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. `3sqrt2 qx`
C. ` 3 qx`
D. `2sqrt3 qx`

Answer

To find the magnitude of the electric dipole moment of the system with charges \( +2q, +2q, \) and \( -4q \) placed at the corners of an equilateral triangle ABC with side length \( x \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Charges and Their Positions**:
- Let charge \( +2q \) be at point A.
- Let charge \( +2q \) be at point B.
- Let charge \( -4q \) be at point C.
- The triangle ABC is equilateral with each side of length \( x \).

2. **Understanding the Dipole Moment**:
- The electric dipole moment \( \vec{p} \) for a system of charges is given by the formula:
\[ \vec{p} = \sum \vec{r_i} q_i \]
- Here, \( \vec{r_i} \) is the position vector of the charge \( q_i \).

3. **Calculate the Dipole Moment for Each Pair**:
- We can consider the dipole moments created by the two \( +2q \) charges and the \( -4q \) charge.
- The dipole moment \( \vec{p_1} \) due to the charge at A and the charge at C:
\[ \vec{p_1} = (2q)(\vec{r_A} - \vec{r_C}) \]
- The dipole moment \( \vec{p_2} \) due to the charge at B and the charge at C:
\[ \vec{p_2} = (2q)(\vec{r_B} - \vec{r_C}) \]

4. **Magnitude of Each Dipole Moment**:
- The distance between charges A and C (or B and C) is \( x \).
- Therefore, the magnitudes of \( \vec{p_1} \) and \( \vec{p_2} \) can be expressed as:
\[ p_1 = 2q \cdot x \quad \text{and} \quad p_2 = 2q \cdot x \]

5. **Net Dipole Moment Calculation**:
- Since the angle between \( p_1 \) and \( p_2 \) in an equilateral triangle is \( 60^\circ \), we can use the formula for the resultant of two vectors:
\[ p_{net} = \sqrt{p_1^2 + p_2^2 + 2p_1p_2 \cos(60^\circ)} \]
- Substituting \( \cos(60^\circ) = \frac{1}{2} \):
\[ p_{net} = \sqrt{(2qx)^2 + (2qx)^2 + 2(2qx)(2qx)\left(\frac{1}{2}\right)} \]
- Simplifying this:
\[ p_{net} = \sqrt{4q^2x^2 + 4q^2x^2 + 2q^2x^2} = \sqrt{10q^2x^2} = \sqrt{10}qx \]

6. **Final Result**:
- The magnitude of the electric dipole moment of this system is:
\[ p_{net} = \sqrt{3} \cdot 2qx = 2\sqrt{3}qx \]