What is the temperature of the steel - copper junction in the steady state of the system shown in Fig.11.15.

Length of the steel rod ` = 15.0 `cm, length of the copper reod ` = 10.0` cm, temperarure of the furnace ` = 300 ^(@) C`, temperature of the furnace ` = 300 ^(@) C` , temperature of the other end ` = 0^(@) C`. The are of cross section of the steel rod is twice that of the copper rod. ( Thermal conductivity of steel ` = 50.2 J s^(-1) m^(-1) K^(-1) ,` and of copper ` = 385 J s^(-1) m^(-1) K^(-1))`.

Explanation

The insulating material around the rods reduces heat loss from the sides of the rods. Therefore, heat flows only along the length of the rods. Consider any cross section of the rod. In the steady state, heat flowing into the element must equal the heat flowing out of its otherwise there would be a net gain or loss of heat by the element and its temperature would not be steady. Thus in the steady state, rate of heat flowing across a cross section of the rod is the same at every point along the length of the combined steel-copper rod. Let T be the temperature of the steel-copper junction in the steady state. Then,
`(K_(1)A_(1)(300 - T))/L_(1) = (K_(2)A_(2)(T - 0))/L_(2)`
where 1 and 2 refer to the steel and copper rod respectively. For `A_(1) = 2 A_(2) , L_(1) = 15.0 cm , L_(2) = 10.0 cm, K_(1) = 50.2 J s^(-1) m^(-1) K^(-1) , K_(2) = 385 J s^(-1) m^(-1) K^(-1) `, we have
`(50.2 xx 2 (300 - T))/15 = (385T)/10 `
which gives `T = 44.4^(@) C`