A lagged stick of cross section area `1 cm^(2)` and length `1`m is initially at a temperature of `0^(@)C`. It is then kept between `2` reservoirs of temperature `100^(@)C "and" 0^(@)C`. Specific heat capacity is `10J//kg^(@)C` and linear mass density is `2kg//m` . Find
(i) Temperature gradient along the rod in steady state
(ii) Total heat absorbed by the rod to reach steady state

Answer

`-100^(@)C//m , (ii) 1000J`

Explanation

(i) Temperature gradient = `(T_(1) - T_(2))/(l) = (100-0)/(1) = 100^(@)C//m`
(ii) Steady state temperature of element dx: `T = 100(1-x)`
Heat absorbed by the element to reach steady state. `dQ = "(dm)s" DeltaT = (gammadx)s(T-0)`
`implies dQ = 20[100(1-x)]dx`
Total heat absorbed by the rod `Q= intdQ = 2000 underset(0)overset(1)int(1-x)dx = 1000J`