Problem 3.79

Near the plane surface of a uniform isotropic dielectric with permittivity \(\varepsilon\) the electric field strength in vacuum is equal to \(E_ {0}\), the vector \(\mathbf{E}_ {0}\) forming an angle \(\theta\) with the normal of the dielectric's surface (Fig. 3.11). Assuming the field to be uniform both inside and outside the dielectric, find: (a) the flux of the vector \(\mathbf{E}\) through a sphere of radius \(R\) with centre located at the surface of the dielectric: (b) the circulation of the vector \(\mathbf{D}\) around the closed path \(\Gamma\) of length \(l\) (see Fig. 3.11) whose plane is perpendicular to the surface of the dielectric and parallel to the vector \(\mathbf{E}_{0}\).