Question

# Three charges are arranged on the vertices of an equilateral triangle as shown in figure. Find the dipole moment of the combination.

Solution

Consider an equilateral triangle of side I. Point A and B have charges ( +mathrm{q} ) where ( mathrm{A} ) has ( -2 mathrm{q} ) charge. The system as two dipoles with ( overrightarrow{mathrm{p}}_{1} ) and

( vec{p}_{2} )

along BA and CA respectively. Hence, the magnitude of ( overrightarrow{mathrm{p}}_{1} )

and

( vec{p}_{2} )

are equal to (qI) and the angle between them is ( 60^{circ} . ) Therefore

the resultant of two is the net dipole moment.

( overrightarrow{mathrm{P}}=sqrt{mathrm{P}_{1}^{2}+mathrm{P}_{2}^{2}+2 mathrm{P}_{1} mathrm{P}_{2} cos 60^{circ}}=sqrt{mathrm{q}^{2} mathrm{I}^{2}+mathrm{q}^{2} mathrm{I}^{2}+2 mathrm{q}^{2} mathrm{I}^{2}(1 / 2)} )

( P=q I sqrt{3} )