  Exact solutions to an Einstein-Maxwell(E-D-M) system with an electric current
are simple models of global cosmic magnetic phenomena in the universe. In this
paper, we consider an E-D-M system with two chiral spinors $ \psi^\pm$ coupled
with a gauge field $ A$ and an electromagnetic field $F=dA$. We construct a
family of exact solutions to the E-D-M system on four-dimensional static
Sasakian spacetimes. The electromagnetic field in the solutions is a contact
magnetic field, and the chiral spinors are induced from Sasakian quasi-Killing
spinors. We find closed universe models whose weak energy condition holds and
open universe models whose weak energy condition is violated. In this study, we
introduce the Sasakian frame, which is used to prospectively discuss the
properties of Sasakian quasi-Killing spinors. A simple description of Sasakian
quasi-Killing spinors of a certain type is obtained.
