Question

Determine the electric dipole moment of the system of three charges, placed on the vertices of an equilateral triangle, as shown in the figure:

• A. $\left(ql \right)\frac{\hat{i} + \hat{j}}{\sqrt{2}}$
• B. $\sqrt{3}ql\frac{\hat{j} - \hat{I}}{\sqrt{2}}$
• C. $-\sqrt{3}ql\hat{j}$
• D. $2ql\hat{j}$

Answer 1

The Correct Option is C

To determine the electric dipole moment of a system with three charges placed at the vertices of an equilateral triangle, as depicted in the situation provided, follow these steps:

Step 1: Understand the System Configuration

Consider an equilateral triangle with side length \( l \), where each vertex hosts a charge. Let the charges at the vertices be \( +q, +q, \) and \( -2q \). For simplicity, place the center of the triangle at the origin for calculation purposes.

Step 2: Define the Position Vectors

Assume the vertices of the triangle are \( A, B, \) and \( C \). The typical coordinates for the vertices of an equilateral triangle, centered at the origin, are:

• \( A \left(0, \frac{l}{\sqrt{3}}\right) \),
• \( B \left(-\frac{l}{2}, -\frac{l}{2\sqrt{3}}\right) \),
• \( C \left(\frac{l}{2}, -\frac{l}{2\sqrt{3}}\right) \).

Step 3: Calculate the Total Dipole Moment

The electric dipole moment \( \mathbf{p} \) is defined as the product of charge \( q \) and displacement vector \( \mathbf{d} \):

\[\mathbf{p} = q_1 \mathbf{r}_1 + q_2 \mathbf{r}_2 + q_3 \mathbf{r}_3\]

Where:

• \( q_1 = +q \), \( \mathbf{r}_1 = \left(0, \frac{l}{\sqrt{3}}\right) \)
• \( q_2 = +q \), \( \mathbf{r}_2 = \left(-\frac{l}{2}, -\frac{l}{2\sqrt{3}}\right) \)
• \( q_3 = -2q \), \( \mathbf{r}_3 = \left(\frac{l}{2}, -\frac{l}{2\sqrt{3}}\right) \)

Substituting the values,

\[\begin{align*} \mathbf{p} &= q \left(0, \frac{l}{\sqrt{3}}\right) + q \left(-\frac{l}{2}, -\frac{l}{2\sqrt{3}}\right) + (-2q) \left(\frac{l}{2}, -\frac{l}{2\sqrt{3}}\right) \\ &= q \left(0, \frac{l}{\sqrt{3}}\right) - q \left(\frac{l}{2}, \frac{l}{2\sqrt{3}}\right) - q \left(l, -l/\sqrt{3}\right) \\ &= -ql \left(\frac{1}{2} + 1\right) \hat{i} + ql \left(\frac{1}{\sqrt{3}} + \frac{1}{2\sqrt{3}} - \frac{2}{2\sqrt{3}}\right) \hat{j} \\ &= -\frac{3}{2}ql \, \hat{i} + (-\sqrt{3}ql) \, \hat{j} \end{align*}\]

Step 4: Conclusion

The final electric dipole moment of the system, therefore, has only a significant \(\hat{j}\) component:

\(-\sqrt{3}ql \, \hat{j}\)

This matches the given correct answer: -\sqrt{3}ql\hat{j}