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Two concentric spheres of radii r₁ and r₂ (r₁<r₂) are kept at temperatures T₁ and T₂ respectively, with a conducting substance filling the space between them. The radial rate of flow of heat through that substance is proportional to

Asked in AIEEE 2005 · Radial conduction through shells

• (1) (r₁r₂)/(r₂-r₁)
• (2) (r₂-r₁)
• (3) (r₂-r₁)/(r₁r₂)
• (4) ln ((r₂)/(r₁))

Step-by-step solution

Why the other options are wrong

• (2) comes from has the heat current growing as the gap is widened. Conduction works the other way: a thicker layer is a bigger resistance, so a wider gap must carry less heat.
• (3) comes from is the thermal resistance itself, (r₂-r₁)/(4π Kr₁r₂), quoted in place of the current. The two are reciprocals, so this option is exactly upside down.
• (4) comes from is the cylindrical answer, where R=1/(2π Kℓ)ln (r₂)/(r₁) because the area there grows as r, not as r². For spheres the integral of (dr)/(r²) gives a difference of reciprocals, not a logarithm.

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