Comparison of Electric and Magnetic Self Forces in a Toroidal Geometry

Magnetic and electric self fields are evaluated for a rectangular beam centered within a closed toroidal chamber of rectangular cross section. The chamber walls are regarded as conductors, at zero electrostatic potential, that exclude the time varying magnetic fields of the beam. Expressions are derived for the axial derivatives of the electric and magnetic fields in the mid-plane, and numerical evaluations of these derivatives are reported for examples relevant to betatron design. Attention is directed to the failure of the electric and magnetic self forces to exhibit, in this toroidal geometry, the strong 1- sq Beta cancellation characteristic of the self fields of a beam in a straight conducting chamber.