Set the normal electric field to zero and cancel its nonzero overall factor:
(1)
Taking the power gives , hence
(2)
The numerical value is independent of the nonzero magnitude of . Beyond this radius the signed normal electric field is positive. Thus its magnitude first falls to zero, rises again and eventually decays; it is not globally monotone.
For , expanding each denominator gives
(3)
This is a dipole rather than a monopole tail. The real-plus-image charge distribution has zero total charge and dipole moment ; in its equatorial plane a dipole produces , confirming the coefficient and sign.
Figure 1.
Image charges and the sign-changing normal field above an infinite conducting plane
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