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If Er is constant, the potential 0 will be linear in r with the assumed form 0(r) = A(r - B) + C, (2.9) where A, B, and C are constants to be determined. If 0 is the potential at the inner cylinder, r = r, then #(r) = A(r - ra) + Oo. (2.10) Using (2.10) in (2.7a) and defining p(r) = po, then #() = aPO (r - ra) + 0, (2.11a) EO Er rapo Or CO (2.11b) Now a suitable equation for p(r) is found by inspection after substituting (2.Ila) into (2.8). The charge density is then given by (2.12) p(r) = rapo The boundary conditions on the outer cylinder, r = rb, have not yet been considered. The assumption that E, is constant over the domain precludes specification of another boundary condition on the outer cylinder. A numerical boundary value problem requires specification of an additional boundary condition to ensure that the solution is unique. Requiring (rb)= 0 determines the geometry of the outer cylinder. Using the boundary condition in (2.11a) and solving for rb yields rb = O60+ rapo ra. (2.13) The analytical solution and geometry of the outer cylinder are fully defined if po, ra, and #o are specified. 29