An uniform metallic sphere heated in the center The temperat

An uniform metallic sphere heated in the center. The temperature is uniform and equals T_1 on the internal spherical surface with the radius R_1, and the temperature is uniform and equals T_2 on the internal spherical surface with the radius R_2. Find the radius on the internal spherical surface in terms of R_1, R_2, T_1, T_2 where the temperature (T_3) equals to the average temperature between T_1 and T_2, T_3 = (T_1 + T_2)/2.

Solution

let T3 be at R3

Since T3 = (T1 + T2)/2

T1 - T3 =k( 1/R1 - 1/R3)

T3 - T2 = K(1/R3 - 1/R2)

adding the two equations

T1 - T2 = k(1/R1 - 1/R2)

K = (T1 - T2) (R2R1)/(R2 - R1)

T3 - T2 + K/R2 = K/R3

=>(T1 - T2)/2 K + 1/R2 = 1/R3

=> R3 = 2 R1 R2 /(R2 + R1)