# Is it meaningful to imagine a sphere uniformly charged with 2e?

If the charge is large, considering charge density is usually meaningful despite the discrete nature of electrical charge. The following sentence is part of a problem in a textbook on electromagnetics:

Imagine a sphere of radius a filled with negative charge of uniform density, the total charge being equivalent to that of two electrons.

The charge is only 2e, and so it seems that the model wouldn't work at all. Is there any real-world application of such a calculation?

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Buckyball with shared orbitals? Maybe QM will allow some sort of psuedo-distributed charge? –  AlanSE Jun 14 '11 at 19:27
Why not? Electrons in metal are not "point-like" and the metal is polarizable. –  Vladimir Kalitvianski Jun 14 '11 at 19:35
As the Fermi velocity of electrons in such a metal is generally quite high (~10^6 m/s), I presume two extra electrons could bounce around in the sphere at such high speeds that it might seem as a uniform charge density. Of course, if we can just say that it is in an eigenfunction of the Hamiltonian, the electrons will be spread about the entire and with it their charge. –  Kasper Meerts Jun 14 '11 at 19:52
The Fermi velocity is a quantum mechanical notion and it does not provide a classical averaging. In metals the extra electrons are not normally localized so with a metallic ball the charge distribution can by spherical. As well as with an axial symmetry (a circular current). –  Vladimir Kalitvianski Jun 14 '11 at 21:21

No, I would suspect there is no particle with such a uniform charge density. As you state, at this level, charges are discrete with quarks having multiples of $$\pm\frac{1}{3}e$$ and leptons having either +/-e or zero charge. The charge density of a proton falls off nearly exponentially, see

http://prl.aps.org/abstract/PRL/v99/i11/e112001

All I found was the charge density of Helium ions which seems to be very roughly constant. But the total charge would be +1. see

http://www.ae.utexas.edu/~lraja/APL_80(4)_Kothnur_et_al.pdf

This is probably not intended to be a real world problem but just a homework problem which keeps the numbers simple. At this scale you won't be able to use classical electrodynamics anyways.

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