  We study the electric field around a continuous one-dimensional loop of
static charge, under the assumption that the charge is distributed uniformly
along the loop. For rectangular or stadium-shaped loops in the plane, we find
that the electric field can undergo a symmetry-breaking pitchfork bifurcation
as the loop is elongated; the field can have either one or three zeros,
depending on the loop's aspect ratio. For knotted charge distributions in
three-dimensional space, we compute the electric field numerically and compare
our results to previously published theoretical bounds on the number of
equilibrium points around charged knots. Our computations reveal that the
previous bounds are far from sharp. The numerics also suggest conjectures for
the actual minimum number of equilibrium points for all charged knots with five
or fewer crossings. In addition, we provide the first images of the
equipotential surfaces around charged knots, and visualize their topological
transitions as the level of the potential is varied.
