17. Think of the quadrupole as composed of two dipoles, each with dipole moment of magnitude p=qd.
The moments point in opposite directions and produce fields in opposite directions at points on the
quadrupole axis. Consider the point P on the axis, a distance zto the right of the quadrupole center and
take a rightward pointing field to be positive. Then, the field produced by the right dipole of the pair
is qd/2πε0(z−d/2)3and the field produced by the left dipole is −qd/2πε0(z+d/2)3. Use the binomial
expansions (z−d/2)−3≈z−3−3z−4(−d/2) and (z+d/2)−3≈z−3−3z−4(d/2) to obtain
E=qd
2πε01
z3+3d
2z4−1
z3+3d
2z4=6qd2
4πε0z4.
Let Q=2qd2. Then,
E=3Q
4πε0z4.
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