  We study the effect of Faraday rotation due to a homogeneous magnetic field
on the polarization of the cosmic microwave background (CMB). Scalar
fluctuations give rise only to parity-even E-type polarization of the CMB.
However in the presence of a magnetic field, a non-vanishing parity-odd B-type
polarization component is produced through Faraday rotation. We derive the
exact solution for the E and B modes generated by scalar perturbations
including the Faraday rotation effect of a uniform magnetic field, and evaluate
their cross-correlations with temperature anisotropies. We compute the angular
autocorrelation function of the B-modes in the limit that the Faraday rotation
is small. We find that primordial magnetic fields of present strength around
$B_0=10^{-9}$G rotate E-modes into B-modes with amplitude comparable to those
due to the weak gravitational lensing effect at frequencies around $\nu=30$
GHz. The strength of B-modes produced by Faraday rotation scales as
$B_0/\nu^2$. We evaluate also the depolarizing effect of Faraday rotation upon
the cross correlation between temperature anisotropy and E-type polarization.

Astrophysics