Question

Three point charges $+q, - 2q$ and $+q$ are placed at points $(x = 0, y = a, z = 0), (x = 0, y = 0, z = 0)$ and $(x = a, y = 0, z = 0)$, respectively. The magnitude and direction of the electric dipole moment vector of this charge assembly are

• A. $\sqrt 2 qa$ along $+ y$ direction
• B. $\sqrt 2\,qa$ along the line joining points $(x = 0, y = 0, z = 0)$ and $(x = a, y = a, z - 0)$
• C. $qa$ along the line joining points $(x = 0, y = 0, z = 0)$ and $(x = a, y = a, z = 0)$
• D. $\sqrt 2\, qa$ along $+ x$ direction

Answer 1

Answer: (B)

As shown in figure, charge $-2q$ is at $O(0, 0, 0), +q$
is at $A (0, a , 0)$ and $+q$ is at $B (a, 0, 0)$.
The given arrangement of charges consists of two dipoles $-q$ and $+q$ with dipole moment $\vec{p}_{1}$ along $+y$ direction and $-q$ and $+q$ with dipole moment $\vec{p}_{2}$ along $+x$ direction
i.e., $P_{1}=qa$ along $OA$
$P_{2}=qa$ along $OB$
Resultant dipole moment $\vec{p}$ is along $OC$ where $(C=a,a,0)$
$P=\sqrt{P_{1}^{2}+P_{2}^{2}}$
$=\sqrt{\left(qa\right)^{2}+\left(qa\right)^{2}}$
$=\sqrt{2}\,qa$