--- title: "Two open-ended thin shells of the same axial length \\(L\\) are centered at the origin along the \\(z\\)-axis: a circular cylinder of constant radius \\(R_0\\) and a truncated cone whose radius varies linearly from \\(R_0 – \\Delta R\\) at \\(z = -L/2\\) to \\(R_0 + \\Delta R\\) at \\(z = +L/2\\), where \\(\\Delta R < R_0\\) and \\(L \\ll R_0\\). Each shell carries a steady, purely azimuthal surface current with identical uniform current per axial length \\(K\\). A magnetic sensor placed at the origin measures a field that is directed purely along the \\(z\\)-axis and has a greater magnitude for the cone than for the cylinder. Which of the following provides the correct physical explanation for these observations?" description: "## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please […]" url: "https://nerd-notes.com/ubq/124818/" date_modified: "2026-09-28T14:11:31+00:00" --- # Two open-ended thin shells of the same axial length \(L\) are centered at the origin along the \(z\)-axis: a circular cylinder of constant radius \(R_0\) and a truncated cone whose radius varies linearly from \(R_0 – \Delta R\) at \(z = -L/2\) to \(R_0 + \Delta R\) at \(z = +L/2\), where \(\Delta R < R_0\) and \(L \ll R_0\). Each shell carries a steady, purely azimuthal surface current with identical uniform current per axial length \(K\). A magnetic sensor placed at the origin measures a field that is directed purely along the \(z\)-axis and has a greater magnitude for the cone than for the cylinder. Which of the following provides the correct physical explanation for these observations? Two open-ended thin shells of the same axial length \(L\) are centered at the origin along the \(z\)-axis: a circular cylinder of constant radius \(R_0\) and a truncated cone whose radius varies linearly from \(R_0 - \Delta R\) at \(z = -L/2\) to \(R_0 + \Delta R\) at \(z = +L/2\), where \(\Delta R < R_0\) and \(L \ll R_0\). Each shell carries a steady, purely azimuthal surface current with identical uniform current per axial length \(K\). A magnetic sensor placed at the origin measures a field that is directed purely along the \(z\)-axis and has a greater magnitude for the cone than for the cylinder. Which of the following provides the correct physical explanation for these observations? ![Two coaxial grayscale geometric figures positioned along horizontal coordinate axes labeled z. On the left is a cylinder of length L centered at z = 0 with a constant radius labeled R_0, showing circular dashed end rims and solid horizontal top and bottom boundaries. On the right is a truncated cone of the same length L centered at z = 0, with a smaller circular left rim of radius R_0 - \Delta R and a larger circular right rim of radius R_0 + \Delta R. A horizontal dashed central axis labeled z passes through the center of both shapes, with the origin z = 0 marked by a small filled dot at the midpoint of each shape. Curved circular arrows around the curved surface of each shell indicate the direction of the azimuthal surface current K. Double-headed dimension arrows indicate the total length L for each shell. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604690-PUlXLW.jpg) - **A.** Applying Ampère's law along the central axis requires the magnetic field to depend exclusively on the enclosed linear current density \(K\), making the ideal field \(\mu_0 K\) for both geometries; the observed increase in the cone occurs because magnetic flux lines are compressed through the narrower aperture, increasing the axial field density near the center. - **B.** Because the radius of the truncated cone varies linearly about the origin, the mean radius of the current rings is identically \(R_0\); since each differential Biot-Savart ring contributes an axial field inversely proportional to its radius, the equal and opposite radius deviations \(\pm \Delta R\) symmetrically cancel in the first-order Taylor expansion, leaving higher-order end fringing as the sole source of the field difference. - **C.** Because current flows along a tapering boundary, the magnetic field vectors produced by individual current segments are tilted toward the central axis; the radial components of these vectors cancel symmetrically around the azimuth, while their axial projections reinforce one another constructively to produce a larger net axial field than the parallel current loops of the cylinder. - **D.** Each circular current slice produces an axial field whose radial components cancel by symmetry, keeping the net field purely along the \(z\)-axis; furthermore, because the on-axis Biot-Savart field of a loop scales inversely with its radius, the nonlinear \(1/R\) dependence ensures that the field enhancement from rings with \(R < R_0\) is strictly greater than the field reduction from rings with \(R > R_0\), yielding a larger net field for the cone. *The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124818/*