The figure shows a system of two concentric spheres of radii r 1 ​ and r 2 ​ and kept at temperature T 1 ​ and T 2 ​ , respectively. The radial rate of flow of heat in a substance between the two concentric spheres, is proportional to:

Options

A. `(r_(1)r_(2))/((r_(2)-r_(1)))`
B. `(r_(2)-r_(1))`
C. `(r_(2)-r_(1))(r_(1)r_(2))`
D. `1n((r_(2))/(r_(1)))`

Answer

A

Explanation

Consider a concentric spherical shell of radius r and thickness dr as shown in the figure. `(r_(1)ltrltr_(2))`
The radial rate of flow of heat through this shell in steady state will be
`H=(dQ)/(dt)=-KA(dT)/(dr)=-K(4pir^(2))(dT)/(dr)`
`rArrint_(r_(1))^(r_(2))(dr)/(r^(2))=-(4piK)/(H)int_(T_(1))^(T_(2))dT`
This on integration and simplification gives,
`H=(dQ)/(dt)=(4piKr_(1)r_(2)(T_(1)-T_(2)))/(r_(2)-r_(1))`