Movement of a charged particle in a magnetic field | 带电粒子在磁场中的运动

📚 Movement of a charged particle in a magnetic field | 带电粒子在磁场中的运动

When a charged particle moves through a magnetic field, it experiences a force known as the Lorentz force (magnetic component). This force is always perpendicular to both the velocity of the particle and the magnetic field, resulting in a curved trajectory. Understanding this motion is fundamental to many technologies, from mass spectrometers to cyclotrons. In this article, we explore the principles governing the movement of charged particles in magnetic fields, including circular and helical paths, and examine important applications.

当带电粒子在磁场中运动时，会受到一个力，即洛伦兹力的磁场分量。这个力始终垂直于粒子的运动速度和磁场方向，导致粒子轨迹发生弯曲。理解这一运动对于许多技术至关重要，从质谱仪到回旋加速器。本文将深入探讨带电粒子在磁场中运动的原理，包括圆周运动和螺旋运动，并考察其重要应用。

1. The Lorentz Force | 洛伦兹力

he motion is perpendicular to the field (θ = 90°) and zero when parallel (θ = 0° or 180°).

带电量为 q 的粒子以速度 v 在磁场 B 中运动，受到的磁场力为矢量叉积 F = q(v x B)。力的大小为 F = q v B sin θ，其中 θ 是速度矢量与磁场矢量之间的夹角。当运动方向与磁场垂直时（θ = 90°），力最大；平行时（θ = 0° 或 180°），力为零。

F = q v B sin θ

This expression is the foundation for analyzing all trajectories of charges in magnetic fields. It applies to positive and negative charges alike, with the direction of the force determined by the charge sign.

该表达式是分析一切电荷在磁场中运动轨迹的基础。它既适用于正电荷，也适用于负电荷，力的方向由电荷符号决定。

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2. Direction of the Force: Right-Hand Rule | 力的方向与右手定则

To find the direction of the magnetic force on a positive charge, use the right-hand rule: point the fingers of your right hand in the direction of the velocity v, then curl your fingers towards the magnetic field B. Your extended thumb points in the direction of the force F. For a negative charge, the force is opposite to the thumb's direction. This rule is a direct consequence of the cross product v x B.

要确定正电荷所受磁场力的方向，可使用右手定则：伸出右手，四指指向速度 v 的方向，然后弯曲四指转向磁场 B 的方向。此时，大拇指所指的方向就是力 F 的方向。对于负电荷，力的方向与大拇指指向相反。这一规则是叉积 v x B 的直接推论。

An alternative that is sometimes used is Fleming's left-hand rule, but the right-hand rule is more universal for moving charges as it mirrors the mathematical cross product. Students often practice with diagrams showing the three mutually perpendicular vectors v, B and F.

有时也可使用弗莱明左手定则，但右手定则对于运动电荷更具普适性，因为它与数学上的叉积一致。学生常通过展示 v、B 和 F 三矢量相互垂直的示意图来练习。

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3. Circular Motion: Conditions and Radius | 圆周运动的条件与半径

If a charged particle enters a uniform magnetic field with velocity exactly perpendicular to the field, the magnetic force is constant in magnitude (qvB) and always perpendicular to the velocity. This provides the centripetal force required for uniform circular motion. The particle travels in a circular path with constant speed.

若带电粒子以完全垂直于磁场的方向进入匀强磁场，则磁场力大小恒定（qvB），且始终垂直于速度。这就提供了匀速圆周运动所需的向心力。粒子将沿圆形路径以恒定速率运动。

qvB = m v² / r

Equating magnetic force to centripetal force, we solve for the radius r of the circular orbit:

令磁力等于向心力，可解出圆周轨道的半径 r：

r = m v / (q B)

This shows that the radius is directly proportional to the particle's momentum mv and inversely proportional to its charge and the magnetic field strength. For a given particle and field, a higher speed results in a larger radius.

这表明半径与粒子的动量 mv 成正比，与电荷及磁场强度成反比。对于给定的粒子和磁场，速度越高，半径越大。

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4. Period and Frequency of Circular Motion | 圆周运动的周期与频率

The time taken for one complete revolution, the period T, is obtained by dividing the circumference by the speed:

走完一整圈所需的时间，即周期 T，可通过周长除以速率得到：

T = 2πr / v = 2πm / (q B)

Notice that the period does not depend on the speed or the radius. Consequently, the angular frequency (cyclotron frequency) is:

注意，周期与速率和半径无关。因此，角频率（回旋频率）为：

ω = 2π / T = q B / m

from speed is a crucial feature exploited in particle accelerators such as the cyclotron.

普通频率为 f = ω/(2π) = qB/(2πm)。这种与速度无关的特性是回旋加速器等粒子加速器所利用的关键特征。

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5. Helical Motion: Velocity with a Parallel Component | 螺旋运动：速度的平行分量

When a charged particle has a velocity component parallel to the magnetic field (v∥), the trajectory becomes a helix. The parallel component remains unchanged because the magnetic force has no component along B. The perpendicular component v⊥ produces circular motion in the plane normal to B. The combination results in a spiral path along the field lines.

