  We show that a unit vector field on an oriented Riemannian manifold is a
critical point of the Landau Hall functional if and only if it is a critical
point of the Dirichlet energy functional. Therefore, we provide a
characterization for a unit vector field to be a magnetic map into its unit
tangent sphere bundle. Then, we classify all magnetic left invariant unit
vector fields on $3$-dimensional Lie groups.
