In this paper, electron trajectories are studied analytically in the rest frame of a plasma bubble using nonevolving, linear, radial electric and azimuthal magnetic fields in a spherical structure. The electron motion is broken into two distinct periods; one where it orbits around the periphery to the rear of the bubble, and one where it performs oscillations within the bubble interior. By using the first period as an initial condition for the second, general scalings are developed for the x-ray radiation produced by the electron oscillations. The equations are also analyzed to give self-trapping conditions for the electron and to examine the sensitivity of the transverse momentum to small variations from an orbit that is a circular arc. The scalings are in reasonable agreement with recent experiments on x-ray generation and predict a peak spectral brightness of S≈6×1027 photons/s mrad mmS≈6×1027 photons/s mrad mm 0.1%BW of radiation with a critical energy of 300 MeV using a single stage accelerator driven by a 120 PW laser.