A long metallic bar is carrying heat from one of its ends to the other end under steady-state. The variation of temperature \theta along the length x of the bar from its hot end is best described by which of the following figures?

Heat and Thermodynamics PYQ — JEE Main 2009 · Shift 0 · Q69 with Solution

View written solutionFree

Correct answer: A

Steady-state heat conduction in a long metallic bar

Let the bar carry heat along the $x$ -direction from the hot end.

In steady state, temperature at each point does not change with time, so the heat current through any cross-section is constant.
Using Fourier’s law

Fourier’s law of conduction gives
$H=-kA\frac{d\theta}{dx}$
where:
- $H$ = heat current,
- $k$ = thermal conductivity,
- $A$ = area of cross-section,
- $\theta$ = temperature,
- $x$ = distance from hot end.
If the bar is uniform and there is no heat loss from the sides, then $H$ , $k$ , and $A$ are constant. Hence,
$\frac{d\theta}{dx}=\text{constant}$
Therefore,
$\theta = \theta_0 - mx$
which is a straight line with negative slope.
Physical interpretation

So, as we move away from the hot end, temperature decreases linearly with distance.

Hence the correct graph is the one showing a linear fall of temperature with $x$ .
Matching with options

The stored correct answer says A, so option A must be the straight-line decreasing graph.
Conclusion

Therefore, the correct option is:
$\boxed{A}$