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In their last homework, some of my students miscalculated a charge to be $10^{20}$ C over a squared meter and I was wondering if there was as much positive charges in the entire Universe. It would do the trick if I could simply estimate the total number of protons (including those in nucleus) in the entire Universe but how to do so in a simple way ?

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I think what you are looking for is the Eddington number, see for example en.wikipedia.org/wiki/Eddington_number. Maybe this is be a good basis to answer your question. –  Bru Apr 1 '13 at 9:37
    
Thanks, that's exactly what I was looking for. –  JJ Fleck Apr 1 '13 at 10:24
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Just in case anyone is misled, it might be worth re-emphasizing that Eddington's argument for the number is probably only good for its comedy value. The following link which is given on that wiki page is the one to follow for a more reasonable justification of the value. –  twistor59 Apr 1 '13 at 12:39
    
The total charge of the universe is actually not a well-defined quantity. There is a discussion of this on p. 457 of Misner, Thorne, and Wheeler for the case of a spatially closed universe. It holds more trivially in the case of a spatially infinite universe, since there's no particular reason for the result to be finite. If you meant the observable universe, then: en.wikipedia.org/wiki/Observable_universe#Matter_content –  Ben Crowell Apr 1 '13 at 15:18

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