   Chapter 16.7, Problem 48E

Chapter
Section
Textbook Problem

The temperature at a point in a ball with conductivity K is inversely proportional to the distance from the center of the ball. Find the rate of heat flow across a sphere S of radius a with center at the center of the ball.

To determine

To find: The rate of heat flow across a sphere S of radius a .

Explanation

Formula used:

SFdS=SFndS (1)

F=Ku (2)

=ix+jy+kz

By taking the center of the ball to be at the center of the radius, then the u(x,y,z) is,

u(x,y,z)=Cx2+y2+z2

Find u .

u=(ix+jy+kz)(Cx2+y2+z2)=ix(Cx2+y2+z2)+jy(Cx2+y2+z2)+kz(Cx2+y2+z2)=iCx(x2+y2+z2)32jCy(x2+y2+z2)32kCz(x2+y2+z2)32=C(x2+y2+z2)32(xi+yj+zk)

Find F

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