A hollow tube has a length l, inner radius `R_(1)` and outer radius `R_(2)` . The material has a thermal conductivity K. Find the heat flowing through the walls of the tube if (a) the flat ends are maintained at temperature `T_(1)` and `T_(2)``(T_(2)gtT_(1))` (b) the inside of the tube is maintained at temperature `T_(1)` and the outside is maintained at `T_(2)` .

Answer

`(2piKl(T_(2)-T_(1)))/(In(R_(2)//R_(1)))`

Explanation

Let an element which is hollow cyl. of Radius and width dr
`dR = (dr)/(K2pirl) R = (1)/(2piKl) underset(R_(1))overset(R_(2))int (dr)/(r ) = (1)/(2piKl)`
`(In r)_(R_(1))^(R_(2)) = (1)/(2piKl)In ((R_(2))/(R_(1)))`
`i_(H) = (T_(2)-T_(1))/((1)/(2piKl)ln(R_(2)//R_(1))) =(2piKl(T_(2)-T_(1)))/(ln(R_(2)//R_(1)))`