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)/(2sqrt(3))`
D. `(-Q)/(sqrt(3))`

Explanation

To solve the problem, we need to determine the value of a charge \( q \) that, when placed at the centroid of an equilateral triangle with three equal charges \( Q \) at its vertices, will reduce the interaction energy of the system to zero.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have three equal charges \( Q \) located at the vertices of an equilateral triangle.
- The centroid of the triangle is the point where we will place the charge \( q \).

2. **Distance from Centroid to Vertex**:
- For an equilateral triangle with side length \( A \), the distance from the centroid to each vertex is given by:
\[ L = \frac{A}{\sqrt{3}} \]

3. **Calculating the Interaction Energy**:
- The interaction energy \( U \) of the system with the three charges \( Q \) at the vertices can be calculated using the formula for potential energy between point charges:
\[ U = k \left( \frac{Q \cdot Q}{A} + \frac{Q \cdot Q}{A} + \frac{Q \cdot Q}{A} \right) = 3 \frac{k Q^2}{A} \]
- Here, \( k \) is Coulomb's constant.

4. **Energy Contribution from Charge \( q \)**:
- When we place charge \( q \) at the centroid, it will interact with each of the three charges \( Q \). The potential energy due to these interactions is:
\[ U_q = 3 \cdot k \frac{Q \cdot q}{L} = 3 \cdot k \frac{Q \cdot q}{\frac{A}{\sqrt{3}}} = 3 \cdot k \frac{Q \cdot q \cdot \sqrt{3}}{A} \]

5. **Setting Total Energy to Zero**:
- To reduce the total interaction energy of the system to zero, we need:
\[ U + U_q = 0 \]
- Substituting the expressions we found:
\[ 3 \frac{k Q^2}{A} + 3 \cdot k \frac{Q \cdot q \cdot \sqrt{3}}{A} = 0 \]
- Dividing through by \( 3k/A \) (assuming \( k \neq 0 \) and \( A \neq 0 \)):
\[ Q + q \cdot \sqrt{3} = 0 \]

6. **Solving for Charge \( q \)**:
- Rearranging gives:
\[ q \cdot \sqrt{3} = -Q \]
- Therefore, the value of charge \( q \) is:
\[ q = -\frac{Q}{\sqrt{3}} \]

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