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title: "A cylindrical resistor of length \\(L\\) and uniform cross-sectional area \\(A\\) has a position-dependent resistivity given by \\(\\rho(x) = \\rho_0\\left(1 + \\dfrac{x}{L}\\right)\\) for \\(0 \\le x \\le L\\), where \\(\\rho_0\\) is a positive constant. It is connected in series at \\(x = L\\) to a second cylindrical resistor of the same cross-sectional area \\(A\\), length \\(L\\), and uniform resistivity \\(2\\rho_0\\) occupying the interval \\(L \\le x \\le 2L\\). A steady current \\(I\\) flows through the combination in the \\(+x\\)-direction, and the end at \\(x = 2L\\) is connected to an electrical ground such that \\(V(2L) = 0\\text{ V}\\).  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 2L\\)?"
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url: "https://nerd-notes.com/ubq/121587/"
date_modified: "2026-08-23T05:24:10+00:00"
---

# A cylindrical resistor of length \(L\) and uniform cross-sectional area \(A\) has a position-dependent resistivity given by \(\rho(x) = \rho_0\left(1 + \dfrac{x}{L}\right)\) for \(0 \le x \le L\), where \(\rho_0\) is a positive constant. It is connected in series at \(x = L\) to a second cylindrical resistor of the same cross-sectional area \(A\), length \(L\), and uniform resistivity \(2\rho_0\) occupying the interval \(L \le x \le 2L\). A steady current \(I\) flows through the combination in the \(+x\)-direction, and the end at \(x = 2L\) is connected to an electrical ground such that \(V(2L) = 0\text{ V}\).

Which of the following best describes the graph of the electric potential \(V(x)\) as a function of position \(x\) for \(0 \le x \le 2L\)?

A cylindrical resistor of length \(L\) and uniform cross-sectional area \(A\) has a position-dependent resistivity given by \(\rho(x) = \rho_0\left(1 + \dfrac{x}{L}\right)\) for \(0 \le x \le L\), where \(\rho_0\) is a positive constant. It is connected in series at \(x = L\) to a second cylindrical resistor of the same cross-sectional area \(A\), length \(L\), and uniform resistivity \(2\rho_0\) occupying the interval \(L \le x \le 2L\). A steady current \(I\) flows through the combination in the \(+x\)-direction, and the end at \(x = 2L\) is connected to an electrical ground such that \(V(2L) = 0\text{ V}\).

Which of the following best describes the graph of the electric potential \(V(x)\) as a function of position \(x\) for \(0 \le x \le 2L\)?

![A horizontal cylindrical conductor of uniform radius extending along an x-axis from x = 0 to x = 2L. A vertical dashed line at x = L divides the cylinder into two segments of equal length L: the left segment from x = 0 to x = L is labeled \rho(x) = \rho_0(1 + x/L), and the right segment from x = L to x = 2L is labeled \rho = 2\rho_0. A horizontal arrow pointing to the right above the cylinder is labeled I. At the rightmost end x = 2L, a line extends downward to a standard electrical ground symbol labeled V = 0. Below the cylinder, a horizontal axis is marked with ticks and labels at x = 0, x = L, and x = 2L. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787462650-EF6I2l.jpg)

- **A.** A curve that is concave upward with an increasingly less negative slope for \(0 \le x \le L\), transitioning smoothly to a straight line with a constant negative slope for \(L \le x \le 2L\).
- **B.** A curve that is concave downward with an increasingly negative slope for \(0 \le x \le L\), followed by a discontinuous step-down jump in potential at \(x = L\), and then a straight line with a constant negative slope for \(L \le x \le 2L\).
- **C.** A curve that is concave downward with an increasingly negative slope for \(0 \le x \le L\), transitioning smoothly with continuous slope to a straight line with a constant negative slope for \(L \le x \le 2L\).
- **D.** A straight line with a constant negative slope for \(0 \le x \le L\), transitioning at a sharp corner to a concave downward curve with an increasingly negative slope for \(L \le x \le 2L\).

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121587/*