  Cosmic strings are a type of cosmic defect formed by a symmetry-breaking
phase transition in the early universe. Individual strings would have gathered
to build a network, and their dynamical motion would induce scalar--, vector--
and tensor--type perturbations. In this paper, we focus on the vector mode
perturbations arising from the string network based on the one scale model and
calculate the time evolution and the power spectrum of the associated magnetic
fields. We show that the relative velocity between photon and baryon fluids
induced by the string network can generate magnetic fields over a wide range of
scales based on standard cosmology. We obtain the magnetic field spectrum
before recombination as
$a^2B(k,z)\sim4\times10^{-16}G\mu/((1+z)/1000)^{4.25}(k/{\rm Mpc}^{-1})^{3.5}$
Gauss on super-horizon scales, and
$a^2B(k,z)\sim2.4\times10^{-17}G\mu/((1+z)/1000)^{3.5}(k/{\rm Mpc}^{-1})^{2.5}$
Gauss on sub-horizon scales in co-moving coordinates. This magnetic field grows
up to the end of recombination, and has a final amplitude of approximately
$B\sim10^{-17\sim -18} G\mu$ Gauss at the $k\sim1\ {\rm Mpc}^{-1}$ scale today.
This field might serve as a seed for cosmological magnetic fields.
