We have examined in detail the synchrotron emission by electrons of energy greater than a few TeV in the earth's magneti field. The photon spectrum lies in the X-ray and &ggr; ray region. As the emission takes place in a narrow cone along the direction of the electron, the photons would be incident nearly along a straight line on a detector. This unique feature provides the signature to identify the electron unambiguously. The mean energy of the photons being proportional to the square of the electron energy allows us to determine the energy accurately. Though it may appear that one needs to know the arrival direction of electrons to obtain its energy, we have shown that an omnidirectional detector can be satisfactorily used to estimate the energy. We also show that the colleting power of the detector is a sensitive function of the area of the detector A, the energy of electron E0, and the number of photons required to identify an electron n&ggr;; asymptotically the collecting power is proportional to A1.43 E0n&ggr;-1.8. An instrument, with an energy threshold for the detection of photons can be used to measure reliably the integral flux of electrons, even if it has limited energy resolution. We have calculated the event rate expected by using an ideal balloon-borne detector capable of detecting above 20 keV at 4 g cm-2 of atmospheric depth over Palestine Texas, and compared with the expected rates using instruments based on currently available techniques of detection.