C 10.9 A sphere with its center in the coordinate origin and with radius a is homo- geneously electrically polarised with polarisation P¯ = P zˆ.
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a. Determine the electric field E¯ on the z axis inside and outside the sphere.
b. Determine the electric displacement D¯ on the z axis inside and outside
c. Give the continuity condition for the normal component of the displace- ment vector D¯ and verify that this is fulfilled in this case.
Hint: Utilize formula (8.31) and apply formulas (10.22) and (10.54).
d. Show that ∇· D¯ = ρ f . Accordingly, the source of D¯ is free charge. Explain why D¯ is finite despite the fact that only a bound charge exists
in this case.