What happens to electrostatic potential energy of a two electron system, if one electrons brought towards another electron?

Options

A. It become zero
B. It decreases
C. It increases
D. It remains same

Answer

To determine what happens to the electrostatic potential energy of a two-electron system when one electron is brought closer to another electron, we can follow these steps:

### Step 1: Understand the Electrostatic Potential Energy Formula
The electrostatic potential energy (U) between two point charges (Q1 and Q2) separated by a distance (r) is given by the formula:
\[ U = k \frac{Q_1 Q_2}{r} \]
where \( k \) is Coulomb's constant.

### Step 2: Identify the Charges
In our case, we have two electrons. The charge of an electron is negative, specifically \( Q_1 = Q_2 = -e \) (where \( e \) is the magnitude of the charge of an electron).

### Step 3: Write the Initial Potential Energy
When two electrons are at a distance \( r \) apart, the initial potential energy (U_initial) can be expressed as:
\[ U_{initial} = k \frac{(-e)(-e)}{r} = k \frac{e^2}{r} \]
This value is positive because both charges are negative, and the product of two negative charges is positive.

### Step 4: Consider Bringing One Electron Closer
Now, if we bring one electron closer to the other, the distance between them decreases. Let’s say we bring the second electron to a distance \( r' \) (where \( r' < r \)).

### Step 5: Write the New Potential Energy
The new potential energy (U_new) when the distance is \( r' \) becomes:
\[ U_{new} = k \frac{(-e)(-e)}{r'} = k \frac{e^2}{r'} \]
Since \( r' < r \), it follows that \( \frac{1}{r'} > \frac{1}{r} \), which means:
\[ U_{new} > U_{initial} \]

### Step 6: Conclusion
As a result, when one electron is brought closer to another electron, the electrostatic potential energy of the system increases. This is because like charges repel each other, and as they get closer, the potential energy associated with their positions increases.

### Final Answer
The electrostatic potential energy of a two-electron system increases when one electron is brought closer to the other.

---