📚 Charged Particles | 带电粒子
Charged particles are central to A-Level Physics because electric and magnetic fields exert forces on them, allowing us to control, accelerate and analyse their motion. This revision guide covers quantisation of charge, Coulomb's law, electric fields, magnetic fields, velocity selectors, the Hall effect and Millikan's oil drop experiment, with the derivations and worked examples that examiners expect.
带电粒子是 A-Level 物理的核心内容，因为电场和磁场能对它们施加力，使我们能够控制、加速并分析它们的运动。本复习指南涵盖电荷的量子化、库仑定律、电场、磁场、速度选择器、霍尔效应以及密立根油滴实验，并给出考官期待的推导过程与例题解析。
1. Charge and the Elementary Charge | 电荷与元电荷
nteger multiple of a single basic amount. This is the principle of quantisation: Q = ne, where n is an integer and e is the elementary charge. The magnitude of e is 1.60 × 10⁻¹⁹ C; a proton carries +e and an electron carries -e.
电荷是物质的基本属性，而在所有已完成的实验中，孤立物体所带的总电荷量始终是某个基本电荷量的整数倍。这就是量子化原理：Q = ne，其中 n 为整数，e 为元电荷。元电荷大小 e = 1.60 × 10⁻¹⁹ C；质子带 +e，电子带 -e。
Charge is conserved in all processes. When objects are rubbed together, electrons transfer from one material to another, making one object negative and the other positive, but the total charge of the isolated system remains constant. No process ever creates or destroys net charge; it is only redistributed.
电荷在所有过程中都是守恒的。物体相互摩擦时，电子从一个物体转移到另一个物体，使一个物体带负电、另一个带正电，但孤立系统的总电荷保持不变。没有任何过程能创生或消灭净电荷，电荷只会被重新分配。
currents involve enormous numbers of charge carriers. In exam questions about ionisation, remember that a singly charged ion has charge ±1.60 × 10⁻¹⁹ C, a doubly charged ion ±3.20 × 10⁻¹⁹ C, and so on.
电荷的 SI 单位是库仑（C）。由于 e 非常小，1 库仑约相当于 6.25 × 10¹⁸ 个电子，这也解释了为什么日常电流涉及数量极其庞大的载流子。在涉及电离的考题中要记住：一价离子带电 ±1.60 × 10⁻¹⁹ C，二价离子带电 ±3.20 × 10⁻¹⁹ C，依此类推。
he electron value is far larger simply because the electron mass (9.11 × 10⁻³¹ kg) is roughly 1836 times smaller than the proton mass (1.67 × 10⁻²⁷ kg).
比值 e/m 称为比荷，是高频考点。质子的比荷约为 9.58 × 10⁷ C kg⁻¹，电子约为 1.76 × 10¹¹ C kg⁻¹。电子比荷远大于质子，原因只是电子质量（9.11 × 10⁻³¹ kg）约为质子质量（1.67 × 10⁻²⁷ kg）的 1/1836。
2. Coulomb's Law | 库仑定律
For two point charges q₁ and q₂ separated by distance r in a vacuum, the force between them is directly proportional to the product of the charges and inversely proportional to the square of the separation. In equation form, the magnitude of the force is written as shown below.
对于真空中相距 r 的两个点电荷 q₁ 和 q₂，它们之间的作用力与电荷量的乘积成正比，与距离的平方成反比。用公式表示时，力的大小如下式所示。
F = q₁q₂ ÷ (4πε₀r²) = k q₁q₂ ÷ r²
1 ÷ (4πε₀). Candidates sometimes lose marks by forgetting to square r or by using the diameter instead of the separation between the centres of the charges.
常数 k 的值为 8.99 × 10⁹ N m² C⁻²，而真空介电常数 ε₀ = 8.85 × 10⁻¹² F m⁻¹，且 k = 1 ÷ (4πε₀)。考生失分常见于忘记将 r 平方，或误把直径当作两电荷中心之间的距离。
Coulomb's law is a vector relationship in disguise. Like charges repel, unlike charges attract, and the two forces form a Newton's third law pair: they are equal in magnitude, opposite in direction and act on different bodies. When more than two charges are present, the resultant force on any one charge is found by adding the individual forces as vectors, usually by resolving into perpendicular components.
库仑定律本质上是一种矢量关系。同种电荷相斥，异种电荷相吸，且这一对力构成牛顿第三定律作用力与反作用力：大小相等、方向相反，分别作用在不同物体上。当存在两个以上电荷时，任一电荷所受的合力需按矢量相加，通常通过分解为互相垂直的分量来求解。
It is instructive to compare the electric force with the gravitational force. Both obey an inverse-square law, but the electric force can be attractive or repulsive, whereas gravity is always attractive. For two protons, the electrostatic repulsion is about 10³⁶ times stronger than their gravitational attraction, which is why gravity only dominates on astronomical scales where huge neutral masses are involved.
