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

Asked in AIEEE 2009 · Temperature profile along a bar

• (1) θ first rises to a maximum and then falls
• (2) θ falls linearly with x
• (3) θ falls steeply at first and then levels off, the curve bending upward
• (4) θ stays nearly flat at first and then falls steeply, the curve bending downward

Step-by-step solution

Why the other options are wrong

• (1) comes from would mean heat flowing uphill in temperature over the first stretch of the bar, which no passive conduction does. Heat flows down the gradient, so θ can only fall as x grows from the hot end.
• (3) comes from is the shape a bar that loses heat from its sides shows, θ-θ₀=(θ₁-θ₀)e^-mx, because the current itself shrinks along the bar. With the sides insulated the current is the same everywhere and the fall is straight.
• (4) comes from has the curvature the wrong way for any conduction problem: a bar in steady state cannot hold its temperature near the source and then drop suddenly, because that would need the heat current to grow with distance.

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