📚 Orbiting Charges | 轨道电荷
When a charged particle moves through a magnetic field, it experiences a force that acts perpendicular to both its velocity and the magnetic field. If the particle enters the field at right angles, this magnetic force provides the centripetal force needed for uniform circular motion, causing the charge to travel in a circular orbit. Understanding orbiting charges is central to many applications in A-Level Physics, including mass spectrometers, cyclotrons, and velocity selectors.
当带电粒子穿过磁场时，会受到一个垂直于其速度和磁场的力。若粒子垂直进入磁场，该磁力便提供匀速圆周运动所需的向心力，使电荷沿圆形轨道运动。理解轨道电荷是A-Level物理中许多应用的核心，包括质谱仪、回旋加速器和速度选择器。
1. Lorentz Force and Circular Motion | 洛伦兹力与圆周运动
is the angle between v and B. When the particle moves perpendicular to the field (θ = 90°), the force simplifies to F = B q v. This force is always perpendicular to the velocity, so it does no work and changes only the direction of motion, not the speed. Consequently, the particle follows a circular path with constant speed.
电荷量为 q 的带电粒子以速度 v 在磁感应强度为 B 的磁场中运动时，会受到磁力 F = B q v sin θ 的作用，其中 θ 是 v 与 B 之间的夹角。当粒子垂直于磁场运动（θ = 90°）时，力简化为 F = B q v。该力始终垂直于速度，因此不做功，只改变运动方向而不改变速率。结果，粒子以恒定速率沿圆形路径运动。
For the circular motion, the magnetic force provides the centripetal force:
对于圆周运动，磁力提供向心力：
B q v = m v² / r
where m is the mass of the particle and r is the radius of the orbit.
其中 m 是粒子质量，r 是轨道半径。
2. Radius of the Orbiting Charge | 轨道电荷的半径
Rearranging the centripetal equation gives the radius of the circular path:
重新整理向心力方程，得到圆形路径的半径：
r = m v / (B q)
The radius is directly proportional to the momentum m v and inversely proportional to the magnetic field strength and the charge. Heavier particles or those with higher speed move in larger circles, while a stronger field or higher charge makes the orbit tighter.
半径与动量 m v 成正比，与磁场强度和电荷量成反比。较重的粒子或速度较高的粒子会以更大的半径运动，而更强的磁场或更大的电荷则使轨道更紧缩。
3. Period and Frequency – Cyclotron Formula | 周期与频率——回旋公式
The frequency of revolution, known as the cyclotron frequency f, is the reciprocal of the period:
绕行一周的周期 T 等于周长除以速率：T = 2π r / v。代入 r 的表达式，得到 T = 2π m / (B q)。旋转频率，即回旋频率 f，是周期的倒数：
T = 2π m / (B q) f = 1/T = B q / (2π m)
Notice that both T and f are independent of the particle's speed and radius, which is a remarkable feature of magnetic orbital motion.
注意，T 和 f 均与粒子速度和半径无关，这是磁场轨道运动的一个显著特点。
4. Independence from Speed – Isochronous Property | 与速度无关——等时性
In a uniform magnetic field, faster particles travel in larger circles, but the time to complete one circle remains the same. This is because the radius increases linearly with speed, so the ratio r/v stays constant. This property is called isochronism and is exploited in the cyclotron, where a fixed-frequency alternating electric field can repeatedly accelerate particles moving in semi-circular paths of increasing radius but constant period.
在均匀磁场中，速度较快的粒子在更大的圆周上运动，但完成一圈的时间保持不变。这是因为半径随速度线性增加，因此比值 r/v 恒定。这一特性称为等时性，并被用于回旋加速器：固定频率的交变电场可以反复加速在半径逐渐增大、周期恒定的半圆形路径上运动的粒子。
5. Velocity Selector – Straight Path through Crossed Fields | 速度选择器——穿过正交场地的直线路径
where the electric force qE and magnetic force Bqv act in opposite directions. These forces balance when qE = Bqv, giving v = E / B. Particles with this exact speed travel straight through; slower or faster ones are deflected. The crossed fields thus act as a gate, allowing only particles of the chosen velocity to pass.
速度选择器利用正交的电场和磁场筛选具有特定速率的粒子。带电粒子进入一个区域，在那里电场力 qE 与磁力 Bqv 方向相反。当 qE = Bqv 时，两力平衡，得出 v = E / B。具有这一准确速率的粒子沿直线穿过；速率较低或较高的粒子则会发生偏转。因此，正交场就像一道门，只允许选定速度的粒子通过。
v = E / B
6. Mass Spectrometer – Measuring Ionic Masses | 质谱仪——测量离子质量
ey emerge with a known speed v = E/B. They then enter a region of uniform magnetic field B' the selector field) where they move in a semicircular path of radius r. From r = m v / (B' q), we can determine the mass-to-charge ratio: m/q = B' dentify isotopes, analyse chemical samples, and measure atomic masses with high precision.
质谱仪用于测定离子的质荷比。离子首先通过速度选择器，因此以已知速率 v = E/B 射出。然后进入一个均匀磁场 B'（通常与
Published by TutorHao | A-Level Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图