Three point charges of `+2q,+2q and -4q` are placed at the corners A, B and C of an equilateral triangle ABC of side x. The magnitude of the electric dipole moment of this system is ?

Options

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

Answer

To find the magnitude of the electric dipole moment of the system of three point charges located at the corners of an equilateral triangle, we can follow these steps:

### Step 1: Identify the Charges and their Positions
We have three charges:
- Charge at A: \( +2q \)
- Charge at B: \( +2q \)
- Charge at C: \( -4q \)

These charges are placed at the corners of an equilateral triangle ABC with side length \( x \).

### Step 2: Break Down the Charges into Dipoles
We can treat the charges \( +2q \) at points A and B as forming a dipole with the charge \( -4q \) at point C. To analyze the dipole moments, we can consider the following:
- The dipole moment \( \vec{P_1} \) formed by the charge at A and the charge at C.
- The dipole moment \( \vec{P_2} \) formed by the charge at B and the charge at C.

### Step 3: Calculate Individual Dipole Moments
The dipole moment \( \vec{P} \) is given by the formula:
\[ \vec{P} = q \cdot d \]
where \( q \) is the charge and \( d \) is the distance between the charges.

For dipole \( P_1 \) (between \( +2q \) at A and \( -2q \) at C):
\[ P_1 = 2q \cdot x \]

For dipole \( P_2 \) (between \( +2q \) at B and \( -2q \) at C):
\[ P_2 = 2q \cdot x \]

### Step 4: Determine the Direction of the Dipoles
The direction of the dipole moment is from the negative charge to the positive charge. Therefore, both dipoles \( P_1 \) and \( P_2 \) point towards the center of the triangle.

### Step 5: Calculate the Resultant Dipole Moment
Since both dipoles are at an angle of \( 60^\circ \) to each other (the angle in an equilateral triangle), we can use the formula for the resultant of two vectors:
\[ P = \sqrt{P_1^2 + P_2^2 + 2P_1P_2 \cos(60^\circ)} \]

Substituting \( P_1 = P_2 = 2qx \) and \( \cos(60^\circ) = \frac{1}{2} \):
\[ P = \sqrt{(2qx)^2 + (2qx)^2 + 2(2qx)(2qx)\left(\frac{1}{2}\right)} \]
\[ = \sqrt{4q^2x^2 + 4q^2x^2 + 4q^2x^2} \]
\[ = \sqrt{12q^2x^2} = 2\sqrt{3}qx \]

### Final Result
Thus, the magnitude of the electric dipole moment of the system is:
\[ \boxed{2\sqrt{3}qx} \]