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)lt theta_(2))`. Calculate the rate at which heat flows radially through the material.

Explanation

Let us drow two spherical shells of radii x and `x+dx` concentric with the given system. Let the temperatures at these shells be theta and `theta+dtheta` respectively. The amount of heat flowing radially inward through the material between x and `x+dx` is
`(DeltaQ)/(Deltat)=(K4pix^(2)d theta)/(dx)` . thus, `K4piint_(theta_(1))^(theta_(2))d theta=(DeltaQ)/(Deltat)int_(r1)^(R2)dx/x^(2)` . or, `K4pi(theta_(2)-theta_(1))=(DeltaQ)/(Deltat)((1)/R_(1)-(1)/R_(2))` . or, `(DeltaQ)/(Deltat)=(4piKr_(1)r_(2)(theta_(2)-theta_(1)))/(r_(2)-r_(1))