Potential energy of electron present in `He^(+)` is :

Options

A. (a) `(e^(2))/(2 piepsi_(0)r)`
B. (b) `(3e^(2))/(4 piepsi_(0)r)`
C. (c) `(-2e^(2))/(4 piepsi_(0)r)`
D. (d) `(-e^(2))/(4 piepsi_(0)r^(2))`

Answer

To find the potential energy of the electron present in the helium ion \( \text{He}^+ \), we can follow these steps:

### Step 1: Understand the System
The helium ion \( \text{He}^+ \) has a nucleus with 2 protons (since helium has an atomic number of 2) and one electron revolving around it. The potential energy of the electron is influenced by the electrostatic force between the positively charged nucleus and the negatively charged electron.

### Step 2: Use the Formula for Potential Energy
The potential energy \( U \) of an electron in the electric field of a nucleus is given by the formula:
\[ U = -\frac{k \cdot Z \cdot e^2}{r} \]
where:
- \( k \) is Coulomb's constant (\( k = \frac{1}{4\pi \epsilon_0} \)),
- \( Z \) is the atomic number (for helium, \( Z = 2 \)),
- \( e \) is the charge of the electron,
- \( r \) is the distance between the nucleus and the electron.

### Step 3: Substitute the Values
Substituting the values into the formula:
\[ U = -\frac{1}{4\pi \epsilon_0} \cdot \frac{2 \cdot e^2}{r} \]

### Step 4: Simplify the Expression
This expression can be simplified to:
\[ U = -\frac{2e^2}{4\pi \epsilon_0 r} \]

### Conclusion
Thus, the potential energy of the electron in the helium ion \( \text{He}^+ \) is given by:
\[ U = -\frac{2e^2}{4\pi \epsilon_0 r} \]

### Final Answer
The potential energy of the electron present in \( \text{He}^+ \) is:
\[ U = -\frac{2e^2}{4\pi \epsilon_0 r} \]

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