Answer to Problem 26

Setting when gives . Then, since , we have



Equation 12

or



Equation 13

Finally substituting



Equation 14

in the expression for , we obtain



Equation 15

It should be noticed that when , is negative; this is consistent with our intuition that heat flows from higher to lower temperatures.

Note too that the temperature distribution is a linear function of .

Finally, to underscore the role of the gradient in this analysis, it is readily verified that the expressions for and given by equations (3.4) and (3.5) satisfy (1.1), Fourier's law of heat conduction.