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Step-by-Step Solution
1. Understanding Steady-State Heat Conduction
In steady-state heat conduction along a metallic bar, the temperature at any given point no longer changes with time. The hot end of the bar remains at a higher temperature $ \theta_{1} $, and the bar conducts heat toward the colder end.
2. Expressing the Heat Flow Rate
For a rod of thermal conductivity $ k $, cross-sectional area $ A $, and length $ x $, under steady-state conditions, the rate of heat flow $ \frac{dQ}{dt} $ is given by:
$ \frac{dQ}{dt} \;=\; \frac{k\,A\,(\theta_{1} - \theta)}{x} \,,
$
where $ \theta $ is the temperature at distance $ x $ from the hot end.
3. Rearranging to Find Temperature Variation
Rearrange the above relationship to isolate the temperature $ \theta $ at distance $ x $:
$
\theta_{1} - \theta \;=\; \frac{x}{k\,A} \frac{dQ}{dt}
\quad \Longrightarrow\quad
\theta \;=\; \theta_{1} \;-\; \frac{x}{k\,A} \frac{dQ}{dt}.
$
This expression shows that $ \theta $ decreases linearly with $ x $. When $ x = 0 $ (at the hot end), $ \theta = \theta_{1} $, and as $ x $ increases, $ \theta $ decreases in a straight-line fashion.
4. Conclusion and Graph
The equation demonstrates a linear drop in temperature from the hot end to the other end along the rod. Graphically, this corresponds to a straight line when plotting $ \theta $ versus $ x $. Therefore, the correct option is the one that shows a linear decrease in temperature with increasing distance $ x $ from the hot end.
Correct Answer Figure
