  The properties of the electric field of a two-body system consisting of a
Reissner-Nordstr\"om black hole and a charged massive particle at rest have
recently been analyzed in the framework of first order perturbation theory
following the standard approach of Regge, Wheeler and Zerilli. In the present
paper we complete this analysis by numerically constructing and discussing the
lines of force of the "effective" electric field of the sole particle with the
subtraction of the dominant contribution of the black hole. We also give the
total field due to the black hole and the particle. As the black hole becomes
extreme an effect analogous to the Meissner effect arises for the electric
field, with the "effective field" lines of the point charge being expelled by
the outer horizon of the black hole. This effect existing at the level of test
field approximation, i.e. by neglecting the backreaction on the background
metric and electromagnetic field due to the particle's mass and charge, is here
found also at the complete perturbative level. We point out analogies with
similar considerations for magnetic fields by Bi{\v c}\'ak and Dvo{\v r}\'ak.
We also explicitly show that the linearization of the recently obtained
Belinski-Alekseev exact solution coincides with our solution in the
Regge-Wheeler gauge. Our solution thus represents a "bridge" between the test
field solution, which neglects all the feedback terms, and the exact two-body
solution, which takes into account all the non-linearity of the interaction.
