Question

The ends of a long bar are maintained at different temperatures and there is no loss of heat from the sides of the bar due to conduction or radiation. The graph of temperature against distance of the bar when it has attained steady state is shown here. The graph shows :

• A. the temperature gradient is not uniform
• B. the bar has uniform cross-sectional area
• C. the cross-sectional area of the bar increases as the distance from the hot end increases
• D. the cross-sectional area of the bar decreases as the distance from the hot end increases

Answer 1

Answer: (A), (D)

the temperature gradient is not uniform
the cross-sectional area of the bar decreases as the distance from the hot end increases

It is clear from the graph that the slope (gradient) $\frac{dT}{dx}$ is not constant as x varies.

From conductive heat transfer, we know that,

$\frac{dQ}{dt}=kA\frac{dT}{dx}$

Since the heat current $\frac{dQ}{dt}$ has to remain constant throughout the length of bar (as there are no losses from sides), $A\frac{dT}{dx}$ is constant. From graph, magnitude of $\frac{dT}{dx}$ increases with x and hence area A decreases with x, i.e. from hot end to cold end.