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A rod of length l and cross-section are...

A rod of length l and cross-section area A has a variable thermal conductivity given by K = `alpha` T, where `alpha` is a positive constant and T is temperature in kelvin. Two ends of the rod are maintained at temperature `T_(1)` and `T_(2)` `(T_(1)gtT_(2))`. Heat current flowing through the rod will be

A

`(Aalpha(T_(1)^(2)-T_(2)^(2)))/(1)`

B

`(Aalpha(T_(1)^(2)+T_(2)^(2)))/(1)`

C

`(Aalpha(T_(1)^(2)+T_(2)^(2)))/(31)`

D

`(Aalpha(T_(1)^(2)-T_(2)^(2)))/(21)`

Text Solution

Verified by Experts

The correct Answer is:
D
H through the rod will be same as through the element
So `H=(-KA)/(dX)dT,Hdx=-KAdT`
or , `int_(0)^(l)Hdx=int_(T_(1))^(T_(2))-alphaTAdT`
`Hl=(-alphaA)(T_(2)^(2)-T_(1)^(2))/(2),H=(alphaA(T_(1)^(2)-T_(2)^(2)))/(2l)`
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