📚 Charged Particles in Electric and Magnetic Fields | 电场与磁场中的带电粒子
Understanding the behaviour of charged particles in electric and magnetic fields is a cornerstone of A-Level Physics. It underpins the design of particle accelerators, mass spectrometers, velocity selectors and Hall‑effect sensors. In the CIE syllabus, you are expected to analyse forces, trajectories and energy changes, and to apply key equations to quantitative problems.
理解带电粒子在电场和磁场中的运动行为是 A-Level 物理的核心。它支撑着粒子加速器、质谱仪、速度选择器和霍尔传感器等实际应用。在 CIE 大纲中，你需要会分析力、轨迹和能量变化，并能将关键方程应用于定量计算。
1. Force on a Charged Particle in an Electric Field | 电场对带电粒子的作用力
F = qE, and its direction is parallel to the field for a positive charge and opposite to the field for a negative charge.
电荷量为 q 的粒子置于强度为 E 的均匀电场中时会受到恒定的电场力。力的大小由 F = qE 给出，正电荷受力方向与电场方向平行，负电荷受力方向与电场方向相反。
F = qE
ential difference V, the work done is W = qV, leading to a gain in kinetic energy of ½mv².
这个力可以对粒子做功，改变其动能。如果粒子通过电势差 V，电场做功 W = qV，从而使粒子获得动能 ½mv²。
2. Motion in a Uniform Electric Field: Parabolic Trajectories | 均匀电场中的运动：抛物线轨迹
ational field. The constant electric force causes a uniform acceleration a = qE/m in the direction of the field, while the velocity perpendicular to the field remains unchanged.
带电粒子垂直射入均匀电场时，其运动类似于重力场中的抛体运动。恒定的电场力在沿电场方向产生匀加速度 a = qE/m，而垂直于电场方向的速度保持不变。
a = qE/m
Taking the initial velocity v₀ along the x‑axis and the electric field along the y‑ parabolic path y = (qE / 2mv₀²) x². This trajectory is important in understanding cathode‑ray tubes and electrostatic deflection.
设初速度 v₀ 沿 x 轴、电场沿 y 轴，位移为 x = v₀t 和 y = ½at² = (qE/2m)t²。消去时间 t 得到抛物线轨迹 y = (qE / 2mv₀²) x²。这是理解阴极射线管和静电偏转的关键路径。
3. Acceleration of Charges: Electron Gun | 电荷的加速：电子枪
kinetic energy: qV = ½mv². Hence the final speed of the electrons is independent of the distance travelled and depends only on the accelerating voltage.
电子枪利用高电势差 V 将电子从静止加速。电场做功全部转化为动能：qV = ½mv²。因此电子的最终速率与加速距离无关，只取决于加速电压。
v = √2qV/m
For an electron, q = e and m = mₑ, giving v = √2eV/mₑ. Thermionic emission from a heated cathode provides the electrons, which are then focused into a beam. This principle is used in electron microscopes and X‑ray tubes.
对于电子，q = e、m = mₑ，可得 v = √2eV/mₑ。加热阴极的热电子发射提供电子，然后聚焦成束。这一原理用于电子显微镜和 X 射线管。
4. Lorentz Force: Magnetic Force on a Moving Charge | 洛伦兹力：磁场对运动电荷的作用力
s given by F = qvB sinθ, where θ is the angle between v and B. The direction is perpendicular to both v and B, as given by Fleming’s left‑hand rule for positive charges (reversed for negative charges).
以速度 v 运动的带电粒子在磁通密度为 B 的磁场中会受到磁场力，即洛伦兹力。其大小为 F = qvB sinθ，其中 θ 是 v 与 B 之间的夹角。方向垂直于 v 和 B，对正电荷可用弗莱明左手定则判定（负电荷反向）。
F = qvB sinθ
vB. If the velocity is parallel or antiparallel to the field, no magnetic force acts. Because the magnetic force is always perpendicular to the velocity, it does no work on the particle; it only changes the direction of motion.
当 θ = 90° 时力最大：F = qvB。如果速度与磁场平行或反平行，则不受磁力作用。由于磁力始终垂直于速度，它不对粒子做功，只改变运动方向。
5. Circular Motion in a Uniform Magnetic Field | 均匀磁场中的圆周运动
required for circular motion. Equating qvB = mv²/r gives the radius of the circular path.
当带电粒子垂直进入均匀磁场时，垂直于速度的磁力提供圆周运动所需的向心力。由 qvB = mv²/r 可得圆周路径的半径。
r = mv/(Bq)
e: T = 2πr/v = 2πm/(Bq). Notice that the period is independent of the speed and radius, a property exploited in cyclotron design.
粒子以恒定速率沿圆弧运动。周期 T（旋转一周的时间）由周长求得：T = 2πr/v = 2πm/(Bq)。注意周期与速率和半径无关，这一特性在回旋加速器设计中得到应用。
6. Deriving Radius and Period | 半径与周期的推导
period, T = 2πr/v, substitute r to obtain T = 2πm/(Bq). The angular frequency ω = 2π/T = Bq/m, also called the cyclotron frequency.
从向心力条件 qvB = mv²/r 出发，整理得 r = mv/(Bq)。动量 p = mv 与半径成正比。周期 T = 2πr/v，将 r 代入得 T = 2πm/(Bq)。角频率 ω = 2π/T = Bq/m，也称为回旋频率。
T = 2πm/(Bq)
These expressions assume the velocity is perpendicular to the magnetic field. If the particle has a velocity component parallel to B, the path becomes a helix. In most A‑Level problems, the perpendicular case is assumed.
