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title: "A thin, nonconducting rod of length \\(L\\) lies along the positive \\(x\\)-axis with its left end at the origin \\(x = 0\\). The rod has a non-uniform linear charge density given by \\(\\lambda(x) = \\beta x\\), where \\(\\beta\\) is a positive constant and \\(0 \\le x \\le L\\). Point \\(P\\) is located on the \\(x\\)-axis at \\(x = D\\), where \\(D > L\\). Assuming the electric potential is zero at infinity, which of the following expressions represents the electric potential \\(V\\) at point \\(P\\)?"
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url: "https://nerd-notes.com/ubq/118137/"
date_modified: "2026-08-04T08:05:24+00:00"
---

# A thin, nonconducting rod of length \(L\) lies along the positive \(x\)-axis with its left end at the origin \(x = 0\). The rod has a non-uniform linear charge density given by \(\lambda(x) = \beta x\), where \(\beta\) is a positive constant and \(0 \le x \le L\). Point \(P\) is located on the \(x\)-axis at \(x = D\), where \(D > L\). Assuming the electric potential is zero at infinity, which of the following expressions represents the electric potential \(V\) at point \(P\)?

A thin, nonconducting rod of length \(L\) lies along the positive \(x\)-axis with its left end at the origin \(x = 0\). The rod has a non-uniform linear charge density given by \(\lambda(x) = \beta x\), where \(\beta\) is a positive constant and \(0 \le x \le L\). Point \(P\) is located on the \(x\)-axis at \(x = D\), where \(D > L\). Assuming the electric potential is zero at infinity, which of the following expressions represents the electric potential \(V\) at point \(P\)?

![A horizontal x-axis with an origin labeled 0. A thin rod lies along the axis from x = 0 to x = L, shaded with increasing dot density from left to right to indicate non-uniform charge. A small segment of width dx is marked at position x on the rod. A point labeled P sits on the x-axis to the right at position x = D. A horizontal arrow below the axis extends from the origin to point P, labeled D. Another arrow extends from the origin to the right end of the rod, labeled L. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830724-CeK7y7.jpg)

- **A.** \(\dfrac{\beta D}{4\pi\varepsilon_0} \ln\left(\dfrac{D}{D-L}\right)\)
- **B.** \(\dfrac{\beta}{4\pi\varepsilon_0} \left[ D \ln\left(\dfrac{D}{D-L}\right) - L \right]\)
- **C.** \(\dfrac{\beta}{4\pi\varepsilon_0} \left[ L - D \ln\left(\dfrac{D}{D-L}\right) \right]\)
- **D.** \(\dfrac{\beta L^2}{8\pi\varepsilon_0 \left(D - \dfrac{1}{2}L\right)}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118137/*