The figure shows a system of two concentric spheres of radii `r_1 and r_2` are 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. `In((r_2)/(r_1))`
B. `(r_2-r_1)/(r_1r_2)`
C. `(r_1-r_2)`
D. `(r_1r_2)/(r_2-r_1)`

Answer

D

Explanation

(d)
Consider a shell of thickness (dr) and of radius(r) and
the temperature of inner and outer surfaces of this shell
be T, `(T-dT)`
`H=(KA[(T-dT)-T])/(dr)=(-KAdT)/(dr)`
`H=-4piKr^2(dT)/(dr) (:' A=4pir^2)`
Then, (H) int_(r_1)^(r_2) 1/(r^2) dr=-4piKint_(T_1)^(T_2) dT`
`H[1/(r_1)-1/(r_2)]=-4piK[T_2-T_1]`
or `H=(-4piKr_1r_2(T_2-T_1))/(r_2-r_1)`