Three equal charge `q_(0)` each are placed at three corners of an equilateral triangle of side 'a'. Find out force acting on one of the charge due to other two charges ?

Answer

To find the force acting on one of the charges due to the other two charges placed at the corners of an equilateral triangle, we can follow these steps:

### Step 1: Identify the Charges and Their Positions
Let the three equal charges be \( q_0 \) placed at the corners of an equilateral triangle with side length \( a \). Denote the charges as \( Q_1, Q_2, \) and \( Q_3 \).

### Step 2: Determine the Forces Acting on One Charge
We will calculate the force acting on charge \( Q_1 \) due to charges \( Q_2 \) and \( Q_3 \). The forces acting on \( Q_1 \) are:
- The force \( F_{12} \) due to charge \( Q_2 \)
- The force \( F_{13} \) due to charge \( Q_3 \)

### Step 3: Calculate the Magnitude of Each Force
Using Coulomb's Law, the magnitude of the force between two point charges is given by:
\[ F = k \frac{|q_1 q_2|}{r^2} \]
For our charges, the distance \( r \) between any two charges is \( a \):
\[ F_{12} = F_{13} = k \frac{q_0^2}{a^2} \]

### Step 4: Determine the Angle Between the Forces
Since the charges are at the corners of an equilateral triangle, the angle \( \theta \) between the forces \( F_{12} \) and \( F_{13} \) is \( 60^\circ \).

### Step 5: Use the Law of Cosines to Find the Resultant Force
The resultant force \( F_1 \) acting on charge \( Q_1 \) can be found using the formula for the resultant of two vectors:
\[ F_1 = \sqrt{F_{12}^2 + F_{13}^2 + 2 F_{12} F_{13} \cos \theta} \]
Substituting \( F_{12} = F_{13} = F \) and \( \theta = 60^\circ \):
\[ F_1 = \sqrt{F^2 + F^2 + 2F^2 \cdot \frac{1}{2}} = \sqrt{2F^2 + F^2} = \sqrt{3F^2} = \sqrt{3} F \]

### Step 6: Substitute the Value of F
Now substituting \( F = k \frac{q_0^2}{a^2} \):
\[ F_1 = \sqrt{3} \left( k \frac{q_0^2}{a^2} \right) \]

### Final Result
Thus, the force acting on charge \( Q_1 \) due to the other two charges is:
\[ F_1 = \sqrt{3} k \frac{q_0^2}{a^2} \]