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. `sqrt(3)ql`
C. zero
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 \( l \), we can follow these steps:

### Step 1: Understand the Configuration
We have an equilateral triangle ABC with:
- Charge \( q \) at point A,
- Charge \( q \) at point B,
- Charge \( -2q \) at point C.

### Step 2: Define the Dipole Moment
The electric dipole moment \( \vec{p} \) for a system of charges is defined as:
\[ \vec{p} = \sum \vec{p_i} = \sum q_i \vec{r_i} \]
where \( q_i \) is the charge and \( \vec{r_i} \) is the position vector of the charge from the origin.

### Step 3: Calculate Individual Dipole Moments
We can calculate the dipole moments due to each charge with respect to a reference point (let's take point C as the origin):
- For charge \( q \) at A (distance \( l \)):
\[ \vec{p_A} = q \cdot l \hat{r}_{AC} \]
- For charge \( q \) at B (distance \( l \)):
\[ \vec{p_B} = q \cdot l \hat{r}_{BC} \]
- For charge \( -2q \) at C (which we consider as the origin, so its contribution is zero):
\[ \vec{p_C} = -2q \cdot 0 = 0 \]

### Step 4: Determine the Direction of Dipole Moments
The dipole moments \( \vec{p_A} \) and \( \vec{p_B} \) will have a certain angle between them. Since triangle ABC is equilateral, the angle between \( \vec{p_A} \) and \( \vec{p_B} \) is \( 60^\circ \).

### Step 5: Calculate the Resultant Dipole Moment
Using the formula for the resultant of two vectors:
\[ |\vec{p}_{net}| = \sqrt{|\vec{p}_A|^2 + |\vec{p}_B|^2 + 2 |\vec{p}_A| |\vec{p}_B| \cos(60^\circ)} \]
Substituting \( |\vec{p}_A| = ql \) and \( |\vec{p}_B| = ql \):
\[ |\vec{p}_{net}| = \sqrt{(ql)^2 + (ql)^2 + 2(ql)(ql) \cdot \frac{1}{2}} \]
\[ = \sqrt{(ql)^2 + (ql)^2 + (ql)^2} = \sqrt{3(ql)^2} = \sqrt{3}ql \]

### Final Answer
Thus, the magnitude of the electric dipole moment of the system is:
\[ |\vec{p}_{net}| = \sqrt{3}ql \]