  Based on tests of a tentative detector for observing geo-synchrotron hard
X-rays generated by primary electrons, we study the feasibility of probing
cosmic electrons above a few TeV to over 10 TeV. Such high energy electrons are
expected to give proof of sources near the Earth (e.g. supernova remnants such
as Vela: age < ~10^5 years located within <~1kpc). The idea itself is rather
old; a high energy electron emits synchrotron X-rays successively in the
geomagnetic field and thus gives several X-rays aligned on a meter scale. This
feature is a clue to overcome the background problem encountered in other
traditional observation methods. We critically examine the feasibility of this
approach assuming a satellite altitudes observation, and find that it is
difficult to derive a precise energy spectrum of electrons but is possible to
get a clear signal of the existence of several TeV electrons if the flux is
comparable to the level predicted by a class of plausible models. For such
observations, an exposure >1 m^2year would be needed. It would be attractive to
incorporate the present scheme in the gamma-ray burst observations.
