# a lagged copper rod of uniform cross-sectional area has a length of 60cm, the free ends of the rod are maintained at 20 degree celsius and 0 degree celsius respectively at a steady state. calculate the temperature at a point 20cm from the high temperature.

Samyak Jain
333 Points
3 years ago
In steady state, heat current (i.e. heat per unit time) flowing throuh any cross section of conductur is same.
We know that dQ/dt  =  K A dT/dx   ...(1), where dQ/dt is heat current, K = thermal coefficient of substance, A is area of cross section of length dx and dT/dx = temprature gradient.
Let point at T1 = 20$\dpi{80} \degree$ C be A, that at T2 = 0$\dpi{80} \degree$ C be B and that at 20cm from A be C with temperture T.  AB = 60 cm.
CB = (60 – 20)cm = 40 cm.
In steady state, we can replace dx by length and dt by $\dpi{80} \Delta$T in (1) of conductor if area of cross-section is uniform.
dQ/dt across AC = dQ/dt across CB.
i.e. KA(T1 – T)/20 cm = KA(T – T2)/40 cm  $\dpi{80} \Rightarrow$  (T – T1)/20 + (T – T2)/40 = 0.    (by Kirchoff’s junction law)
2 T – 2 T1 + T – T2 = 0  or  T = (2T1 + T2) / 3  =  (2.20$\dpi{80} \degree$ C + 0$\dpi{80} \degree$ C)/3
T = (40 / 3)$\dpi{80} \degree$ C .