Electric charges `q, q, –2q` are placed at the corners of an equilateral triangle ABC of side l. The magnitude of electric dipole moment of the system is

Options

A. `ql`
B. `2ql`
C. `sqrt(3ql)`
D. `4ql`

Answer

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

### Step 1: Identify the Charges and Their Positions
- Place charge \( +q \) at point A, charge \( +q \) at point B, and charge \( -2q \) at point C of the equilateral triangle ABC.

### Step 2: Define the Dipole Moments
- We can form two dipole moments:
- **Dipole Moment \( \mathbf{p_1} \)** between charges at A and C:
\[ \mathbf{p_1} = q \cdot \mathbf{l} \]
where \( \mathbf{l} \) is the vector pointing from C to A.

- **Dipole Moment \( \mathbf{p_2} \)** between charges at B and C:
\[ \mathbf{p_2} = q \cdot \mathbf{l} \]
where \( \mathbf{l} \) is the vector pointing from C to B.

### Step 3: Determine the Direction of Dipole Moments
- The dipole moment is directed from the negative charge to the positive charge. Thus:
- \( \mathbf{p_1} \) is directed from C to A.
- \( \mathbf{p_2} \) is directed from C to B.

### Step 4: Calculate the Magnitudes of the Dipole Moments
- The magnitudes of both dipole moments are:
\[ |\mathbf{p_1}| = q \cdot l \]
\[ |\mathbf{p_2}| = q \cdot l \]

### Step 5: Calculate the Angle Between the Dipole Moments
- The angle between \( \mathbf{p_1} \) and \( \mathbf{p_2} \) is \( 60^\circ \) because ABC is an equilateral triangle.

### Step 6: Use the Formula for the Resultant Dipole Moment
- The resultant dipole moment \( \mathbf{p}_{\text{system}} \) can be calculated using the formula:
\[ |\mathbf{p}_{\text{system}}| = \sqrt{|\mathbf{p_1}|^2 + |\mathbf{p_2}|^2 + 2 |\mathbf{p_1}| |\mathbf{p_2}| \cos(60^\circ)} \]
Substituting the values:
\[ |\mathbf{p}_{\text{system}}| = \sqrt{(ql)^2 + (ql)^2 + 2(ql)(ql)(\frac{1}{2})} \]

### Step 7: Simplify the Expression
- Simplifying the equation:
\[ |\mathbf{p}_{\text{system}}| = \sqrt{q^2l^2 + q^2l^2 + q^2l^2} = \sqrt{3q^2l^2} = \sqrt{3}ql \]

### Final Result
- Therefore, the magnitude of the electric dipole moment of the system is:
\[ |\mathbf{p}_{\text{system}}| = \sqrt{3}ql \]