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title: "A liquid conductor with uniform resistivity \\(\\rho\\) flows through a tapered pipe of length \\(L\\). The pipe has a circular cross-section whose radius varies with position \\(x\\) along the pipe as \\(r(x) = R_0\\left(1 – \\dfrac{x}{2L}\\right)\\) for \\(0 \\le x \\le L\\). A steady total electric current \\(I\\) is maintained through the liquid along the \\(x\\)-axis, carried by charge carriers of charge \\(q\\) and uniform concentration \\(n\\). Which of the following correctly gives the drift speed \\(v_d(L)\\) of the charge carriers at \\(x = L\\) in terms of their initial drift speed \\(v_d(0)\\) at \\(x = 0\\), and the total potential difference \\(\\Delta V\\) across the pipe?"
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url: "https://nerd-notes.com/ubq/118414/"
date_modified: "2026-08-04T08:09:58+00:00"
---

# A liquid conductor with uniform resistivity \(\rho\) flows through a tapered pipe of length \(L\). The pipe has a circular cross-section whose radius varies with position \(x\) along the pipe as \(r(x) = R_0\left(1 – \dfrac{x}{2L}\right)\) for \(0 \le x \le L\). A steady total electric current \(I\) is maintained through the liquid along the \(x\)-axis, carried by charge carriers of charge \(q\) and uniform concentration \(n\). Which of the following correctly gives the drift speed \(v_d(L)\) of the charge carriers at \(x = L\) in terms of their initial drift speed \(v_d(0)\) at \(x = 0\), and the total potential difference \(\Delta V\) across the pipe?

A liquid conductor with uniform resistivity \(\rho\) flows through a tapered pipe of length \(L\). The pipe has a circular cross-section whose radius varies with position \(x\) along the pipe as \(r(x) = R_0\left(1 - \dfrac{x}{2L}\right)\) for \(0 \le x \le L\). A steady total electric current \(I\) is maintained through the liquid along the \(x\)-axis, carried by charge carriers of charge \(q\) and uniform concentration \(n\). Which of the following correctly gives the drift speed \(v_d(L)\) of the charge carriers at \(x = L\) in terms of their initial drift speed \(v_d(0)\) at \(x = 0\), and the total potential difference \(\Delta V\) across the pipe?

![A horizontal tapered cylinder centered along a horizontal dashed x-axis extending from x = 0 to x = L. At x = 0 on the left, the circular end of the cylinder has radius R_0, drawn as a vertical line from the central axis to the top outer edge labeled R_0. The cylinder continuously tapers down to a smaller circular cross-section of radius R_0 / 2 at x = L on the right, labeled with a vertical radius line labeled R_0 / 2. The region inside the cylinder is shaded light grey representing a liquid conductor. A thick horizontal arrow pointing to the right along the central axis is labeled I. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830997-3G7RAv.jpg)

- **A.** \(v_d(L) = 4 v_d(0)\) and \(\Delta V = \dfrac{2\rho I L}{\pi R_0^2}\)
- **B.** \(v_d(L) = 2 v_d(0)\) and \(\Delta V = \dfrac{2\rho I L}{\pi R_0^2}\)
- **C.** \(v_d(L) = 4 v_d(0)\) and \(\Delta V = \dfrac{\rho I L}{\pi R_0^2}\)
- **D.** \(v_d(L) = 2 v_d(0)\) and \(\Delta V = \dfrac{\rho I L}{\pi R_0^2}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118414/*