Three charges +q, +q and -2q are placed at the vertices of an equilareral triagle. What is the dipole moment of the system?

Options

A. `sqrt(3)qa`
B. `(sqrt(3)qa)/2`
C. `sqrt(2)qa`
D. `(qa)/2`

Explanation

To find the dipole moment of the system with charges +q, +q, and -2q placed at the vertices of an equilateral triangle, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Configuration**:
- Let the vertices of the equilateral triangle be labeled as A, B, and C.
- Place charge +q at vertex A, +q at vertex B, and -2q at vertex C.

2. **Consider the Dipole Moment**:
- 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 Individual Dipole Moments**:
- We can treat the -2q charge at C as two -q charges for the purpose of calculating dipole moments:
- From A to C, the dipole moment \( \vec{p_1} \) can be calculated as:
\[ \vec{p_1} = q \cdot \vec{d_{AC}} \quad \text{(from A to C)} \]
- From B to C, the dipole moment \( \vec{p_2} \) can be calculated as:
\[ \vec{p_2} = q \cdot \vec{d_{BC}} \quad \text{(from B to C)} \]

4. **Determine the Length of the Sides**:
- Let the length of each side of the triangle be \( L \).

5. **Calculate the Magnitudes of the Dipole Moments**:
- The magnitude of the dipole moment for each charge configuration is:
\[ p_1 = q \cdot L \quad \text{(for charge at A)} \]
\[ p_2 = q \cdot L \quad \text{(for charge at B)} \]

6. **Combine the Dipole Moments**:
- The dipole moments \( \vec{p_1} \) and \( \vec{p_2} \) are at an angle of 60 degrees to each other (since they are from the same vertex to the opposite vertex in an equilateral triangle).
- The resultant dipole moment \( \vec{P}_{net} \) can be calculated using the formula for the resultant of two vectors:
\[ P_{net} = \sqrt{p_1^2 + p_2^2 + 2p_1p_2 \cos(60^\circ)} \]
- Substituting \( p_1 = p_2 = qL \):
\[ P_{net} = \sqrt{(qL)^2 + (qL)^2 + 2(qL)(qL) \cdot \frac{1}{2}} \]
\[ = \sqrt{(qL)^2 + (qL)^2 + (qL)^2} = \sqrt{3(qL)^2} = qL\sqrt{3} \]

7. **Final Result**:
- The net dipole moment of the system is:
\[ P_{net} = qL\sqrt{3} \]