Question

Three point-charges of +2q, +2q and -4q are placed at the corner A, B, and C of an equilateral triangle ABC of side $x$. What is the magnitude of the electric dipole moment of this system?

Answer 1

A system consisting of a positive and a negative charge of equal magnitude q, separated by a distance d is called an electric dipole. For an electric dipole, we define a new vector, called the electric dipole moment. The magnitude of the dipole moment vector p is the magnitude of the charge q times the distance d between them, $\overrightarrow p = qd$ . The vector points from the negative towards the positive charge. Split -4q into two charges of -2q quantities each. Calculate the magnitude of the resultant electric dipole moment.

Complete step by step solution:
Step 1: consider the following figure.

First, split -4q into two equal parts -2q and -2q. Now we have two electric dipoles. The electric dipole moment is a vector quantity. Therefore, express the electric dipole moment formula.
$\overrightarrow {\therefore p} = qd$
Step 2: The length of the side of the triangle is $x$ . if we assume two electric dipole moment $\overrightarrow {{p_1}}$ and $\overrightarrow {{p_2}}$ from A to B and A to C respectively. Then
$\overrightarrow {{p_1}} = 2qx$ and
$\overrightarrow {{p_2}} = 2qx$
Step 3: Now we have two vectors from A to B and A to C. since the triangle is equilateral, that is, all three sides are of equal length and thus each angle is 60 degrees. Therefore, the angle between the two vectors is 60 degrees. The net electric dipole moment is
$\therefore {\overrightarrow p _{net}} = \sqrt {{p^2} + {p^2} + 2pp\cos \theta }$ , where $\theta$ is the angle between the two vectors.
Substitute 60 for $\theta$ .
$\therefore {\overrightarrow p _{net}} = \sqrt {{p^2} + {p^2} + 2pp\cos 60}$
$\Rightarrow {\overrightarrow p _{net}} = \sqrt {2{p^2} + pp}$
$\Rightarrow {\overrightarrow p _{net}} = \sqrt {3{p^2}}$
$\Rightarrow {\overrightarrow p _{net}} = \sqrt 3 p$
$p$ is the product of a charge of the dipole and the length of the dipole. Therefore,
$\therefore {\overrightarrow p _{net}} = \sqrt 3 \times 2q \times x$
$\Rightarrow {\overrightarrow p _{net}} = \sqrt {12} qx$

Hence the magnitude of the electric dipole moment is $\sqrt {12} qx$ coulomb-meter.

Note: An electric dipole consists of two equal and opposite charges placed very close to each other. Therefore, it is important to split -4q negative charge into two equal charges of -2q so that we can think of the system as a two electric dipole system. While calculating net electric dipole moment we should take care of the direction of each electric dipole moment.