Three equal charges `Q` are placed at the three vertices of an equilateral triangle. What should be the va,ue of a charge, that when placed at the centroid, reduces the interaction energy of the system to zero ?

Options

A. `(-Q)/(2)`
B. `(-Q)/(3)`
C. `(-Q)/(2 sqrt(3))`
D. `(-Q)/(sqrt(3))`

Answer

To solve the problem of determining the value of a charge that, when placed at the centroid of an equilateral triangle with three equal charges \( Q \), reduces the interaction energy of the system to zero, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- We have three equal charges \( Q \) placed at the vertices \( A \), \( B \), and \( C \) of an equilateral triangle.
- The centroid \( G \) of the triangle is where we will place the additional charge \( q \).

2. **Distance from Centroid to Vertices**:
- The distance from the centroid \( G \) to any vertex (say \( A \)) of the triangle can be calculated as:
\[ d = \frac{a}{\sqrt{3}} \]
where \( a \) is the length of the side of the triangle.

3. **Calculating the Interaction Energy**:
- The potential energy \( U \) of the system with three charges \( Q \) at the vertices can be given by:
\[ U = k \left( \frac{Q^2}{a} + \frac{Q^2}{a} + \frac{Q^2}{a} \right) = \frac{3kQ^2}{a} \]
where \( k \) is Coulomb's constant.

4. **Introducing the Charge at the Centroid**:
- When we place a charge \( q \) at the centroid, it will interact with each of the three charges \( Q \). The potential energy due to these interactions is:
\[ U' = 3 \cdot k \cdot \frac{Qq}{d} = 3 \cdot k \cdot \frac{Qq}{\frac{a}{\sqrt{3}}} = \frac{3\sqrt{3}kQq}{a} \]

5. **Condition for Zero Interaction Energy**:
- To reduce the total interaction energy of the system to zero, we need:
\[ U + U' = 0 \]
Substituting the expressions for \( U \) and \( U' \):
\[ \frac{3kQ^2}{a} + \frac{3\sqrt{3}kQq}{a} = 0 \]

6. **Solving for \( q \)**:
- Rearranging the equation gives:
\[ \frac{3\sqrt{3}kQq}{a} = -\frac{3kQ^2}{a} \]
- Dividing both sides by \( \frac{3k}{a} \) (assuming \( k \neq 0 \) and \( a \neq 0 \)):
\[ \sqrt{3}q = -Q \]
- Thus, we find:
\[ q = -\frac{Q}{\sqrt{3}} \]

### Final Answer:
The value of the charge \( q \) that, when placed at the centroid, reduces the interaction energy of the system to zero is:
\[ q = -\frac{Q}{\sqrt{3}} \]