An example I am reading has the following setup: a dielectric slab of dielectric constant εr exists between z=0 and z=d, whilst an external electric field E0=E0k is applied with k a unit vector in the z direction. This setup exists for all x and y. The rest of space is a vacuum.
The aim is to compute the electric displacement, the electric field, the polarization and the bound charges of the system.
One of the first steps is to conclude that outside of the slab, the electric field is E0. There is no explanation for this, and it seems sort of obvious - for example assuming a parallel plate capacitor produces the uniform field and then placing the slab inside this capacitor, the field in the vacuum would remain the same as what it was in the capacitor without the slab. I can prove this using Gauss' law. However how can I actually prove it for this problem without such an analogy - ideally mathematically but any simple physical reasoning will do. When I think of the situation without the capacitor analogy, I'm wondering how it is we would know that the polarization of the dielectric would not have any influence on the vacuum field.