Question

Two thin conectric shells made of copper with radius `r_(1)` and `r_(2) (r_(2) gt r_(1))` have a material of thermal conductivity `K` filled between them. The inner and outer spheres are maintained at temperature `T_(H)` and `T_(C)` respectively by keeping a heater of power `P` at the centre of the two sphers. Find the value of `P`.

Answer 1

Heat flowing per second through each cross-section of the sphere `=P = i`. Thermal resistance of the spherical sheel of radius `x` and thickness `dx`.
`dR = (dx)/(K.4pix^(2))`
`rArr R = underset(r_(1))overset(r_(2))int (dx)/(4pix^(2)K) = (1)/(4piK) ((1)/(r_(1))-(1)/(r_(2)))`
thermal current
`i = P = (T_(H)-T_(C))/(R) = (4piK(T_(H)-T_(C))r_(1)r_(2))/((r_(2)-r_(1)))`