  The Fermi Bubbles are giant Galactic structures observed in both gamma-rays
and microwaves. Recent studies have found support for the hypothesis that the
gamma-ray and microwave emission can both be understood as arising from a hard
cosmic-ray electron population within the volume of the Bubbles, via inverse
Compton scattering and synchrotron radiation respectively. The relative rates
of these processes are set by the relative energy density of the interstellar
radiation field and the magnetic field within the Bubbles; consequently, under
the hypothesis of a common origin, the combination of the gamma-ray and
microwave measurements can be used to estimate the magnetic field within the
Bubbles. We revisit the consistency of this hypothesis on a
latitude-by-latitude basis, using data from Fermi, WMAP and Planck; estimate
the variation of the electron spectrum within the Bubbles; and infer bounds on
the magnetic field within the Bubbles as a function of distance from the
Galactic plane. We find that while the microwave and gamma-ray spectra are
generally consistent with the leptonic hypothesis for few-microGauss magnetic
fields, there appears to be a preference for spectral hardening in the
microwaves at mid-latitudes (especially in the |b|~ 25-35 degree range) that is
not mirrored in the gamma rays. This result may hint at a non-leptonic
contribution to the gamma-ray spectra; however, the discrepancy can be
reconciled in purely leptonic models if the cutoff energy for the electrons is
lower in this latitude range and the spectrum below the cutoff is harder.
