  We calculate the effective Lagrangian for a magnetic field in spinor, scalar
and vector QED. Connections are then made to $SU(N_C)$ Yang--Mills theory and
QCD. The magnetization and the corresponding effective charge are obtained from
the effective Lagrangian. The renormalized vacuum magnetization will depend on
the renormalization scale chosen. Regardless of this, the effective charge
decreasing with the magnetic field, as in QCD, corresponds to anti- screening
and asymptotic freedom. In spinor and scalar QED on the other hand, the
effective charge is increasing with the magnetic field, corresponding to
screening. Including effects due to finite temperature and density, we comment
on the effective charge in a degenerate fermion gas, increasing linearly with
the chemical potential. Neglecting the tachyonic mode, we find that in hot QCD
the effective charge is decreasing as the inverse temperature, in favor for the
formation of a quark-gluon plasma. However, including the real part of the
contribution from the tachyonic mode, we find instead an effective charge
increasing with the temperature. Including a thermal gluon mass, the effective
charge in hot QCD is group invariant (unlike in the two cases above), and
decreases logarithmically in accordance to the vacuum renormalization group
equation, with the temperature as the momentum scale.
