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A rod of length l with thermally insulat...

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`.

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The correct Answer is:
A, B

`H = TD/R = (-dT)/((dx)//(KA))`
`= -(dT)/(dx) (a/T A) = "constant" …….(i)`
`:. int _(T_1)^(T_2) -(dT)/T = H/(aA) int_(0)^(l) dx`
`In (T_1/T_2) = (Hl)/(aA)`
or `H = (aA)/l In (T_1//T_2)`
Substituting in Eq. (i), we have
`(aA)/l In (T_1/T_2) = -(dT)/(dx) (aA)/T`
or `int_(T_1)^T (dT)/T = -(In (T_1/T_2))/l int_0^x dx`
`:. In (T/T_1) = -x/l In (T_1//T_2) = In (T_1/T_2)^(-x/l)`
or `T/T_1 = (T_1/T_2)^(-x/l)`
or `T=T_1 (T_1/T_2)^(-x/l) = T_1(T_2/T_1)^(x/l)`.
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