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title: "A positively charged particle moves with velocity \\(\\vec{v}\\) in a region containing a non-uniform magnetic field \\(\\vec{B}\\) aligned generally along the \\(z\\)-axis (a magnetic bottle configuration). Near the “neck” of the bottle at \\(z > 0\\), the magnetic field lines converge toward the \\(z\\)-axis, introducing a radial magnetic field component \\(B_r < 0\\) directed toward the axis. The particle spirals around the \\(z\\)-axis with an azimuthal velocity component \\(v_\\theta\\) while traveling in the \\(+z\\)-direction with axial speed \\(v_z\\). As the particle approaches the neck, its axial speed \\(v_z\\) decreases to zero and the particle reflects back toward \\(z = 0\\). Which of the following explanations correctly accounts for the axial deceleration of the particle and the work done by the magnetic field during this reflection?"
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url: "https://nerd-notes.com/ubq/118590/"
date_modified: "2026-08-04T08:11:31+00:00"
---

# A positively charged particle moves with velocity \(\vec{v}\) in a region containing a non-uniform magnetic field \(\vec{B}\) aligned generally along the \(z\)-axis (a magnetic bottle configuration). Near the “neck” of the bottle at \(z > 0\), the magnetic field lines converge toward the \(z\)-axis, introducing a radial magnetic field component \(B_r < 0\) directed toward the axis. The particle spirals around the \(z\)-axis with an azimuthal velocity component \(v_\theta\) while traveling in the \(+z\)-direction with axial speed \(v_z\). As the particle approaches the neck, its axial speed \(v_z\) decreases to zero and the particle reflects back toward \(z = 0\). Which of the following explanations correctly accounts for the axial deceleration of the particle and the work done by the magnetic field during this reflection?

A positively charged particle moves with velocity \(\vec{v}\) in a region containing a non-uniform magnetic field \(\vec{B}\) aligned generally along the \(z\)-axis (a magnetic bottle configuration). Near the "neck" of the bottle at \(z > 0\), the magnetic field lines converge toward the \(z\)-axis, introducing a radial magnetic field component \(B_r < 0\) directed toward the axis. The particle spirals around the \(z\)-axis with an azimuthal velocity component \(v_\theta\) while traveling in the \(+z\)-direction with axial speed \(v_z\). As the particle approaches the neck, its axial speed \(v_z\) decreases to zero and the particle reflects back toward \(z = 0\). Which of the following explanations correctly accounts for the axial deceleration of the particle and the work done by the magnetic field during this reflection?

![A horizontal magnetic field configuration symmetric about a central horizontal z-axis running from left to right. At z=0 on the left, magnetic field lines are parallel to the z-axis and widely spaced. Moving toward z=L on the right, the magnetic field lines slope inward toward the z-axis, converging to form a narrow neck. A positively charged particle trajectory is shown as a helical coil spiraling around the z-axis from left to right. Near the neck, velocity vectors v_z pointing right along the z-axis and v_theta pointing tangent to the spiral in the azimuthal direction are labeled. A magnetic field vector B slopes inward and to the right, showing an inward radial component B_r pointing toward the z-axis. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831091-4xOcxW.jpg)

- **A.** The converging magnetic field lines create an axial gradient in \(B_z\) that exerts a direct magnetic force along the \(-z\)-direction. Because this force opposes the particle's displacement, the magnetic field performs negative work on the particle, reducing its total kinetic energy until it momentarily stops and reflects.
- **B.** As the particle moves toward the neck, the spatial variation in magnetic field induces a retarding axial electric field. This induced electric field performs negative work on the particle, decreasing its kinetic energy until its axial velocity reaches zero.
- **C.** The inward radial field component \(B_r\) exerts a retarding axial force \(F_z\). The magnetic field does negative work on the axial component of motion while doing an equal amount of positive work on the azimuthal component of motion, resulting in a net work of zero.
- **D.** The inward radial field component \(B_r\) interacts with the particle's azimuthal velocity \(v_\theta\) via \(q(\vec{v}_\theta \times \vec{B}_r)\) to produce a retarding axial force \(F_z\). The magnetic force performs zero work on the particle because \(\vec{F}_B\) is perpendicular to \(\vec{v}\) at every instant, continuously converting axial kinetic energy into transverse kinetic energy until \(v_z = 0\).

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118590/*