Figure 2.32 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 gt gt 1` , and contrast your results with that due to an electric dipole, and an electric monopole (i.e., a single charge).

Explanation

Potential at point P
`V=k[(q)/(r-a)-(q)/(r)-(q)/(r)+(q)/(r+a)]`
`= kq[(1)/(r-a)-(2)/(r)+(1)/(r+a)]`
`=kq[(r(r+a)-2(r^(2)-a^(2))+r(r-a))/(r(r^(2)-a^(2)))]`
`=kq[(r^(2)+ra-2r^(2)+2a^(2)+r^(2)-ra)/(r(r^(2)-a^(2)))]`
`=kq[(2a^(2))/(r(r^(2)-a^(2)))]`
`=(kQ)/(r(r^(2)-a^(2)))`
where Q=Quadrupole moment
If `r gt gt ` a and `a^(2)` is neglected with compare to `r^(2)`
`V = (kQ)/(r^(3))`
`:. V prop (1)/(r^(3))` for quadrupole
` V prop (1)/(r^(2))` for dipole
and `V prop (1)/(r)` for monopole