Two thin metallic spherical shells of radii `r_(1)` and `r_(2)` `(r_(1)lt r_(2))` are placed with their centres coinciding. A material of thermal conductivity `K` is filled in the space between the shells. The inner shells is maintained at temperature `theta_(1)` and the outer shell at temperature `theta_(2)` `(theta_(1)lttheta_(2))`. Calculate the rate at which heat flows radially through the material.

Options

A. `(R_(1)+R_(2))/(2)`
B. `(R_(1)R_(2))/(R_(1)+R_(2))`
C. `(2R_(1)R_(2))/(R_(1)+R_(2))`
D. `R_(1)+(R_(2))/(2)`

Answer

C

Explanation

`theta_(1) - theta_(2) = Delta theta (theta_(1)-theta)/(overset(R)underset(R_(1))int(dr)/(K4pir^(2))) =(theta_(1)-theta_(2))/(overset(R_(2))underset(R_(1))int(dr)/(K4pir^(2)))`
`(Delta theta//2)/((1)/(4piK_(1))[(1)/(R_(1))-(1)/(R)]) =(Delta theta)/((1)/(4piK_(1))((1)/(R_(1))-(1)/(R_(2))))`
`rArr R = (2R_(1)R_(2))/(R_(1)+R_(2))`