  We consider the minimal discrete and continuous energy problems on the unit
sphere $\mathbb{S}^d$ in the Euclidean space $\mathbb{R}^{d+1}$ in the presence
of an external field due to finitely many localized charge distributions on
$\mathbb{S}^d$, where the energy arises from the Riesz potential $1/r^s$ ($r$
is the Euclidean distance) for the critical Riesz parameter $s = d - 2$ if $d
\geq 3$ and the logarithmic potential $\log(1/r)$ if $d = 2$. Individually, a
localized charge distribution is either a point charge or assumed to be
rotationally symmetric. The extremal measure solving the continuous external
field problem for weak fields is shown to be the uniform measure on the sphere
but restricted to the exterior of spherical caps surrounding the localized
charge distributions. The radii are determined by the relative strengths of the
generating charges. Furthermore, we show that the minimal energy points solving
the related discrete external field problem are confined to this support. For
$d-2\leq s<d$, we show that for point sources on the sphere, the equilibrium
measure has support in the complement of the union of specified spherical caps
about the sources. Numerical examples are provided to illustrate our results.
