Non-radial silent magnetizations on a fixed spherical shell are exactly those whose angular coefficients satisfy one finite set of radial moment conditions per degree; if the shell is allowed to shrink, only radial (and in 2D, degree-one) magnetizations remain.
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A converse to generalized Runcorn's theorem
Non-radial silent magnetizations on a fixed spherical shell are exactly those whose angular coefficients satisfy one finite set of radial moment conditions per degree; if the shell is allowed to shrink, only radial (and in 2D, degree-one) magnetizations remain.