## Electric Displacement

Dielectric may not be neutral even when unpolarised. If the dielectric carries a charge density of free charges representing a net surplus or deficit of electrons in the atoms of the dielectric and is the charge density due to the polarizing effect of an electric field then the total charge density is given by The macroscopic electric field is related to the total charge density, and in the presence of matter Gauss's Law becomes We can rearrange this expression to give If we define then this equation becomes  is a new vector field, called the electric displacement. Since we can write The last equation is really Gauss's Law, apart from a constant factor, modified to include polarization charges. This can be further illustrated using the integral form of Gauss's Law: which becomes on substituting (1). has no clear meaning, but is useful because it makes many problems very easy to solve. It has the property that the normal component of at a dielectric boundary is continuous, as illustrated in the diagram below.  so  