Dieletric Boundary Conditions (Parallel Plate Capacitor)

[jegues]

Homework Statement

See figure attached for problem statement, as well the solution.

Homework Equations

The Attempt at a Solution

I'm confused as to how he is writing these equations from the boundary conditions.

What I understand as the boundary condition for D is,

$$\hat{n} \cdot \vec{D_{1}} - \hat{n} \cdot \vec{D_{2}} = \rho_{s}$$

With the normal vector directed from region 1 to region 2.

With this I only generated 2 equations, as there is only 2 boundarys; one being from d1 to d2 and the other from d2 to d3.

He labels,

$$\rho_{s0}, \rho_{s1}, \rho_{s2}, \rho_{s3}$$

I'm confused as to what these surface charge densities pertain to? Is the surface charge density with subscript 0 and 3 the charge on the plates of the capacitor? Are 1 and 2 the surface charge densities on the faces of the dielectric material?

He states by conservation of charge that,

$$\rho_{s0} = -\rho_{s3}, \quad \rho_{s1} = -\rho_{s2}$$

This doesn't seem obvious to me at all. Can someone show me how he is drawing such a conclusion? (Is there some math behind it?)

 

[jegues]

Bump, still looking for some help on this confusion please!