4.4.1 Analysis of the Lorentz Field (newly added)

Here, we are interested in calculating the field from the free ends of dipoles i.e. Lorentz field El, lined along the cavity wall in the direction of applied field, as shown below. This charge density arises from the bound charges and is determined by the normal component of polarization/dielectric displacement vector P and is written as 

Now, since each element dA contributes to the field, according to Coulomb’s law, the radial field intensity is

Each dA’s angular position is between θ and θ+dθ and for each dA element, there is another dA element on the other side of the sphere which produces same but opposite horizontal field component.
Horizontal components cancel each other and vertical components dE₂ cosθ  survives

So the total field intensity is

The field intensity is parallel to the applied field and actually strengthens it. Now we can also rewrite dA as