Consider a closed spherical surface surrounding a non-uniformly charged insulating material. How does the electric field behave inside the surface, and what implications does this have for field lines emanating from the surface according to Gauss's Law?

Inside the closed spherical surface the electric field can be non‑zero because the insulating material contains charge that is not uniformly spread, so the field inside depends on the local charge density. Gauss’s Law says the total outward flux from the surface equals the total charge inside divided by the permittivity, so if the enclosed charge is positive the field lines must point outward from the surface, and if it is negative they point inward. Thus, even though the surface is closed, the field lines that cross the surface can originate from charges inside or terminate on them, reflecting the net charge inside. For example, if a single positive charge sits inside the sphere, the field lines start at that charge, cross the spherical surface outward, and leave it, producing a net outward flux equal to the charge divided by ε₀. This shows that a closed surface does not guarantee a zero field inside; the field depends on the enclosed charge distribution.