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 with three point 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 (0, 0).
- Let charge \(+2q\) be at point B \((x, 0)\).
- Let charge \(-4q\) be at point C \(\left(\frac{x}{2}, \frac{x\sqrt{3}}{2}\right)\) (the coordinates of C are derived from the properties of an equilateral triangle).

2. **Calculate the Dipole Moment for Each Pair:**
- The dipole moment \( \vec{p} \) is given by the formula:
\[ \vec{p} = q \cdot \vec{d} \]
where \(q\) is the charge and \(\vec{d}\) is the displacement vector from the negative charge to the positive charge.

3. **Calculate the Dipole Moment for Charges at A and C:**
- For the charges at A and C:
- The effective dipole moment due to \(+2q\) at A and \(-4q\) at C can be calculated as:
\[ \vec{p}_{AC} = (+2q) \cdot \left(\frac{x}{2}, \frac{x\sqrt{3}}{2}\right) = (q \cdot x, q \cdot x\sqrt{3}) \]

4. **Calculate the Dipole Moment for Charges at B and C:**
- For the charges at B and C:
- The effective dipole moment due to \(+2q\) at B and \(-4q\) at C can be calculated as:
\[ \vec{p}_{BC} = (+2q) \cdot \left(-\frac{x}{2}, -\frac{x\sqrt{3}}{2}\right) = (-q \cdot x, -q \cdot x\sqrt{3}) \]

5. **Sum the Dipole Moments:**
- The total dipole moment \( \vec{p}_{total} \) is the vector sum of \( \vec{p}_{AC} \) and \( \vec{p}_{BC} \):
\[ \vec{p}_{total} = \vec{p}_{AC} + \vec{p}_{BC} \]
- This results in:
\[ \vec{p}_{total} = (q \cdot x - q \cdot x, q \cdot x\sqrt{3} - q \cdot x\sqrt{3}) = (0, 0) \]

6. **Magnitude of the Electric Dipole Moment:**
- Since the total dipole moment sums to zero, the magnitude of the electric dipole moment of this system is:
\[ |\vec{p}_{total}| = 0 \]

### Final Answer:
The magnitude of the electric dipole moment of this system is \(0\).