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title: "A long, tightly wound solenoid of length \\(L\\) and radius \\(R\\) (where \\(L \\gg R\\)) carries a steady current \\(I\\). Deep inside the solenoid near its geometric center, the magnetic field along the central axis is uniform with magnitude \\(B_0 = \\mu_0 n I\\), where \\(n\\) is the number of turns per unit length. At the center of one of the open ends of the solenoid, the magnetic field magnitude on the axis is measured to be \\(\\frac{1}{2} B_0\\). Which of the following provides the correct physical explanation for this measurement?"
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url: "https://nerd-notes.com/ubq/118564/"
date_modified: "2026-08-04T08:11:23+00:00"
---

# A long, tightly wound solenoid of length \(L\) and radius \(R\) (where \(L \gg R\)) carries a steady current \(I\). Deep inside the solenoid near its geometric center, the magnetic field along the central axis is uniform with magnitude \(B_0 = \mu_0 n I\), where \(n\) is the number of turns per unit length. At the center of one of the open ends of the solenoid, the magnetic field magnitude on the axis is measured to be \(\frac{1}{2} B_0\). Which of the following provides the correct physical explanation for this measurement?

A long, tightly wound solenoid of length \(L\) and radius \(R\) (where \(L \gg R\)) carries a steady current \(I\). Deep inside the solenoid near its geometric center, the magnetic field along the central axis is uniform with magnitude \(B_0 = \mu_0 n I\), where \(n\) is the number of turns per unit length. At the center of one of the open ends of the solenoid, the magnetic field magnitude on the axis is measured to be \(\frac{1}{2} B_0\). Which of the following provides the correct physical explanation for this measurement?

![A long horizontal solenoid is shown in cross-section. The helical coils are wrapped tightly around a cylindrical core. Point P_center sits on the central axis at the geometric midpoint, with a rightward vector arrow labeled B_0. Point P_end sits on the central axis at the rightmost open boundary of the solenoid, with a rightward vector arrow labeled B_end = 1/2 B_0. No other labels or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831082-iyFbvh.jpg)

- **A.** An Amperian loop placed at the open end encloses only half as many current-carrying turns as an identical loop placed at the center, causing the line integral of the magnetic field to be halved.
- **B.** Self-inductance near the open ends produces a back electromotive force that reduces the local current density in the end turns to half of its value at the center.
- **C.** An infinitely long solenoid can be viewed as two semi-infinite solenoids joined end-to-end; by axial symmetry, each semi-infinite solenoid contributes equally to the magnetic field at their junction, so the field at an open end is half that of an infinite solenoid.
- **D.** Magnetic charges accumulate on the open boundary surfaces of the solenoid, creating a reverse magnetic field that linearly decreases the net axial field from \(B_0\) at the center to zero at the end.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118564/*