8 A region is filled with a material that conducts heat Let

8. A region is filled with a material that conducts heat. Let T(x) be the (scalar) tem- perature at the point z. Suppose that in this region region T(2)--chr when xO. At the origin we have T(0-0. (a) Suppose we are at a point a which is not 0. What is the magnitude of this gradient? (b) What happens to the magnitude of the gradient as our point moves far from the origin? (c) At non-zero a, where is the gradient pointing? (d) Relatively speaking, is it hot or cold at the origin? In which direction will heat flow? Give a brief explanation.

Solution

(a) T(x) = - e^-lxl . x / lxl

for x > 0 for x < 0

T(x) = - e^-x .x / x T(x) = - e^x . x / (-x)

T(x) = - e^-x T(x) = e^x

(b) magnitude for point far from the origin

in positive x-axis - e^-x in negative x-axis e^x

it tend to zero it tend to zero

(c) At non-zero x , gradient is pointing in both positive and negative x-axis for positive and negative value of x respectively.

(d) At origin T(0) = 0 it means temperaure is zero so it is cold at origin. And in both positive and negative x-axis direction there is cold because value will be nearly equal to zero.