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title: "A thin, nonconducting rod of length \\(L\\) lies along the \\(z\\)-axis, centered at the origin, and carries a total positive charge \\(Q\\) distributed uniformly along its length. A detector measures the electric field magnitude \\(E(r)\\) as a function of distance \\(r\\) along the positive \\(x\\)-axis, which forms the perpendicular bisector of the rod. The measured field magnitude is well approximated by a \\(1/r\\) dependence when \\(r \\ll L\\), but transitions smoothly to a \\(1/r^2\\) dependence when \\(r \\gg L\\). Which of the following statements provides the physically correct explanation for this transition?"
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url: "https://nerd-notes.com/ubq/124540/"
date_modified: "2026-09-28T14:08:34+00:00"
---

# A thin, nonconducting rod of length \(L\) lies along the \(z\)-axis, centered at the origin, and carries a total positive charge \(Q\) distributed uniformly along its length. A detector measures the electric field magnitude \(E(r)\) as a function of distance \(r\) along the positive \(x\)-axis, which forms the perpendicular bisector of the rod. The measured field magnitude is well approximated by a \(1/r\) dependence when \(r \ll L\), but transitions smoothly to a \(1/r^2\) dependence when \(r \gg L\). Which of the following statements provides the physically correct explanation for this transition?

A thin, nonconducting rod of length \(L\) lies along the \(z\)-axis, centered at the origin, and carries a total positive charge \(Q\) distributed uniformly along its length. A detector measures the electric field magnitude \(E(r)\) as a function of distance \(r\) along the positive \(x\)-axis, which forms the perpendicular bisector of the rod. The measured field magnitude is well approximated by a \(1/r\) dependence when \(r \ll L\), but transitions smoothly to a \(1/r^2\) dependence when \(r \gg L\). Which of the following statements provides the physically correct explanation for this transition?

![A vertical thick solid line segment of length L is centered at the origin of a coordinate system. A vertical dashed axis labeled z passes through the rod, with +L/2 at the top end and -L/2 at the bottom end. A horizontal solid axis extends to the right and is labeled x. A point P is marked on the positive x-axis at distance r from the origin. Two straight dashed lines extend from point P to the top end (+L/2) and bottom end (-L/2) of the rod, forming an angle subtended at P. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604514-0jMJXs.jpg)

- **A.** At small distances (\(r \ll L\)), mirror symmetry across the bisector ensures that axial field components cancel identically, confining the field lines to two dimensions in the \(xy\)-plane; at large distances (\(r \gg L\)), this cancellation breaks down across the rod's length, allowing axial field components to dominate and steepening the radial decay to \(1/r^2\).
- **B.** When \(r \ll L\), the rod subtends a nearly constant angular span near \(\pi\) radians, so the length of the rod that effectively contributes to the perpendicular field grows proportionally to \(r\), offsetting one factor of distance in Coulomb's law; when \(r \gg L\), the entire fixed length subtends a shrinking angle proportional to \(1/r\), leaving the total charge to act from a single distance and restoring \(1/r^2\) geometric dilution.
- **C.** When \(r \ll L\), a spherical Gaussian surface of radius \(r\) centered at the origin encloses a charge \(q_{\text{enc}} = 2(Q/L)r\) that grows linearly with radius, balancing the \(r^2\) growth of the surface area in Gauss's law; when \(r > L/2\), the Gaussian sphere encloses the constant total charge \(Q\), causing the field from Gauss's law to scale strictly as \(Q/r^2\).
- **D.** When \(r \ll L\), the electric potential varies logarithmically because the zero-potential reference is set at the surface of the rod, resulting in an electric field that scales as the derivative \(d(\ln r)/dr = 1/r\); when \(r \gg L\), the potential reference must be transferred to infinity, which alters the mathematical functional form of the potential from \(\ln(r)\) to \(1/r\) and the field to \(1/r^2\).

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124540/*