Four charges of the 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 Y.
The system of charges forms an electric quadrupole.
It can be considered that the system of 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
PZ = r - a
Electrostatic potential caused by the system of three charges at point P is given by,
V=14πϵ0[qXP−2qYP+qZP]=14πϵ0[qr+a−2qr+qr−a]=q4πϵ0[r(r−a)−2(r+a)(r−a)+r(r+a)r(r+a)(r−a)]=q4πϵ0[r2−ra−2r2+2a2+r2+ra]r(r2−a2)]=q4πϵ0[2a2r(r2−a2)]=2qa24πϵ0r3(1−a2r2)
since ra>>1∴ar<<1a2r2 is taken as negligible. ∴V=2qa24πϵ0r3
It can be inferred that potential, V ∝ 1r3
However, it is known that for a dipole, V ∝ 1r2
And, for a monopole, V ∝ 1r