这些表达式假设速度垂直于磁场。如果粒子有平行于磁场的速度分量，路径会变成螺旋线。在大多数 A‑Level 习题中，都假设垂直情形。
7. Velocity Selector: Balancing Electric and Magnetic Forces | 速度选择器：平衡电场力与磁场力
or a particle to travel in a straight line, the net force must be zero: qE = qvB, giving v = E/B.
速度选择器利用互相垂直的电场和磁场，只允许特定速度的粒子无偏转通过。要粒子沿直线运动，合力必须为零：qE = qvB，得 v = E/B。
v = E/B
In practice, charged particles enter a region where E and B are crossed. Particles moving too fast or too slow are deflected and hit the walls. By adjusting E or B, one can select the desired speed. Velocity selectors are integral to Bainbridge mass spectrometers.
实践中，带电粒子进入电场和磁场交叉的区域。速度过快或过慢的粒子会发生偏转并打在壁上。通过调节 E 或 B，可以选择所需的速度。速度选择器是班布里奇质谱仪的关键组成部分。
8. Mass Spectrometry: The Bainbridge Spectrometer | 质谱法：班布里奇质谱仪
at emerging ions have velocity v = Eₛ/Bₛ. They then enter a uniform magnetic field B and move in a semicircle of radius r = mv/(Bq). Combining these yields the ion mass.
班布里奇质谱仪可以测定离子的质量。离子首先通过交叉电场 Eₛ 和磁场 Bₛ 的速度选择器，出射离子具有速度 v = Eₛ/Bₛ。接着进入均匀磁场 B 并沿半径 r = mv/(Bq) 的半圆运动。结合两式可得离子质量。
m = (B Bₛ q r) / Eₛ
Since q, Eₛ, Bₛ and B are known, and r can be measured from the position of impact on a photographic plate or detector, the mass m is determined. This technique can separate isotopes of the same element.
由于 q、Eₛ、Bₛ 和 B 已知，r 可通过打在照相底片或探测器上的位置测出，从而确定质量 m。这种技术可以分离同一种元素的同位素。
9. Time‑of‑Flight (TOF) Mass Spectrometry | 飞行时间（TOF）质谱
al difference V, giving them kinetic energy qV = ½mv². The ions then drift through a field‑free region of length d to a detector. The time taken t = d/v is measured.
在飞行时间质谱仪中，离子通过已知电势差 V 加速，获得动能 qV = ½mv²。离子随后在长度为 d 的无场漂移区飞向探测器，测得其飞行时间 t = d/v。
m = (2qV t²) / d²
Because t depends on mass, lighter ions arrive earlier. The mass spectrum is obtained by recording arrival times. TOF mass spectrometers are widely used in biochemical analysis and have the advantage of detecting very large molecules.
因为飞行时间与质量有关，较轻的离子先到达。通过记录到达时间得到质谱。飞行时间质谱仪广泛用于生化分析，具有能检测大质量分子的优点。
10. The Hall Effect | 霍尔效应
When a current‑carrying conductor or semiconductor is placed in a perpendicular magnetic field, the charge carriers experience a magnetic force, causing them to accumulate on one side. This creates a transverse electric field and a measurable Hall voltage VH.
当载流导体或半导体置于垂直磁场中时，电荷载流子受到磁力作用，在一侧积累，从而产生横向电场和可测量的霍尔电压 VH。
VH = B I / (n e t)
Here, B is the magnetic flux density, I the current, n the number density of charge carriers, e the elementary charge, and t the thickness of the sample. The sign of VH indicates whether the charge carriers are positive or negative, making the Hall effect a powerful tool for studying semiconductors.
式中 B 为磁通密度，I 为电流，n 为载流子数密度，e 为基元电荷，t 为样品厚度。霍尔电压的符号揭示了载流子是正还是负，使得霍尔效应成为研究半导体的有力工具。
11. Determining Specific Charge (e/m) | 测定荷质比（e/m）
One classic method to measure e/m for an electron uses a fine‑ ld B at right angles. The circular path radius r is measured, and from r = mv/(eB) and v = √2eV/m, the specific charge is derived as follows.
测量电子荷质比 e/m 的经典方法之一是使用亥姆霍兹线圈的细束管。电子经电压 V 加速后垂直进入均匀磁场 B，测量圆周半径 r，由 r = mv/(eB) 和 v = √2eV/m 可推导比荷。
e/m = 2V / (B² r²)
directly confirms the particle nature of cathode rays and was historically crucial in the discovery of the electron.
通过测量 V、B 和 r，学生可以得到接近公认值 e/m = 1.76×10¹¹ C kg⁻¹ 的结果。该实验直接证实了阴极射线的粒子性，历史上对电子的发现至关重要。
12. Summary of Key Equations | 关键方程总结
The table below collects the most important equations for charged particles in fields, as required for CIE A‑Level Physics. These should be memorised, but more importantly, you must understand the physical principles from which they are derived.
下表汇总了 CIE A‑Level 物理中带电粒子在场中的最重要方程。这些公式需要熟记，但更重要的是理解其背后的物理原理。
| Relationship | Equation | Conditions |
|---|---|---|
| Electric force | F = qE | Uniform field |
| Magnetic force | F = qvB sinθ | Point charge |
| Radius in B‑field | r = mv/(Bq) | v ⟂ B |
| Period in B‑field | T = 2πm/(Bq) | v ⟂ B |
| Velocity selector | v = E/B | Straight line |
| Kinetic energy gain | <
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