At
low temperatures (below
about 50 K), the
thermal conductivity of a metal is proportional to the absolute
temperature; that is
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Solution: Click For PDF Version Let the variation of temperature along the length of the rod be described by the function ,
where x is the length
measured from the end which is kept at temperature
 .
Under the assumption that there is no heat loss from the surface of the rod, heat flow H will be independent of x. Thermal conductivity varies with temperature and is given by the function
where a has a constant value that depends on the metal through which heat is being conducted. From the law of thermal conduction, we have
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,
where a is a constant with a numerical value that depends on the
particular metal. We have to show that the
rate of heat flow through the rod of length L and cross-sectional
area A whose ends are at temperatures
is
given by
