When an electron of charge , e and mass , m moves with velocity v around the nuclear charge 'Ze' describing the circular orbit, the potential energy of the electron is

Options

A. `Ze^(2) //r`
B. `-Ze^(2) //r`
C. `Ze^(2) //r^2`
D. `mv^2//r`

Answer

To find the potential energy of an electron moving in a circular orbit around a nuclear charge 'Ze', we can use the formula for electric potential energy between two point charges.

### Step-by-Step Solution:

1. **Identify the Charges**:
- The charge of the electron (Q1) is \( -e \).
- The nuclear charge (Q2) is \( Ze \).

2. **Use the Formula for Potential Energy**:
The potential energy (U) between two point charges is given by the formula:
\[ U = \frac{k \cdot Q1 \cdot Q2}{r} \]
where \( k \) is Coulomb's constant, \( r \) is the distance between the charges.

3. **Substitute the Values**:
Substituting the values of Q1 and Q2 into the formula:
\[ U = \frac{k \cdot (-e) \cdot (Ze)}{r} \]

4. **Simplify the Expression**:
This simplifies to:
\[ U = \frac{-k \cdot Ze^2}{r} \]

5. **Express in Terms of Proportionality**:
Since \( k \) is a constant, we can express the potential energy in terms of proportionality:
\[ U \propto \frac{-Ze^2}{r} \]

6. **Final Result**:
Therefore, the potential energy of the electron in the circular orbit around the nuclear charge 'Ze' is:
\[ U = -\frac{k \cdot Ze^2}{r} \]