Polarization in a dielectric

If an electrical field E is applied between two parallel metal plates separated by a fixed distance (a parallel-plate capacitor), one plate acquires a positive charge and the other a negative charge (Fig. 26.1a). The charge per unit area of the plate is the charge density a. 

Since the charge is uniformly distributed on each plate, the field is perpendicular to the plates, E = E, and the tangential components of the field are zero. 

If we draw a gaussian surface, S, which encloses both plates, then the total charge enclosed by this surface is q = aA -h ( — (t)A = 0, where A is the area of the gaussian surface parallel to the plates. Then, by Gauss s law, Eq. (A.II.14), 

Since the tangential components are zero, there is no contribution to the flux from the surfaces perpendicular to the metal plates. This leaves 

But we showed that E on the outside surface is zero, so the first integral is zero. Since the charge density is uniform, the E on the inside surface is constant everywhere on the surface and can be removed from the integral. The equation becomes EA = (tA/cq, or 

---