把电力与万有引力作比较很有启发性。两者都遵循平方反比定律，但电力既可以是引力也可以是斥力，而万有引力永远是引力。对两个质子而言，静电斥力约为它们之间万有引力的 10³⁶ 倍，这正是引力只在涉及巨大中性质量的天文尺度上才占主导的原因。
3. Electric Field Strength and Field Lines | 电场强度与电场线
rge placed at that point, E = F ÷ Q, measured in newtons per coulomb (N C⁻¹), which is equivalent to volts per metre (V m⁻¹).
电场是带电粒子会受到力的区域。某点的电场强度 E 定义为放置于该点的单位正电荷所受的力，即 E = F ÷ Q，单位为牛顿每库仑（N C⁻¹），也等价于伏特每米（V m⁻¹）。
ntial difference across the plates and d is their separation. Because E is constant, a charge moving between the plates experiences a constant force and therefore a constant acceleration, exactly analogous to projectile motion under gravity.
在两块平行带电板之间电场是匀强的，其大小由 E = V ÷ d 给出，其中 V 是两板间的电势差，d 是板间距离。由于 E 恒定，电荷在板间运动时受到恒力作用，因而加速度恒定，与重力场中的抛体运动完全类似。
ength falls off with distance according to E = Q ÷ (4πε₀r²), obtained by combining Coulomb' ile the potential V follows a 1/r law; this distinction is a favourite exam discriminator.
在单个点电荷周围电场是径向的，其强度随距离按 E = Q ÷ (4πε₀r²) 衰减，该式由库仑定律与电场强度定义式结合得到。注意 E 遵循平方反比规律，而电势 V 遵循 1/r 规律，这一区别是考官偏爱的区分点。
Field lines are drawn with arrows showing the direction of the force on a positive charge, and their density indicates field strength. Field lines never cross, they leave and enter conductors perpendicular to the surface, and inside a conductor in electrostatic equilibrium the field is zero. Where several charges are present, the resultant field is the vector sum of the individual fields.
电场线用箭头表示正电荷所受力的方向，其疏密程度表示场强大小。电场线永不相交，离开或进入导体时与导体表面垂直，而在静电平衡的导体内部场强为零。当存在多个电荷时，合场强等于各分场强的矢量和。
4. Motion of a Charged Particle in a Uniform Electric Field | 带电粒子在匀强电场中的运动
d, so the acceleration is a = QV ÷ (md). This acceleration is independent of the particle's speed, which means the horizontal and vertical motions can be treated quite separately.
在匀强电场中，电荷 Q 所受的力为 F = QE = QV ÷ d，因此加速度 a = QV ÷ (md)。该加速度与粒子速率无关，这意味着水平方向与竖直方向的运动可以完全分开处理。
e vertical deflection is then y = ½at², giving the expression below for a particle of charge Q and mass m.
若粒子以速率 v 垂直进入电场，并在板间水平飞行距离 L，则其在电场中的时间 t = L ÷ v。竖直偏转量为 y = ½at²，对于电荷量为 Q、质量为 m 的粒子可得到下式。
y = QVL² ÷ (2mdv²)
This result explains how a cathode-ray oscilloscope works: the electron beam is deflected by a voltage applied to the Y plates, and the spot on the screen moves in proportion to that voltage. The same physics underlies inkjet printers, where charged droplets are steered onto paper by deflection plates, and electrostatic precipitators used to remove ash from power-station flue gases.
这一结果解释了阴极射线示波器的工作原理：电子束被施加在 Y 板上的电压偏转，屏幕上的光点位移与该电压成正比。同样的物理原理也支撑着喷墨打印机（带电墨滴被偏转板引导到纸上）以及用于去除电厂烟道灰分的静电除尘器。
Exam tip: always state whether you are dealing with an electron or a positive ion, because the direction of the force reverses for a negative charge. Also check whether gravity can be neglected. For an electron, the electric force is typically 10¹⁴ times its weight, so gravity is ignored; for a charged oil droplet in Millikan's experiment, gravity is not negligible.
应试提示：务必说明处理的是电子还是正离子，因为负电荷受力方向相反。还要判断重力能否忽略。对电子而言，电场力通常约为其重力的 10¹⁴ 倍，故可忽略重力；而在密立根实验中带电油滴所受重力不可忽略。
5. Electric Potential, Potential Difference and Energy | 电势、电势差与能量
easured in volts. Potential difference between two points is the energy transferred per unit charge as charge moves between them, and it is this quantity that a voltmeter measures.
某点的电势是把单位正试探电荷从无穷远处移到该点所做的功，即 V = W ÷ Q，单位为伏特。两点之间的电势差等于电荷在两点间移动时每单位电荷转移的能量，正是这个量由电压表测得。
and if the field alone accelerates the particle from rest, this work appears as kinetic energy.