当带电粒子的速度存在平行于磁场的分量（v∥）时，轨迹变为螺旋线。平行分量保持不变，因为磁力在 B 方向无分量。垂直分量 v⊥ 在垂直于 B 的平面内产生圆周运动。两者叠加导致粒子沿磁场线做螺旋前进。

The radius of the helix is determined solely by the perpendicular component: r = m v⊥/(q B). The pitch p, the distance advanced parallel to B during one period, is given by:

螺旋的半径仅由垂直分量决定：r = m v⊥/(q B)。螺距 p，即在一个周期内平行于 B 前进的距离，为：

p = v∥ T = 2π m v∥ / (q B)

Helical motion is common in nature, for example, when charged particles from the solar wind spiral along the Earth's magnetic field lines towards the poles.

螺旋运动在自然界中很常见，例如太阳风中的带电粒子会沿地球磁场的磁感线螺旋向两极运动。

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6. Magnetic Force Does No Work | 磁场力不做功

the dot product of perpendicular vectors is zero. Therefore, a magnetic field alone cannot change the kinetic energy or speed of a charged particle; it can only alter the direction of motion.

由于磁力始终垂直于瞬时速度，磁场做的功为零。从数学上看，元功 dW = F · ds = F · v dt = 0，因为垂直矢量的点积为零。因此，仅靠磁场无法改变带电粒子的动能或速率；它只能改变运动方向。

This principle explains why the speed remains constant in both circular and helical motion. Any acceleration produced is purely centripetal. This contrasts with electric fields, which can do work and change kinetic energy.

这一定理解释了为何在圆周运动和螺旋运动中速率都保持不变。产生的加速度纯粹是向心加速度。这与电场不同，电场可以做功并改变动能。

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7. Application: Mass Spectrometer | 应用：质谱仪

A mass spectrometer separates ions based on their mass-to-charge ratio (m/q). Ions are first accelerated through a potential difference V, gaining kinetic energy:

质谱仪根据离子的质荷比（m/q）对其进行分离。离子首先通过电势差 V 被加速，获得动能：

(1/2) m v² = q V     ⇒     v = √2 q V / m

d move in a semicircle of radius r. Combining r = mv/(qB) with the expression for v gives:

然后它们垂直进入匀强磁场 B，并沿半径为 r 的半圆运动。将 r = mv/(qB) 与 v 的表达式联立可得：

r = (1/B) √2 m V / q     ⇒     m/q = B² r² / (2 V)

By measuring the impact positions on a detector, one can determine m/q. This technique is widely used in chemistry and physics to identify isotopes and molecular compositions.

通过测量离子在探测器上的落点位置，便可确定 m/q。该技术广泛用于化学和物理中，用以识别同位素和分子组成。

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8. Application: Cyclotron | 应用：回旋加速器

evolution T = 2πm/(qB) is independent of speed, so the alternating voltage applied across the gap between dees can be set to a fixed frequency f = qB/(2πm).

回旋加速器利用匀强磁场与交变电场的组合来加速带电粒子。两个空心 D 形电极（D 型盒）置于磁场中。在 D 形盒内，粒子因磁力沿半圆路径运动。回转周期 T = 2πm/(qB) 与速率无关，因此加在 D 形盒间隙上的交变电压可以设置为固定频率 f = qB/(2πm)。

Each time the particle crosses the gap, it is accelerated by the electric field, gaining kinetic energy and moving in a slightly larger semicircle. The process repeats, gradually boosting the particle's energy. However, relativistic effects limit the cyclotron at very high speeds, where the increasing mass changes the period. This led to the development of the synchrotron.

粒子每次穿过间隙时，都会被电场加速，动能增加，并在稍大的半圆上运动。这一过程不断重复，逐步提高粒子能量。但由于相对论效应，在极高速度下，粒子质量增大使周期改变，回旋加速器受到限制。这促使了同步加速器的诞生。

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9. Velocity Selector: Crossed Electric and Magnetic Fields | 速度选择器：正交电场与磁场

A velocity selector is a region of crossed electric and magnetic fields that permits only particles with a specific speed to pass through undeflected. The electric force qE acts in one direction, while the magnetic force qvB acts in the opposite direction. When the net force is zero, a particle travels in a straight line:

速度选择器是一个正交电场和磁场的区域，仅允许具有特定速率的粒子无偏转地通过。电场力 qE 沿某一方向作用，磁场力 qvB 沿相反方向作用。当合力为零时，粒子直线运动：

qE = q v B     ⇒     v = E / B

Particles moving too fast or too slow are deflected onto the walls and filtered out. This device is essential in many experiments, including mass spectrometers, to ensure a monochromatic beam of particles before entering further analysis.

运动过快或过慢的粒子会发生偏转，撞到壁上被滤除。该装置在许多实验（包括质谱仪）中至关重要，可确保粒子束在进入进一步分析之前具有单一的速率。

Although this setup involves an electric field, understanding the equilibrium between magnetic and electric forces deepens the appreciation of how charged particle trajectories can be controlled.

虽然该装置包含电场，但理解电场力与磁场力的平衡过程，加深了我们对带电粒子运动轨迹可控性的认识。

Published by TutorHao | Physics Revision Series | aleveler.com

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