Accuracy of Continuous Charge Distributions In E&M, it is common to represent the charge on a body with a continuous, scalar charge density function. In reality though, the body contains discrete charges. I understand why this approximation can be used to, say, calculate the electric field outside of a charged body -- averaging the charge over the body serves is a good approximation because actual charges are so close together. 
However, I will often see problems that ask you to calculate the electric field inside of a charged body (say a solid sphere of charge). Why can we get away with using the continuous charge distribution here? For the solid sphere of charge, using the continuous charge distribution, we would find that the electric field is spherically symmetric. However, isn't this not true? On a spherical shell within the body, you could be extremely close to a charge (making your E-field infinite) or in between two charges (making your E-field finite). Why is this approximation acceptable?
 A: Even though we generally refer to electrons as point particles this isn't the case. Electrons are quantum particles and they don't have a precise position. Instead they are delocalised over some region of space. In the case of metals this delocalisation is extreme as in principle the electrons in the conduction bands (where the extra charges would go) are delocalised over the whole piece of metal. In practice they would be delocalised over the distance between scattering events, which is small compared to the size of the metal but still large compared to atomic scales.
In insulators the charges will be more localised as they are probably associated with individual molecules or possibly defects in the structure. But this still means they are delocalised over roughly the size of a molecule, which is of course still infinitely bigger than a point particle.
The point of all this is that the smooth charge density is a remarkably good approximation due to the delocalisation of the electrons.
A: This approximation is accepted because physicists need only the approximate yet accurate enough values of any measurable physical quantity.
I mean who is gonna go near a charge so close to just measure its electric field variations.
By approximating charges as continuous distributions in spherical shell or solid sphere gives a much more reasonable and understandable formulas that can be derived easily from the fundamental laws.But if you go for the true picture,calculating the immense variation of electric field between molecules and try to derive up a picture of the field,i am so sure that it ain't gonna be nice to draw nor will anyone be able to understand it's significance.
If you go by these reasonable approximations,it won't be much different from the actual picture (as long as you stay quite away from the charges).
I suggest you read R P Feynman's lectures to see the real beauty of physics.
