Fig shows a charge array known as an 'electric quadrupole'. For a point on the axis of the quadrupole, obtain the dependence of potential on r for `r//a gtgt 1`, and contract your results with that due to an electric dipole and an electric monopole (i.e, a single charge).

Explanation

Four charges of same magnitude are placed at points X, Y, Y and Z respectively, as shown in the following figure.

A point is located at P, Which is r distance away from point Y.
The system of charges forms an electric quadrupole.
It can be considered that the system of the electric quadrupole has three charges.
Charge +q placed at point X
Charge `-2q` placed at point Y
Charge `+q` placed at point Z
`XY = YZ = a`
`YP= r`
`PX = r+a`
Electrostatic potential caused by the system of three charges at point P is given by,
`V = (1)/(4pi in_(0))[(q)/(XP) - (2q)/(YP)+(q)/(ZP)]`
`= (1)/(4pi in_(0))[(q)/(r+a) -(2q)/(r)+(a)/(r-a)]`
`= (Q)/(4pi in_(0))[(r(r-a)-2(r+a)(r-a)+r(r+a))/(r(r+a)(r-a))]`
`= (q)/(4pi in_(0))[(r^(2)-ra-2r^(2)+2a^(2)+r^(2)+ra)/(r(r^(2)-a^(2)))] = (q)/(4pi in_(0))[(2a^(2))/(r(r^(2)-a^(2)))]`
`= (2qa^(2))/(4pi in_(0) r^(3)(1-(a^(2))/(r^(2))))`
Since `(r)/(a) gt gt 1`
`:. (a)/(r) lt lt 1`
`(a^(2))/(r^(2))` is taken as negligible.
`:. V = (2qa^(2))/(4pi in_(0) r^(3))`
It can be inferred that potential, `V prop (1)/(r^(3))`
However, it is known that for a diople, `V prop (1)/(r^(3))`
And , for a monopole, `V prop (1)/(r)`