If the permittivities are nearly the same, so that is small, the qualitative approach to…

If the permittivities are nearly the same, so that  is small, the qualitative approach to determining the field distribution given in connection with Fig. 6.6.7 can be made quantitative. That is, if  is small, the polarization charge induced by the imposed field can be determined to a good approximation and that charge, in turn, used to find the change in the applied field. Consider the following approximate approach to finding the fields in and around the dielectric cylinder of Example 6.6.2.

(a) In the limit where  is zero, the field is equal to the applied field, both inside and outside the cylinder. Write this field in polar coordinates.

(b) Show that this field gives rise to at the surface of the cylinder.

(c) Find the field due to this induced polarization surface charge and add it to the imposed field to show that, with the first-order contribution of the induced polarization surface charge, the field is

(d) Expand the exact fields given by (21) and (22) to first order in  and show that they are in agreement with this result.