The closed solution and verification of the electromagnetic field of a vertical dipole over the sphere

The closed solution of the electromagnetic filed of the vertical dipole over a perfect conducting sphere is introduced, the problem of discontinuity encountered in the series expression of the direct field is solved. Computation result is compared with the direct summation of the series form solutions of the electromagnetic filed over the sphere. Based on the same electromagnetic model, an approximate method is verified by the closed solution. If the vertical dipole and the observation point are near the surface of the sphere and the frequency is not too high, then the computation results of the two methods are quite the same. If the vertical dipole and the observation point are not near the surface of the sphere and the frequency is high, then the behavior of interference is gradually obtained by the closed solution but not by the approximate method.