The rate of flow of heat depends on the nature of material, cross-sectional area and…
Physics · JEE Advanced · NTA Exams — Thermodynamics
The rate of flow of heat depends on the nature of material, cross-sectional area and temperature gradient. If a material of conductivity k has set up a temperature gradient in x direction, then rate of heat flow at cross-section having area A is \(\frac{\mathrm{dQ}}{\mathrm{dt}}=\mathrm{k} \mathrm{~A} \frac{\mathrm{dT}}{\mathrm{dx}}\)
Between any two points of conductor, if temperature difference is \(\Delta \mathrm{T}\) and rate of heat flow is H, then the resistance or opposition offered by the material to flow is defined as =\(\frac{\Delta \mathrm{T}}{\mathrm{H}}\). Now consider two very thin concentric metallic shells A and B of radii R1 and R2 (R2 > R1) and temperature T1 and T2 (T1 > T2) respectively. The hollow space between them is filled with sand of thermal conductivity k.
The temperature T of a point at distance r from the centre in steady state condition where R1 < r < R2 is
- \[\left(\mathrm{T}_{1}-\mathrm{T}_{2}\right)\left(\frac{\mathrm{R}_{2}\left(\mathrm{r}-\mathrm{R}_{1}\right)}{\mathrm{r}\left(\mathrm{R}_{2}-\mathrm{R}_{1}\right)}\right)\]
- \[\mathrm{T}_{1}-\left(\mathrm{T}_{1}-\mathrm{T}_{2}\right)\left(\frac{\mathrm{R}_{2}\left(\mathrm{r}-\mathrm{R}_{1}\right)}{\mathrm{r}\left(\mathrm{R}_{2}-\mathrm{R}_{1}\right)}\right)\]
- \[\left(\mathrm{T}_{1}-\mathrm{T}_{2}\right)\left(\frac{\mathrm{R}_{1}\left(\mathrm{r}-\mathrm{R}_{1}\right)}{\mathrm{r}\left(\mathrm{R}_{2}-\mathrm{R}_{1}\right)}\right)\]
- \[\mathrm{T}_{1}-\left(\mathrm{T}_{1}-\mathrm{T}_{2}\right)\left(\frac{\mathrm{R}_{1}\left(\mathrm{r}-\mathrm{R}_{2}\right)}{\mathrm{r}\left(\mathrm{R}_{2}-\mathrm{R}_{1}\right)}\right)\]
Answer
(B) T _ 1 - ( T _ 1 - T _ 2 ) ( R _ 2 ( r - R _ 1 ) r ( R _ 2 - R _ 1 ) )
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