Three identical charges are fixed at the corners of an equilateral triangle . If the force between any two charges is `F`, then the net electric force on each will be

Options

A. `sqrt(2) F`
B. `2 F`
C. `sqrt(3) F`
D. `3 F`

Explanation

Let's solve the problem step by step.

### Step 1: Understand the Problem
We have three identical charges, \( Q \), placed at the corners of an equilateral triangle. The force between any two charges is \( F \). We need to find the net electric force on each charge.

### Step 2: Diagram and Forces
Consider an equilateral triangle with charges \( Q \) at vertices A, B, and C. The side length of the triangle is \( d \).

1. The force between charges at A and B is \( F \).
2. The force between charges at B and C is \( F \).
3. The force between charges at A and C is \( F \).

### Step 3: Resolve Forces into Components
For a charge at vertex A, the forces due to charges at B and C need to be resolved into components.

1. The force \( F \) due to charge at B acts along the line AB.
2. The force \( F \) due to charge at C acts along the line AC.

Since AB and AC form a \( 60^\circ \) angle (equilateral triangle), we resolve these forces into x and y components.

### Step 4: Components of Forces
1. The force \( F \) along AB can be resolved into:
- \( F \cos(30^\circ) \) along the x-axis
- \( F \sin(30^\circ) \) along the y-axis

2. The force \( F \) along AC can be resolved into:
- \( F \cos(30^\circ) \) along the x-axis (opposite direction)
- \( F \sin(30^\circ) \) along the y-axis

### Step 5: Net Force Calculation
Since the x-components of the forces along AB and AC cancel each other out, we only need to consider the y-components.

1. The y-component of the force due to charge at B is \( F \sin(30^\circ) \).
2. The y-component of the force due to charge at C is \( F \sin(30^\circ) \).

Both y-components add up because they are in the same direction:
\[ F_{\text{net}} = 2 \times F \sin(30^\circ) \]

### Step 6: Substitute Values
We know that \( \sin(30^\circ) = \frac{1}{2} \):
\[ F_{\text{net}} = 2 \times F \times \frac{1}{2} = F \]

### Step 7: Final Answer
The net electric force on each charge is:
\[ F_{\text{net}} = F \]