Figure shows a thick shell made of electrical conductivity □ and has inner & outer radii of 10 cm & 20cm respectively and is filled with ice inside it. Its inside and outside surface are kept at diffe

Question

Figure shows a thick shell made of electrical conductivity □ and has inner & outer radii of 10 cm & 20cm respectively and is filled with ice inside it. Its inside and outside surface are kept at different potentials by a battery of internal resistance $\frac{2}{π} Ω& ε = 5 V$ . Find the value of $σ$ for which ice melts at maximum possible rate if $25%$ of heat generated by shell due to joule heating is used to melt ice.

TestHub · Accepted Answer

Shell resistance: R s ​ = ∫ 0.1 0.2 ​ σ 4 π r 2 d r ​ = 4 πσ 1 ​ ( 0.1 1 ​ − 0.2 1 ​ ) = 4 πσ 5 ​ . Joule heat in the shell, P = ( R s ​ + r ) 2 ε 2 R s ​ ​ , is maximum when R s ​ equals the internal resistance r = π 2 ​ Ω (maximum power transfer). Setting 4 πσ 5 ​ = π 2 ​ gives σ = 8 5 ​ S/m .

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