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I'm reading FLP vol II, and I would appreciate some help to understand the argument supporting Figure 6-6.

Basically they claim if a sphere has non-uniform charge distribution whose surface density is proportional to the cosine of polar angle, then this surface charge distribution is equivalent to two solid spheres with equal-but-opposite uniform volume density, separated by a small gap.

Intuitively this is reasonable, but I can't prove this rigorously.

Similarly, in section 14-4, similar argument was applied to cylindrical non-uniform surface charge distributions (whose surface density is proportional to the cosine of azimuth angle, the claim is it's equivalent to two equal-but-opposite solid uniform volume-charged cylinders separated by a small gap).

Any help would be greatly appreciated

Thanks