  The cosmological magnetic field is one of the important physical quantities
which affect strongly the cosmic microwave background (CMB) power spectrum.
Recent CMB observations have been extended to higher multipoles $l\gtrsim$1000,
and they resultantly exhibit an excess power than the standard model prediction
in cosmological theory which best fits the Wilkinson Microwave Anisotropy Probe
(WMAP) data at lower multipoles $l\lesssim$900. We calculate the CMB
temperature anisotropies generated by the power-law magnetic field at the last
scattering surface (LSS) in order to remove the tension between theory and
observation at higher multipoles and also place an upper limit on primordial
magnetic field. In our present calculation we take account of the effect of
ionization ratio exactly without approximation. This effect is very crucial to
precisely estimate the effect of the magnetic field on CMB power spectrum. We
consider both effects of the scalar and vector modes of magnetic field on the
CMB anisotropies, where current data are known to be insensitive to the tensor
mode which we ignore in the present study. In order to constrain the primordial
magnetic field, we evaluate likelihood function of the WMAP data in a wide
range of parameters of the magnetic field strength $|\mathbf{B}|_\lambda$ and
the power-law spectral index $n$, along with six cosmological parameters in
flat Universe models, using the technique of the Markov Chain Monte Carlo(MCMC)
method. We find that the upper limit at $2\sigma$ C.L. turns out to be
$|\mathbf{B}_\lambda|\lesssim 3.9 $nG at 1 Mpc for any $n_B$ values, which is
obtained by comparing the calculated result including the Sunyaev-Zeldovich(SZ)
effect with recent WMAP data of the CMB anisotropies.
