The method of images with dielectrics

Problem:

A charge q is situated at the point x = h > 0, y = z = 0 outside a homogeneous dielectric which fills the region x < 0.
(a)  Write the electric fields just outside and just inside the dielectric in terms of the charge q and surface charge density σb of bound charges on the surface of the dielectric.
(b)  Express σb in terms of the electric field just inside the dielectric.  Let ε be the dielectric constant of the dielectric.
(c)  By using the equations obtained in (a) and (b), show that
σb = -[1/(2π)][(ε - 1)/( ε +1)][qh/(h2 + y2 + z2)3/2].
(d)  Calculate the electric field E' due to σb at the position (h,0,0) of the charge q.  Show that it can be interpreted as the field of an image charge q' situated at the point (-h,0,0).
(e) Show that the charge q experiences the force F = -[(ε - 1)/(ε + 1)] (q2/4h2)i.

Solution:

Problem:

Two spherical clouds, each of 1 km diameter, are separated by a distance of 2 km (center-to-center).  Each cloud is uniformly charged with charge density ρ0.  They are floating with their centers 1 km above a fresh-water mountain lake of non-conducting pure water with dielectric constant ε = 1.7ε0.  The bottom surface of each cloud is at a potential of 106 volts compared to the lake immediately below it.
(a)  What is the charge density ρ0?
(b)  What is the electrostatic force on each cloud?
(c)  What is the index of refraction of the lake water?

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Solution: