Charge Inside a conducting hollow sphere

Let two concentric hollow charged conducting spheres of radius R1 and R2. If charge on inner sphere is Q1 and for outer it is Q2, wouldn't the resulting electric field due to the whole system be '0' inside inner sphere because charges at the surface of inner sphere will rearrange themselves to do so... because it's the property of a conductor?

This should also happen for outer sphere and electric field inside it will also be 0.

Is this what really happens?

Since the problem's geometry consists of two concentric spheres, one can apply Gauss' law, $$\oint_S \vec E\cdot n\;dA={Q_{enclosed}\over \epsilon_0};$$ to great effect in analyzing the electric field in the respective regions. Here $$\hat n$$ is the unit normal to the surface $$S$$.

For example, in the region of space inside the smaller sphere we can imagine a spherical Gaussian surface that is concentric with the others, Gauss law tells us that" $$\oint_S \vec E\cdot \hat n\;dA=0,$$ since the net charge enclosed by the surface $$S$$ is zero. The vanishing flux of the electric field implies that the charges on the inner surface of the smaller conducting sphere indeed arrange themselves in such a way as to produce no electric field in the region of space bounded by the smaller conductor. On the other hand, if one posits another concentric spherical Gaussian surface in the region of space between the two conducting spheres, one gets that: $$\oint_S \vec E\cdot n\;dA=Q_1/\epsilon_0.$$ Here it is easy to see that there is a net non-vanishing electric flux through the Gaussian surface, so the field in this region is not zero. However, the non-zero field's source is the charge on the smaller conductor, there is no contribution from the charges on the outer conductor.

1. The net electric field inside the inner conductor will be zero. You are correct there.

2. The electric field outside the inner sphere and inside the outer sphere (R1<r<R2) will be non-zero. The electric field in this region will be solely due to the inner sphere's charge. The contribution of the electric field in this region due to the outer sphere will be zero, but the net electric field in this region will be non-zero.

I hope this helps. Let me know in the comments, if you need any further explanation, I will improve my answer then.

Thank you.

The electrons in the inner sphere will rearrange themselves so as to shield the inside from any static electric field produced by charges outside it.

The electrons in the outer sphere will also do the same---but there is no charge outside the outer sphere in the first place. The outer sphere does not shield the region inside itself from electric fields produced by charges inside (indeed, that would be impossible, considering Gauss's law). So there's an electric field in the region between the two spheres, produced by the charge on the inner sphere.