Target field approach for spherical coordinates

Using the method of separation of variables and expanding the magnetic field generated by currents flowing on a sphere by means of Fourier-Legendre expansion, we can obtain the inverse relationship between the current density and magnetic flux density. The current density required to generate a specified target field inside the sphere may be evaluated. This approach develops Turner´s target field method and can generate axially directed field in large usable volumes inside the sphere, which can satisfy the requirement of high-energy physics, medical treatment, spatial detection etc. Using this method we can generate uniform or gradient magnetic field that vary linearly along the z or y direction. The commercial finite element method software also verifies the validation of this approach.