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In electrostatics, we usually consider charge to be continuous on any body, to calculate the electric field of the body. For eg. I had proved the Shell Theorem taking an infinitesimal charge of $dq$ on the sphere. But we also know that charges are quantized (In terms of electrons and protons). They are finite and not infinitesimal. So is considering them continuous an approximation or is there a reason to why we can do it?

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Considering the charge to be continuous is an approximation, but since the electron charge is $1.60217657 \times 10^{-19}$ coulombs it's a very good approximation. –  John Rennie Jul 9 '13 at 14:42
    
It's an approximation, used because we want to be able to use calculus for these problems. As the number of charges increases, this approximation becomes better. –  Will Jul 9 '13 at 14:42
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I'm surprised there is no mention of the Thomson problem yet. As the question refers to the shell theorem with a finite number of charges, this is answered exactly by that topic. I mean, yes, the shell theorem is an approximation, but physics has also solved the generalized problem. You can look up the exact amount of error that the continuum assumption introduces. en.wikipedia.org/wiki/Thomson_problem –  AlanSE Jul 9 '13 at 14:54
    
@Qmechanic hm, at least the (fluid dynamic) context in the as duplicates linked to questions seems to be quite different from the contest of this question, even though the answers may finally boil down to the same explanation. Not sure, if the question is really a duplicate or just related ...? –  Dilaton Jul 9 '13 at 19:29
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marked as duplicate by Dan, Qmechanic Jul 10 '13 at 23:37

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Yes, the continuity of the electric charge is an approximation that is valid whenever the relevant charges are much greater than the elementary charge (of the electron, or the proton). When we deal with numbers like $1,234,567\,e$, it doesn't really matter that it should have been $1,234,567.8\, e$: very large numbers may be approximated by a nearby integer while the error is small.

This approximation is perfectly OK for macroscopic circuits. For example, if the currents are of order 1 Ampere, then it means that 1 Coulomb goes through the wire each second. But one Coulomb is about $10^{19}$ elementary charges, a number much greater than one, so the relative error is just $10^{-19}$ or so if we approximate the number of elementary charges by the nearest integer.

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