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 alphaT_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(dT)/(dx)`
`idx=-kAdT`
`iint_0^ldx=-Aalphaint_(T_1)^(T_2)TdT`
`il=Aalpha((T_2^2-T_1^2))/(2)impliesi=(Aalpha(T_1^2-T_2^2))/(2l)`