The surface of a prolate ellipsoid (a> b= c) obeys the equation x² b2 y² z2 + + b2 a² - 1. (1) A conductor with this shape is placed in a uniform electric field E = Eoz. In this problem, you will calculate the induced dipole moment in the nearly spherical limit e < 1, where e = √1 − b²/a² is the eccentricity of the ellipsoid. e= (a) Show the surface of the ellipsoid can be written in spherical polar coordinates as r(0) = b/√√1 – e² cos² 0. (b) Find the potential outside the ellipsoid by demanding that, to order e², the potential (r, 0) is constant on the surface defined by r(0). You will need to keep terms up through = 3. What is the induced dipole moment on the ellipsoid? l pz (c) The exact expression for the dipole moment is p₂ = 0V Eo/n(2), where V = Tab² is the volume of the ellipsoid and the appropriate depolarizing factor is 1+e n(z) 1 = e2 2e3 In 1 - e -2e). (2) Expand this expression for pz to O(e²) and compare with (b). See Landau and Lifshitz, Course of Theoretical Physics Vol 8: Electrodynamics of Continuous Media, Elsevier, Amsterdam (2008) §4 for further details.

Fig: 1