Three identical charges are placed at the vertices of an equilateral triangle. The force experienced by each charge, (if `k = 1/4piepsi_(0)`) is

Options

A. `2K(q^(2))/(r^(2))`
B. `(Kq^(2))/(2r^(2))`
C. `sqrt(3)K(q^(2))/(r^(2))`
D. `(Kq^(2))/(sqrt(2)r^(2))`

Answer

To solve the problem of calculating the force experienced by each charge placed at the vertices of an equilateral triangle, we can follow these steps:

### Step 1: Understand the Configuration
We have three identical charges, say \( Q \), placed at the vertices of an equilateral triangle. Let the distance between each pair of charges be \( R \).

### Step 2: Calculate the Force Between Two Charges
Using Coulomb's Law, the force \( F \) between any two charges \( Q \) separated by a distance \( R \) is given by:
\[ F = k \frac{Q^2}{R^2} \]
where \( k = \frac{1}{4 \pi \epsilon_0} \).

### Step 3: Identify Forces Acting on One Charge
Consider one charge (let's say charge at vertex A). It experiences forces due to the other two charges (at vertices B and C). Let the force exerted by charge B on charge A be \( F_1 \) and the force exerted by charge C on charge A be \( F_2 \).

Since the charges are identical and symmetrically placed, we have:
\[ F_1 = F_2 = F = k \frac{Q^2}{R^2} \]

### Step 4: Determine the Angle Between Forces
The angle between the forces \( F_1 \) and \( F_2 \) is \( 60^\circ \) because the triangle is equilateral.

### Step 5: Calculate the Resultant Force
To find the net force \( F \) acting on charge A due to charges B and C, we can use the law of cosines or the parallelogram law. The resultant force can be calculated as:
\[ F = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos(60^\circ)} \]
Substituting \( F_1 = F_2 = F \):
\[ F = \sqrt{F^2 + F^2 + 2 F^2 \cdot \frac{1}{2}} = \sqrt{2F^2 + F^2} = \sqrt{3F^2} = \sqrt{3}F \]

### Step 6: Substitute for \( F \)
Now substituting \( F = k \frac{Q^2}{R^2} \):
\[ F = \sqrt{3} \cdot k \frac{Q^2}{R^2} \]

### Final Answer
Thus, the force experienced by each charge is:
\[ F = \sqrt{3} \cdot k \frac{Q^2}{R^2} \]