Halliday, Resnick & Walker — Electric Fields: Problem 15

15 Electric quadrupole. Figure 22-30 shows a generic electric quadrupole. It consists of two dipoles with dipole moments that are equal in magnitude but opposite in direction. Show that the value of E on the axis of the quadrupole for a point P a distance z from its center (assume z >> d) is given by \( E = \frac{3Q}{4\pi\varepsilon_0 z^4} \), in which \( Q (= 2qd^2) \) is known as the quadrupole moment of the charge distribution.