  The main subject of this paper is equilibrium problems on an unbounded
conductor $\Sigma$ of the complex plane in the presence of a weakly admissible
external field. An admissible external field $Q$ on $\Sigma$ satisfies, along
with other mild conditions, the following growth property at infinity:
$$\lim_{|x| \rightarrow \infty}(Q(x) - \log |x|) = +\infty.$$ This condition
guarantees the existence and uniqueness of the equilibrium measure in the
presence of $Q$, and the compactness of its support. In the last 10-15 years,
several papers have dealt with weakly admissible external fields, in the sense
that $Q$ satisfies a weaker condition at infinity, namely, $$\exists
M\in(-\infty,\infty],\quad\liminf_{|x| \rightarrow \infty}(Q(x) - \log |x|) =
M.$$ Under this last assumption, there still exists a unique equilibrium
measure in the external field $Q$, but the support need not be a compact subset
of $\Sigma$ anymore. In most examples considered in the literature the support
is indeed unbounded. Our main goal in this paper is to illustrate this topic by
means of a simple class of external fields on the real axis created by a pair
of attractive and repellent charges in the complex plane, and to study the
dynamics of the associated equilibrium measures as the strength of the charges
evolves. As one of our findings, we exhibit configurations where the support of
the equilibrium measure in a weakly admissible external field is a compact
subset of the real axis. To achieve our goal, we extend some results from
potential theory, known for admissible external fields, to the weakly
admissible case. These new results may be of independent interest. Finally, the
so--called signed equilibrium measure is an important tool in our analysis. Its
relationship with the (positive) equilibrium measure is also explored.
