question_answer

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   

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 \[x,\text{ }y\]and \[z\]and plot the charges with the given   coordinates as shown. O is the origin at which \[-2\text{ }q\]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 \[P=(a,a,0).\]The magnitude of resultant dipole moment is \[p'=\sqrt{{{p}^{2}}+{{p}^{2}}}=\sqrt{{{(qa)}^{2}}+{{(qa)}^{2}}}\] \[=\sqrt{2}\,qa\]