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title: "A cylindrical conducting rod of length \\(L\\) and uniform cross-sectional area \\(A\\) carries a steady current \\(I\\) flowing in the \\(+x\\)-direction from \\(x = 0\\) to \\(x = L\\). The electric potential along the rod is fixed at \\(V(0) = V_0\\) and \\(V(L) = 0\\). The resistivity of the material varies linearly along its length according to \\(\\rho(x) = \\rho_0 \\left(1 + \\dfrac{x}{L}\\right)\\), where \\(\\rho_0\\) is a positive constant. Which of the following best describes the graph of the electric potential \\(V(x)\\) as a function of position \\(x\\) for \\(0 \\le x \\le L\\)?"
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url: "https://nerd-notes.com/ubq/118447/"
date_modified: "2026-08-04T08:10:11+00:00"
---

# A cylindrical conducting rod of length \(L\) and uniform cross-sectional area \(A\) carries a steady current \(I\) flowing in the \(+x\)-direction from \(x = 0\) to \(x = L\). The electric potential along the rod is fixed at \(V(0) = V_0\) and \(V(L) = 0\). The resistivity of the material varies linearly along its length according to \(\rho(x) = \rho_0 \left(1 + \dfrac{x}{L}\right)\), where \(\rho_0\) is a positive constant. Which of the following best describes the graph of the electric potential \(V(x)\) as a function of position \(x\) for \(0 \le x \le L\)?

A cylindrical conducting rod of length \(L\) and uniform cross-sectional area \(A\) carries a steady current \(I\) flowing in the \(+x\)-direction from \(x = 0\) to \(x = L\). The electric potential along the rod is fixed at \(V(0) = V_0\) and \(V(L) = 0\). The resistivity of the material varies linearly along its length according to \(\rho(x) = \rho_0 \left(1 + \dfrac{x}{L}\right)\), where \(\rho_0\) is a positive constant. Which of the following best describes the graph of the electric potential \(V(x)\) as a function of position \(x\) for \(0 \le x \le L\)?

![A horizontal cylindrical rod of length L and cross-sectional area A lies along the x-axis from x = 0 to x = L. An arrow labeled I points to the right inside the rod. The left end at x = 0 is labeled V_0 and the right end at x = L is labeled 0. Below the rod, a formula label reads \rho(x) = \rho_0(1 + x/L). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831011-yAvviD.jpg)

- **A.** Decreases linearly from \(V_0\) at \(x = 0\) to \(0\) at \(x = L\), having a constant negative slope throughout the rod.
- **B.** Decreases from \(V_0\) at \(x = 0\) to \(0\) at \(x = L\) with concave-upward curvature, such that the magnitude of the slope decreases as \(x\) increases.
- **C.** Decreases from \(V_0\) at \(x = 0\) to \(0\) at \(x = L\) with concave-downward curvature, such that the magnitude of the slope increases as \(x\) increases.
- **D.** Increases from \(V_0\) at \(x = 0\) to a maximum value at \(x = L/2\) before decreasing to \(0\) at \(x = L\).

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118447/*