  Recently, we had numerically shown that, for a non-minimal coupling that is a
simple power of the scale factor, scale invariant magnetic fields arise in a
class of bouncing universes. In this work, we {\it analytically} evaluate the
spectrum of magnetic and electric fields generated in a sub-class of such
models. We illustrate that, for cosmological scales which have wavenumbers much
smaller than the wavenumber associated with the bounce, the shape of the
spectrum is preserved across the bounce. Using the analytic solutions obtained,
we also illustrate that the problem of backreaction is severe at the bounce.
Finally, we show that the power spectrum of the magnetic field remains
invariant under a two parameter family of transformations of the non-minimal
coupling function.
