  Electron scattering induces a polarization in the cosmic microwave background
(CMB) signal measured in the direction of a galaxy cluster due to the presence
of a quadrupole component in the CMB temperature distribution. Measuring the
polarization towards distant clusters provides the unique opportunity to
observe the evolution of the CMB quadrupole at moderate redshifts, z~0.5-3. We
demonstrate that for the local cluster population the polarization degree will
depend on the cluster celestial position. There are two extended regions in the
sky, which are opposite to each other, where the polarization is maximal,
0.1(tau/0.02) microK in the Rayleigh-Jeans part of the CMB spectrum (tau being
the Thomson optical depth across the cluster) exceeding the contribution from
the cluster transverse peculiar motion if v_t<1300 km/s. One can hope to detect
this small signal by measuring a large number of clusters, thereby effectively
removing the systematic contribution from other polarization components
produced in clusters. These polarization effects, which are of the order of
(v_t/c)^2 tau, (v_t/c) tau^2 and (kT_e/m_ec^2) tau^2, as well as the
polarization due to the CMB quadrupole, were previously calculated by Sunyaev
and Zel'dovich for the Rayleigh-Jeans region. We fully confirm their earlier
results and present exact frequency dependencies for all these effects. The
polarization is considerably higher in the Wien region of the CMB spectrum.
