  In regions with strongly varying electron density, radio emission can be
magnified significantly by plasma lensing. In the presence of magnetic fields,
magnification in time and frequency will be different for two circular
polarizations. We show how these effects can be used to measure or constrain
the magnetic field parallel to the line of sight, $B_\parallel$, as well as its
spatial structure, $\sigma_{B_\parallel}$, in the lensing region. In addition,
we discuss how generalized Faraday rotation can constrain the strength of the
perpendicular field, $B_\perp$. We attempt to make such measurements for the
Black Widow pulsar, PSR~B1957+20, in which plasma lensing was recently
discovered. For this system, pressure equilibrium suggests $B\gtrsim 20\,$G at
the interface between the pulsar and companion winds, where the radio eclipse
starts and ends, and where most lensing occurs. We find no evidence for
large-scale magnetic fields, with, on average, $B_\parallel=0.02\pm0.09\,$G
over the egress lensing region. From individual lensing events, we strongly
constrain small scale magnetic structure to $\sigma_B<10\,$mG, thus excluding
scenarios with a strong but rapidly varying field. Finally, from the lack of
reduction of average circular polarization in the same region, we rule out a
strong, quasi-transverse field. We cannot identify any plausible scenario in
which a large magnetic field in this system is concealed, leaving the nature of
the interface between the pulsar and companion winds an enigma. Our method can
be applied to other sources showing plasma lensing, including other eclipsing
pulsars and fast radio bursts, to study the local properties of the magnetic
field.
