Helicon Oscillations in a Sphere

The helicon-wave problem is solved exactly for a sphere having an isotropic conductivity. The method involves the construction of a complete set of solutions of the helicon wave equation in spherical coordinates, a coordinate system in which the equation is not separable. Fitting the boundary condition at the surface of the sphere results in an expression for the induced ac magnetic multipole moments in the form of a ratio of two infinite determinants. Numerical results are presented for the induced dipole moment as a function of frequency and the strength of the applied dc magnetic field. A classification of the resonances in the dipole moment is proposed, and a table of resonant frequencies is given. Representative results for the induced octupole moment are displayed. The even-$l 2^l$-pole moments are zero. A preliminary comparison with the resonant frequencies observed by Rose with Na spheres in a field of 50 kG and with our own data on a small K sphere in a field of 2.5 kG shows an excellent agreement.