Why is electric field inside a cavity of a non-conducting uniformly charged sphere not zero? Consider a charged non conducting solid sphere of uniform charge density, and it as hole of some radius at the centre. 
Now suppose I apply Gauss law.

As there is no charge inside the cavity, no charge is enclosed by Gaussian sphere, so electric flux is zero, hence electric field is zero.
But this is not true according to sources like this, textbook etc..
What am I missing here?
My question about how to have nonzero field in a region where the flux through the boundary of the region vanish? As mentioned by– rob
 A: Gauss's Law only gives results for the integral over a whole, closed surface. This does not mean the electric field is zero. It simply means that all field lines entering the volume also exit it at some other point.
A: In the video the electric field lines are as shown in blue in the diagram below.  

Let the edge of the cavity be the Gaussian surface which has no charge within it.  
Consider small areas $\Delta A$ on either side of the cavity as shown in the diagram in red.
Because of the symmetrical nature of the situation you can imagine that the electric flux entering the cavity through area $\Delta A$ is the same as the electric flux leaving the area $\Delta A$ on the right hand side.
This means that the net flux through those two surfaces is zero.  
Doing the same for the whole Gaussian surface leads to the result that the net flux though the surface is zero commensurate with the fact that there is no charge within the surface.  
A: Just to add to what Noah said, the law says that the total flux through the surface is zero. To make this point clear, imagine a region of uniform electric field $\vec{E} = E_0\hat{x}$ along the x axis. Consider a cube of unit area faces with two of its opposite faces normal to the field. From the RHS of Gauss law we know that the flux has to be zero as there are no charges enclosed by the cube. The LHS says 
$\sum_{n=1}^6 \vec{E}\cdot \vec{A}_{n} = E_0\hat{x}\cdot\hat{x} + E_0\hat{x}\cdot-\hat{x} = 0$
Thus the flux is zero by explicit calculation as well. Remember that the electric field and area are vectors. So the relative directions are very important. If flux is zero all you can be sure of is that the extent of field lines entering is equal to that of exiting. 
A: The field of a uniformly charged shell is zero inside the shell in agreement with Gauss's law, unless there is an external field. 
A uniformly charged sphere with a concentric spherical cavity is a concentric sum of such shells, so the field in the cavity vanishes. 
