  A recently found (gr-qc/0303036) 2-index, symmetric, trace-free,
divergence-free tensor is introduced for arbitrary source-free electromagnetic
fields. The tensor can be constructed for any test Maxwell field in Einstein
spaces (including proper vacuum), and more importantly for any Einstein-Maxwell
spacetime. The tensor is explicitly given and analyzed in some special
situations, such as general null electromagnetic fields, Reissner-Nordstr\"om
solution, or classical electrodynamics. We present an explicit example where
the conserved currents derived from the energy-momentum tensor using symmetries
are trivial, but those derived from the new tensor are not.
