Faraday Cage Explain (Gauss Law) I was given the following explanation of Faraday cage: There is no electric field inside a hollow, charged conducting shell because a Gaussian surface inside the shell must have zero flux through it since it encloses zero charge. In the picture below, ${\bf S}$ is the Gaussian surface.

But I am not convinced by this explanation. There is clearly a field outside the shell since the shell is charged. What if I consider a closed surface ${\bf T}$ outside the Faraday cage like in the image below:

Again, the charge enclosed by ${\bf T}$ is zero so shouldn't this mean that there is zero flux through ${\bf T}$ and hence no electric field inside ${\bf T}$?
 A: All electric flux start at a positive charge (a.k.a source of flux) and end at a negative charge (a.k.a sink). Since the Gaussian surface you considered outside the charged shell houses neither positive nor negative charges, there can't be any net electric flux going into or coming out of the Gaussian surface of your choice. 
That means all electric flux that goes in must come out.
A: In the second case, the netto flux is also zero. There is an equal amount electric field going in the volume as out the volume. Hence, if you take the surface integral over the volume, the result is zero. 
A: 
Again, the charge enclosed by T is zero so shouldn't this mean that there is zero flux through T and hence no electric field inside T?

Let's say we have a point charge $q$ at the origin $O$. Let $S$ be any closed surface that doesn't contain $q$. Electric field is non-zero everywhere (It's $\vec{E}(\vec{r})= \frac{kq}{r^2} \hat{r}$ : we know this). And yet, what we find is that $\oint_S \vec{E} \cdot d\vec{S} = 0$ for such a closed surface even though electric field is non-zero inside $S$. The same applies to your scenario for the second closed surface you drew outside the cage.
So yes, as you say, the argument to show that the electric field inside a cavity of a conductor that contains no charge is zero, involves more reasoning than what has been told to you. Please refer to the following picture that is taken from Griffiths' Electrodynamics book.

A: If T is also a hollow conductive shell then there is zero flux through T and hence no electric field inside T.
This happens because the external electric field, due to the charge distribution in S, produces a displacement of free charges in T (a conductor) which generates an charge re-distribution in T.
This new distribution of charge in T produces a local electric field in T which opposes to the external electric field caused by S. The result is zero net flux in T (equilibrium is re-established).
A: If you have a conducting wire with all the charge is confined to flow within the (regardless of the potential across it).
Now, bend that wire into a loop.  Is there current outside or inside of the loop?
What if you extend that into a 3d shell?
