question_answer

Three point charges \[\text{+}\,\text{q,}\,\text{-}\,\text{2}\,\text{q}\] and \[\text{+}\,\text{q}\] are placed at points \[(x=\,o\text{,}\,y=a,z=o\,)\] \[(x=\,o\text{,}\,y=0,z=o\,)\] and \[(x=\,a\text{,}\,y=0,z=o\,),\] 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

Correct Answer: B

Solution :

Key Idea Electric dipole moment is a vector Quantity directed from negative charge to the similar positive charge. Choose the three coordinate axes as x, y and z and plot the charges with the given coordinates as shown. O is the origin at which -2q charge is placed. The system is equivalent to two dipoles along x and y-directions respectively. The dipole moments of two dipoles are shown in figure.                 The resultant dipole moment will be directed along OP where \[P\equiv (a,a,0),\]The magnitude of resultant dipole moment is \[P'=\sqrt{{{P}^{2}}+{{P}^{2}}}=\sqrt{{{(qa)}^{2}}+{{(aq)}^{2}}}\]                 \[=\sqrt{2}qa\]