Finding volume and surface densities of bound charge

[Askhat15]

Homework Statement

A slab of material has parallel faces. One coincides with the xy plane (z = 0), while the other is given by z = zt . The material has a nonuniform polarization P = P(1 + αz)zˆ where P and α are constants. Fin the volume and surface densities of bound charges[/B]

The Attempt at a Solution

I think that

holds for this case. However, I couldn't any further. Please, help asap. Will appreciate so much![/B]

 

[Korak Biswas]

Locally the surface density of bound charge is $\vec P. \hat n$. Here $\vec P$ is polarization and $\hat n$ is the unit vector along the direction of local surface area. Now integrate it over the whole surface to get the total bound surface charge. Similarly the local volume density of surface charge is $- \vec \nabla.\vec P$. Integrate it over the whole volume to get the total bound volume charge.