_1} - \frac{1}{r_2}\right) \]