Mathematical Methods in the Physical Sciences · Problem 7.14

FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas

Mathematics — Methods · spherical-shell-temperature

Problem

Find the steady-state temperature distribution in a spherical shell of inner radius 1 and outer radius 2 if the inner surface is held at 0∘0^\circ and the outer surface has its upper half at 100∘100^\circ and its lower half at 0∘0^\circ. HintHint: r=0r = 0 is not in the region of interest, so the solutions r−l−1r^{-l-1} in (7.9) should be included. Replace clrlc_l r^l in (7.11) by (alrl+blr−l−1)(a_l r^l + b_l r^{-l-1}).

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