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title: "A cylindrical metal rod of uniform cross-sectional area \\(A\\) and length \\(L\\) is thermally insulated along its sides and connects a hot reservoir at temperature \\(T_H\\) at position \\(x = 0\\) to a cold reservoir at temperature \\(T_C\\) at position \\(x = L\\). The system has reached a steady state of heat transfer, and the metal has a thermal conductivity \\(k(T)\\) that increases with increasing temperature. Which of the following correctly describes the graph of temperature \\(T\\) as a function of position \\(x\\) along the rod, along with the correct physical reasoning?"
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url: "https://nerd-notes.com/ubq/116257/"
date_modified: "2026-08-03T11:46:29+00:00"
---

# A cylindrical metal rod of uniform cross-sectional area \(A\) and length \(L\) is thermally insulated along its sides and connects a hot reservoir at temperature \(T_H\) at position \(x = 0\) to a cold reservoir at temperature \(T_C\) at position \(x = L\). The system has reached a steady state of heat transfer, and the metal has a thermal conductivity \(k(T)\) that increases with increasing temperature. Which of the following correctly describes the graph of temperature \(T\) as a function of position \(x\) along the rod, along with the correct physical reasoning?

A cylindrical metal rod of uniform cross-sectional area \(A\) and length \(L\) is thermally insulated along its sides and connects a hot reservoir at temperature \(T_H\) at position \(x = 0\) to a cold reservoir at temperature \(T_C\) at position \(x = L\). The system has reached a steady state of heat transfer, and the metal has a thermal conductivity \(k(T)\) that increases with increasing temperature. Which of the following correctly describes the graph of temperature \(T\) as a function of position \(x\) along the rod, along with the correct physical reasoning?

![A horizontal cylindrical rod of length L is positioned between two rectangular heat reservoirs. The left reservoir is labeled T_H at position x = 0. The right reservoir is labeled T_C at position x = L. A horizontal line below the rod represents the x-axis with origin x = 0 at the left end and x = L at the right end. An arrow pointing to the right along the rod indicates the heat current H. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785757589-ZDwNzz.jpg)

- **A.** The graph is concave up (steeper near the hot end) because heat flows faster near the hot reservoir where the thermal conductivity is higher.
- **B.** The graph is linear because the rate of heat flow must be uniform throughout the rod in a steady state.
- **C.** The graph is concave down (steeper near the cold end) because maintaining a constant rate of heat flow requires a larger magnitude temperature gradient where the thermal conductivity is lower.
- **D.** The graph is concave down (steeper near the cold end) because thermal energy accumulates near the cold end due to the lower thermal conductivity.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/116257/*