A rod of length l with thermally insulated lateral surface consists of material whose heat conductivity coefficient varies with temperature as `k= a//T`, where a is a constant. The ends of the rod are kept at temperatures `T_1 and T_2`. Find the function T(x), where x is the distance from the end whose temperature is `T_1`.

Answer

`T(x) = T_(1) ((T_(2))/(T_(1)))^(x//l); q = (alpha//l) ln ((T_(2))/(T_(2)))`

Explanation

`(dQ)/(dt) =- kA (dT)/(dx)`
`((dQ)/(dt)) int dx =- int (A alpha dT)/(T)`
`((dQ)/(dt)) x = A alpha ln .(T_(1))/(T(x))`
`((dQ)/(dt)) l = A alpha ln. (T_(1))/(T_(2))`
`rArr (1)/(4) ((dQ)/(dt)) = ((alpha)/(l))ln ((T_(1))/(T_(2))) = q`
or `(l)/(x) = ("ln"(T_(1))/(T_(2)))/("ln"(T_(1))/(T(x))) or ((T_(1))/(T(x)))^(l) =((T_(1))/(T_(2)))^(x)`
or `T(x) = T_(1) ((T_(2))/(T_(1)))^(x//l)`
`q = (1)/(A) (dQ)/(dt)`