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

As is clear from fig, an electric quradruple may be regarded as a system of three charges, `+q, -2q and +q` at A, B and C respectively.
Let `AC = 2a`. We have to calculate electric potential at any point P where `BP = r`. Using superposition principle, potential at P is given by
`V = (1)/(4pi in_(0)) [(q)/(r + a) - (2q)/(r) + (q)/(r - a)] = (q)/(4pi in_(0)) [(r(r-a) - 2(r+a) (r-a) + r(r+a))/(r(r^(2) - a^(2)))]`
`= (q[r^(2) - ra - 2r^(2) + 2a^(2) + r^(2) + ra])/(4pi in_(0) r(r^(2) - a^(2))) = (q.2a^(2))/(4pi in_(0) r^(3) (1-a^(2)//r^(2)))`
when `(r)/(a) gtgt 1, , r gt gt a or a lt lt r`
`:. (a^(2))/(r^(2))` is negligibly small. `V = (q.2a^(2))/(4pi in_(0) r^(3))`, clearly, `V prop (1)/(r^(3))`
In case of an electric dipole , `V prop (1)/(r^(2))`, and in case of an electric monopole (i.e, a single charge). `V prop (1)/(r)`