When to use which representation for an electric field
In class we covered three types of possibilities to evaluate the electric field for static problems. Unfortunately, most physics textbooks cover these ways without addressing the question of applicability when they introduce these equations due to historical motivation.
One possibility is <PERSON>'s law:
$$\int_{\partial V} \langle E, n \rangle dS = \int_V \frac{\rho}{\varepsilon_0} dx$$ This can only be used, when $E$ does by geometrical consideration not depend on the variables that the surface integral depends on(like in spherical symmetry, when it is not a function of the angles, but just of the radius), then it can be used to determine the electric field by a charged sphere. What we can get then, is an electric field on the surface of some volume that contains a particular charge.
Then we have:
$$ E(x) = \frac{1}{4 \pi \varepsilon_0} \int_{\mathbb{R}^3} \frac{\rho(x')(x-x')}{||x-x'||^3} dx'.$$
This is more general in the sense that it does not require any symmetry or reduction to charges inside some volume.
And finally we have $$\Delta \phi(x) = -\frac{\rho(x)}{\varepsilon_0},$$
which is the most general way to think about electrostatic problems and contains (assuming that the solution fulfills all boundary conditions) all information.
I hope my understanding was correct so far. 903
In that case, my question is: How do I distinguish boundary value problems from problems where the first two equation apply?
Many problems are like: Given a sphere with a particular charge distribution, calculate the field everywhere around! What does this mean, does this mean that I should think of this as a ball of charges assembling in vacuum and the material around being vacuum too? Cause if I assume that this is a metallic sphere or a sphere made out of some material which is different from the material (probably varcuum) around, then this would give me an interface which means that my first two equations don't apply anymore, as I would get boundary conditions. ( I am talking now about this in a strict theoretical sense, maybe they would give me the right result, but for the wrong reasons). Or do these two equations also apply, if the ball is an insulator? 903