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.