The correct option is
A R2−R12Given: Heat flows radially outward through a spherical shell of outside radius R2 and inner radius R1. The temperature of inner surface of shell is Δ1 and that of outer is Δ2.
To find the radial distance from centre of shell where the temperature is just half way between Δ1 and Δ2
Solution:
By using the current analogy of thermal flow,
change in temperature = potential difference
And, heat flow = current
conductivity = thermal conductivity
And Resistance = thermal resistance
Here
So,
Δ1−Δ2Rshell=H as V2−V1R=I
⟹H=Δ1−Δ2lKA⟹H=KA(Δ1−Δ2)R2−R1.......(i)
as length will be difference in the radius.
Let us consider any cross sectional area, A
This same H current will flow through all points across the shell
Let x be the distance from the center where the temperature is half, so
((Δ2+Δ12)−Δ2)R2−xKA=H⟹KA(Δ2+Δ1−2Δ22)R2−x=KA(Δ1−Δ2)R2−R1⟹(Δ1−Δ22)R2−x=(Δ1−Δ2)R2−R1⟹(Δ1−Δ2)2(R2−x)=(Δ1−Δ2)R2−R1⟹R2−x=R2−R12⟹x=R2−R2−R12⟹x=R2−R12
is the radial distance from centre of shell where the temperature is just half way between Δ1 and Δ2