  We study the amplification of weak magnetic field with Weak-value technique
based on Faraday magneto-optic effect. By using a different scheme to perform
the Sagnac interferometer with the probe in momentum space, we have
demonstrated the new weak measure protocol to detect the small weak magnetic
field by amplifying the phase shift of Faraday magneto-optic effect. At the
given the maximum incident intensity of the initial spectrum, the detection
limit of the intensity of the spectrometer and the accuracy of detecting weak
magnetic field, we can theoretically give the appropriate pre-selection,
post-selection and others optical structure before experiment. Our numerical
results show our scheme with Weak-value technique is effective and feasible to
detect weak magnetic field with magnetic field intensity lower than $10^{-10}$
T.
