The free‐electron laser (FEL)–electrostatic electron cyclotron resonance maser (EECRM) hybrids, a new kind of FEL comprised of a relativistic electron beam and an electrostatic centrifugal field, are presented and studied in this paper. In a fluid dynamical linear theory, the dispersion equation for transverse electromagnetic (TEM) mode operation is derived, analyzed, and numerically solved. It is found both analytically and numerically that FEL instability occurs near beam resonance [(&ohgr;l−klv∥)2−&ohgr;2p(&ggr;−2∥−&bgr;2∥&bgr;2⊥&Jgr;˙20/ &OHgr;2l)=0]. However, FEL instability near cyclotron resonance (&ohgr;s−ksv∥=±(2−&bgr;2⊥)1/2&Jgr;˙0) is not found in the TEM mode analysis.