A point charge +Q is placed at the centroid of an equilateral triangle. When a second charge +Q is placed at a vertex of the triangle, the magnitude of the electrostatic force on the central charge is 8 N. The magnitude of the net force on the central charge when a third charge +Q is placed at another vertex of the triangle is

Options

A. zero
B. `4N`
C. `4(sqrt2) N`
D. `8 N`

Explanation

To find the magnitude of the net force on the central charge when a third charge +Q is placed at another vertex of the triangle, we can follow these steps:

### Step 1: Understand the Configuration
We have an equilateral triangle with a charge +Q at the centroid and two other charges +Q placed at two vertices. The forces acting on the central charge due to these vertex charges will be symmetrical.

### Step 2: Analyze the Forces
When the second charge +Q is placed at one vertex, it exerts a force of 8 N on the central charge. This force acts along the line connecting the vertex charge and the central charge.

### Step 3: Introduce the Third Charge
When we place the third charge +Q at the other vertex of the triangle, it will also exert a force of 8 N on the central charge, directed towards the centroid.

### Step 4: Determine the Angle Between Forces
The angle between the forces exerted by the two vertex charges on the central charge is 120 degrees (since the angle between the lines connecting the centroid to the vertices of an equilateral triangle is 60 degrees, and we have two forces acting in opposite directions).

### Step 5: Calculate the Resultant Force
To find the net force \( F_{net} \) on the central charge due to both vertex charges, we can use the formula for the resultant of two forces:
\[ F_{net} = \sqrt{F_1^2 + F_2^2 + 2F_1F_2 \cos(\theta)} \]
where:
- \( F_1 = 8 \, \text{N} \)
- \( F_2 = 8 \, \text{N} \)
- \( \theta = 120^\circ \)

Substituting the values:
\[ F_{net} = \sqrt{8^2 + 8^2 + 2 \cdot 8 \cdot 8 \cdot \cos(120^\circ)} \]
Since \( \cos(120^\circ) = -\frac{1}{2} \):
\[ F_{net} = \sqrt{64 + 64 - 64} = \sqrt{64} = 8 \, \text{N} \]

### Conclusion
The magnitude of the net force on the central charge when a third charge +Q is placed at another vertex of the triangle is **8 N**.