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Employing linearized Vlasov-Maxwell system, the Weibel instability embeddedin an ambient magnetic field is discussed for a semi-relativistic bi-Maxwelliandistribution hoping such a scenario occurs in some relativistic environmentse.g., gamma-ray burst sources and relativistic jet sources, supernovae, andgalactic cosmic rays where the perpendicular temperature is much dominated overthe parallel . The dispersion relations are analyzed analytically along withthe graphical representation and the estimates of the growth rate are presentedalong with the instability threshold condition in the limiting cases i.e.,xi<=1 (resonant case) and xi>>1 (non-resonant case). It is observed that therelativistic effect suppresses the instability and also lowers the thresholdfor the instability to set in. The ambient magnetic field contribution toinstability appears only in non-resonant case resulting in reduction of growthrate. However, the effect of the ambient magnetic field is diminished as we gofrom the weak relativistic regime to the highly relativistic one. We also notethat the ambient magnetic field generates real oscillations and thereletivistic effect reduces these oscillations. Further for field free casei.e., B0=0, the growth rates for Weibel instabilities are also presented in asemi-relativistic bi- Maxwellian plasma for both the limiting cases.

Whistler instability in a semi-relativistic bi-Maxwellian plasma