A hole of radius *r*_{1} is made centrally in a uniform circular disc of thickness *d* and radius *r*_{2}. The inner surface (a cylinder a length *d* and radius *r*_{1}) is maintained at a temperature θ_{1} and the outer surface (a cylinder of length *d* and radius *r*_{2}) is maintained at a temperature θ_{2} (θ_{1} > θ_{2}). The thermal conductivity of the material of the disc is K. Calculate the heat flowing per unit time through the disc.

#### Solution

Let `(d theta)/(dt)` be the rate of flow of heat.

Consider an annular ring of radius *r *and thickness *dr*.

Rate of flow of heat is given by

`(d theta)/(dt) = K (2pird)`

Rate of flow of heat is constant.

∴ `(d theta)/(dt) = i`

`i = - k ( 2pir.d) (d theta)/(dr)`

`int_{r_1}^{r_2} dr/r = (2piKd)/l int_{theta_1}^(theta_2) d theta`

`["ln" (r) ]_{r1}^{r2} = (2pikd)/l [ theta_2 - theta 1]`

`i =(2pikd(theta_2 -theta_1))/("ln"(r_2/r_1)`