A rod of length l and cross sectional area A has a variable conductivity given by `K=alphaT`, where `alpha` is a positive constant T is temperatures in Kelvin. Two ends of the rod are maintained at temperatures `T_1` and `T_2(T_1gtT_2)`. Heat current flowing through the rod will be

Options

A. `(A alpha(T_(1)^(2)-T_(2)^(2)))/l`
B. `(A alpha(T_(1)^(2)-T_(2)^(2)))/l`
C. `(A alpha(T_(1)^(2)+T_(2)^(2)))/(3l)`
D. `(A alpha(T_(1)^(2)-T_(2)^(2)))/(2l)`

Answer

D

Explanation

Heat current : `i=-kA (d T)/(dx)`
`idx=-kAdT`
` i int_(0)^(t) dx=-A alpha int_(T_(1))^(T_(2)) T dT`
`rArr il=-A alpha((T_(2)^(2)-T_(1)^(2)))/3rArr i=(A alpha(T_(1)^(2)-T_(2)^(2)))/(2l)`