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December 18th, 2014, 02:54 AM   #1
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Is the charge distribution for an electric field unique?

If the electric field and boundary conditions are known exactly for a region of space, is it true that there exists only one charge distribution in that region of space that could have produced it?

My understanding of the uniqueness theorem in electrostatics is that for a given charge distribution and boundary conditions for a volume, there exists only one (unique) solution to Poisson's equation, and thus the electric field in that volume is uniquely determined. Does the arrow point the other way, too? If we know the field and boundary conditions, is the charge distribution uniquely determined in the volume? Is there a simple example that illustrates why or why not?
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December 18th, 2014, 05:56 AM   #2
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I'm not sure, but I think you are right. Comparing different charge distributions with equivalent electric fields boils down to considering purely radial systems, such as the charge sphere or the point charge. It is possible to show that a point charge has exactly the same electric field as a uniform charge density sphere with the same overall charge (fun task; this is true also with gravitational fields for spherical masses). However, this equivalence is only true outside the sphere; inside the sphere the charge differs from that of a point particle.
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