  The primordial magnetic fields (PMFs) produced in the early universe are
expected to be the origin of the large-scale cosmic magnetic fields. The PMFs
are considered to leave a footprint on the cosmic microwave background (CMB)
anisotropies due to both the electromagnetic force and gravitational
interaction. In this paper, we investigate how the PMFs affect the CMB
anisotropies on smaller scales than the mean-free-path of the CMB photons. We
solve the baryon Euler equation with Lorentz force due to the PMFs, and we show
that the vector-type perturbations from the PMFs induce the CMB anisotropies
below the Silk scale as $\ell>3000$. Based on our calculations, we put a
constraint on the PMFs from the combined CMB temperature anisotropies obtained
by Planck and South Pole Telescope (SPT). We have found that the
highly-resolved temperature anisotropies of the SPT 2017 bandpowers at $\ell
\lesssim 8000$ favor the PMF model with a small scale-dependence. As a result,
the Planck and SPT's joint-analysis puts a constraint on the PMF spectral index
as $n_B<-1.14$ at 95% confidence level (C.L.), and this is more stringent
compared with the Planck-only constraint $n_B<-0.28$. We show that the PMF
strength normalized on the co-moving 1 Mpc scale is also tightly constrained as
$B_{1\mathrm{Mpc}}<1.5$ nG with Planck and SPT at 95% C.L., while
$B_{1\mathrm{Mpc}}<3.2$ nG only with the Planck data at 95% C.L. We also
discuss the effects on the cosmological parameter estimate when including the
SPT data and CMB anisotropies induced by the PMFs.
