Electric field inside a hollow charged sphere

Question

1. We have read that electric flux is the measure of electric field lines going radially outward from a surface... Now in case of positively charged hollow sphere we can some-how understand that electric flux and field be zero but if we take the case of negatively charged sphere we see field lines directed towards surface from centre (we can assume diagram also). So now can we say the electric flux is non zero in that case?

2. Also when we move inside the hollow sphere apart from centre then we find displacement vector different (we can assume the vertical components cancel out each other but the horizontal components doesn't necessarily cancel out each other) so how can we say the net electric field is zero at every point inside hollow sphere? Is there any experimental or scientific evidence?? We knew very well that net electric flux doesn't necessarily defines electric field at a point even though we solve the net electric field inside hollow sphere with gauss law. (which gives indigestible answer) Someone pls explain what is the correct answer and correct method and any additional knowledge about that topic which can be helpful in understanding the concept.

Answer 1

1. For a negatively charged sphere the field lines point inward toward the center. Those lines basically come from infinity and stop at the surface of the sphere. The flux hitting the outside of the sphere is non-zero, but just inside the surface it's 0, consistent with Gauss' law.

2. You're correct that Gauss' law only gives the total surface integral of the E-field. Because we're dealing with a uniformly charged sphere, you can use symmetry arguments to say that the E field can only depend on r, not $\theta$ or $\phi$. Thus any gaussian surface that is itself a sphere integrates as $\int \vec{E} \cdot d\vec{A} = E_r A = \frac{1}{\epsilon_0}q_{enc} $. $E_r$ must be 0 for any point inside the sphere; $E_{\theta}$ and $E_{\phi}$ must be 0 due to azimuthal and polar symmetry of the charge distribution.