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I'm a mathematician self-studying physics for fun and I'm trying to wrap my head around a simple concept, which is a consequence of approximations. Basically, by Coulomb's law, if I have a point charge and look at the electric field, then the electric field grows without a bound when I come close enough to the charge. On the other hand, if I have a flat infinite plane with a continuous uniform surface charge, then the electric field is uniform and independent of the distance to the plane and is discontinuous at the boundary. Ultimately, all charge is discrete and if we distribute discrete point charges on a plane, then obviously the magnitude of the electric field again grows without bound again when we come close enough to one the point charges.

Now my issue with some of the arguments in e.g. Griffith's electrodynamics textbook is that some results can be somewhat easily proven by just saying that "yeah, everything is actually discrete, so just apply superposition", which sometimes avoids some pesky integrals. On the other hand, as the example above shows, we can get contradicting results between the continuous and discrete cases.

My somewhat fuzzy question is mainly the following: When should one be careful of suddenly making conclusions using discrete arguments when otherwise working in the macro scale assuming continuous charge distributions?

I assume part of not losing sleep over these things is learning to think like a physicist.

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I think there is no contradiction. You are just applying an approximation (that the charge distribution is continuous) outside its applicable range. A continuous charge distribution os only a good approximation for a discrete distribution if you are far enough away.

"yeah, everything is actually discrete, so just apply superposition"

When should one be careful of suddenly making conclusions using discrete arguments when otherwise working in the macro scale assuming continuous charge distributions?

You seem to think that the principle of superposition only applies to discrete charges, but the superposition principle holds for both discrete and continuous charge distributions. The reason why the net effect of two individual electric fields on a charge is the same as is the sum of their individual effects is because maxwells equations are linear. For example consider the gauss law $$\rho=\rho_1+\rho_2=\nabla \cdot E_1+\nabla \cdot E_2=\nabla \cdot (E_1+E_2)=\nabla \cdot E$$ That is only possible because the divergence is a linear operator. To describe point charges, you need delta distributions, which means the discrete case is actually mathematically more complicated than the continuous one. Of course maxwells equations are only justified by experiment, so you might as well justify the superposition principle for electric fields directly by experiments.

If you want to make a transition from the discrete to the continuous case you could use $N$ disrete charges, each with charge $q/N$, such that the total charge stays the same as you take the limit $N$ to infinity. I don't see how that would break the superposition principle however.

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