# Is there a limit to the charge of a body?

Can a body be limitlessly charged? Suppose a sphere of radius $R$. Can it be charged limitlessly?

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There is a conjectured limit $Q \leq 2M$ where $M$ is the mass in certain units. This corresponds to an "extremal" black hole. Charged black holes haven't been seen in nature, but the extremal limit is a strong conjecture. Unfortunately the wiki page en.wikipedia.org/wiki/Extremal_black_hole is rather sparse at the moment. Hopefully an expert can find you better references, but googling for "extremal black hole" would be a start. –  Michael Brown Jan 19 '13 at 14:36
Just realised that you are asking physics.stackexchange.com/questions/51626/… again. This makes my black hole discussion a bit of a digression. –  Michael Brown Jan 19 '13 at 14:48

A real body (a metallic sphere) will discharge in vacuum much earlier than at $Q$ sufficient to produce electron-positron pairs. If the body is charged negatively, the excess of electrons will be emitted due to cold emission. If the body is charged positively, the excess of positively charged nuclei will also leave the body due to strong repulsion. The barrier that holds charges together in a neutral body is overcome if the body is charged. The electric field strength on the surface should be of the order of the intra-atomic electric field $E\propto e^2/a_0^2$ or even less. So my estimation is $Q_{max}<e\frac{R^2}{a_0^2}.$

Factually it is the same mechanism as a discharge due to ionization of the air - the ions are always present in the body.

Any inhomogeneity of the surface will essentially decrease the maximum $Q$.

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I don't think your estimate is right--for a 10 m body, you'd get something like $Q < (10\, m)\frac{10^{-19}\,C}{10^{-10}\,m} = 10\,nC$, and we can clearly charge capacitors much more than this. –  Jerry Schirmer Jan 19 '13 at 16:44
I think you could look at something like the ionization energies for atoms. I'm sure you could doctor an atom so that it had a super high ionization energy. –  Jerry Schirmer Jan 19 '13 at 17:12
@JerrySchirmer: You are, of course, right. I just used the smallest potential energy of an electron in an atom and compared it with the potential energy in the charge $Q$. One can imagine tunneling at smaller field strength (introduce a tunneling factor). I made a typo in my answer - it is ratio of squares that is at $e$. Thanks to you, I corrected it. –  Vladimir Kalitvianski Jan 19 '13 at 17:18

If it is in air (or any other substance), there is a limit where the electric field of the object is going to be enough to ionize the surrounding medium, and the resulting current will drain the object of its charge.

Similarly, if the object is immersed in vacuum, you will eventually have an electric field sufficient to "polarize the vacuum" by creating electron-positron pairs that will do the same thing. So a spherical object will definitely have a maximum charge that it can hold.

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The process you mention in the second para is called Schwinger pair production in case the OP wants to look it up. –  Michael Brown Jan 19 '13 at 14:49
@MichaelBrown: the body will explode well before reaching such a field strength. –  Vladimir Kalitvianski Jan 19 '13 at 16:35