Three point charges q are placed at three vertices of an equilateral triangle of side a. Find magnitude of electric force on any charge due to the other two.

Explanation

To find the magnitude of the electric force on any charge due to the other two charges placed at the vertices of an equilateral triangle, we can follow these steps:

### Step 1: Identify the Charges and Configuration
We have three point charges, each of magnitude \( q \), placed at the vertices of an equilateral triangle with side length \( a \). Let's label the charges as \( A \), \( B \), and \( C \).

### Step 2: Calculate the Force Between Two Charges
The electric force between any two point charges can be calculated using Coulomb's Law, which states:
\[ F = k \frac{q_1 q_2}{r^2} \]
For our case, the force between charge \( A \) and charge \( B \) (denoted as \( F_{AB} \)) is:
\[ F_{AB} = k \frac{q \cdot q}{a^2} = k \frac{q^2}{a^2} \]
Similarly, the force between charge \( A \) and charge \( C \) (denoted as \( F_{AC} \)) is also:
\[ F_{AC} = k \frac{q^2}{a^2} \]

### Step 3: Determine the Angles
In an equilateral triangle, the angle between the lines connecting any charge to the other two charges is \( 60^\circ \).

### Step 4: Calculate the Net Force on Charge A
To find the net force acting on charge \( A \) due to charges \( B \) and \( C \), we need to consider the vector addition of the forces \( F_{AB} \) and \( F_{AC} \).

Using the cosine rule for vector addition:
\[ F_{\text{net}} = \sqrt{F_{AB}^2 + F_{AC}^2 + 2 F_{AB} F_{AC} \cos(60^\circ)} \]

Since \( F_{AB} = F_{AC} = F \):
\[ F_{\text{net}} = \sqrt{F^2 + F^2 + 2 F^2 \cdot \frac{1}{2}} \]
\[ = \sqrt{F^2 + F^2 + F^2} = \sqrt{3F^2} = \sqrt{3}F \]

### Step 5: Substitute the Value of F
Substituting \( F = k \frac{q^2}{a^2} \):
\[ F_{\text{net}} = \sqrt{3} \left(k \frac{q^2}{a^2}\right) \]
Thus, the magnitude of the electric force on any charge due to the other two charges is:
\[ F_{\text{net}} = k \frac{q^2}{a^2} \sqrt{3} \]

### Final Answer
The magnitude of the electric force on any charge due to the other two is:
\[ F_{\text{net}} = k \frac{q^2 \sqrt{3}}{a^2} \]

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