电荷 Q 通过电势差 V 时所做的功为 W = QV；若仅由电场把粒子从静止加速，则该功全部转化为动能。
QV = ½mv² → v = √2QV ÷ m
Equipotential surfaces are always perpendicular to field lines, and no work is done when a charge moves along an equipotential. A positive charge accelerates naturally from high potential to low potential, while a negative charge accelerates in the opposite sense, which is a very common source of sign errors in calculations.
等势面总是与电场线垂直，电荷沿等势面移动时不做功。正电荷自发地从高电势向低电势加速，而负电荷则相反方向加速，这是计算中极常见的符号错误来源。
⁹ J. Working in electronvolts often avoids long strings of powers of ten, but remember to convert back to joules whenever a formula requires SI units.
电子伏特是处理此类问题的便捷单位：1 eV 等于电子经过 1 V 电势差加速后获得的动能，即 1.60 × 10⁻¹⁹ J。使用电子伏特常可避免冗长的十的幂，但每当公式要求 SI 单位时，务必换算回焦耳。
6. Magnetic Force on a Moving Charged Particle | 磁场对运动带电粒子的作用力
sinθ, where B is the magnetic flux density in tesla, Q the charge, v the speed and θ the angle between the velocity and the field. The force is maximum when the particle moves perpendicular to the field and zero when it moves parallel to it.
带电粒子只有在运动时才受到磁场力，其大小为 F = BQv sinθ，其中 B 为磁通密度（单位特斯拉），Q 为电荷量，v 为速率，θ 为速度与磁场之间的夹角。当粒子垂直磁场运动时力最大，平行于磁场运动时力为零。
⁻¹, so B = F ÷ (Qv) in that special case. The direction of the force is found with Fleming's left-hand rule, and for a negative charge such as an electron the direction is reversed.
1 特斯拉定义为：使电荷量 1 C 的粒子以 1 m s⁻¹ 垂直通过磁场时受到 1 N 力的磁通密度，故在该特殊情形下 B = F ÷ (Qv)。力的方向由弗莱明左手定则确定，而对于电子等负电荷，方向需要反向。
A crucial feature of the magnetic force is that it is always perpendicular to the velocity, so it does no work on the particle. The speed and kinetic energy therefore remain constant; only the direction of motion changes. This is completely different from the electric force, which can speed a particle up or slow it down.
磁场力的一个关键特性是它始终垂直于速度，因此对粒子不做功。粒子的速率与动能保持不变，改变的只是运动方向。这与电场力截然不同，电场力可以使粒子加速或减速。
7. Circular Motion of a Charged Particle in a Magnetic Field | 带电粒子在磁场中的圆周运动
If a particle enters a uniform magnetic field perpendicular to the field lines, the constant perpendicular force acts as a centripetal force and the particle follows a circular path. Equating the magnetic force to the centripetal requirement gives the radius of that circle.
若粒子垂直进入匀强磁场，恒定的垂直力充当向心力，粒子便沿圆形轨道运动。将磁场力与向心力所需条件相等，即可求出圆半径。
BQv = mv² ÷ r → r = mv ÷ (BQ)
ame field. The time for one complete orbit follows from T = 2πr ÷ v, and substituting for r gives the mass-independent result below.
半径随动量增大而增大，随磁感应强度与电荷量增大而减小，这正是能量更高的粒子在同一磁场中弯曲得更少的原因。绕行一周的时间由 T = 2πr ÷ v 求得，代入 r 后得到与质量无关的下式。
T = 2πm ÷ (BQ) and f = BQ ÷ (2πm)
The fact that the period does not depend on speed or radius is the operating principle of the cyclotron, in which particles are repeatedly accelerated across a gap by an alternating voltage whose frequency matches this cyclotron frequency. Because the frequency is independent of energy, a fixed-frequency supply keeps the particles in step as their radius grows.
周期与速率、半径无关这一事实正是回旋加速器的工作原理：粒子被频率与该回旋频率匹配的交变电压反复在缝隙中加速。由于频率与能量无关，固定频率的电源可在半径不断增大的过程中始终保持粒子与电压同步。
In exams, the direction of curvature is often worth a mark. Work out the direction of the force with Fleming's left-hand rule, or simply note that negative charges curve one way and positive charges the other way in the same field. Also remember to quote the mass of the proton or electron if the question does not supply it, and to keep track of powers of ten when calculating r.
考试中，弯曲方向的判断常常占一分。用弗莱明左手定则求出力的方向，或者简单记住：在同一磁场中，负电荷与正电荷的弯曲方向相反。此外，若题目未给出质子或电子的质量，要记得引用其数值，并在计算 r 时仔细处理十的幂。
8. Velocity Selector and Mass Spectrometer | 速度选择器与质谱仪
e direction; only particles whose speed satisfies QE = BQv pass through undeflected. The selected speed is therefore independent of charge and mass.
速度选择器使用相互垂直的电场与磁场。电场力 QE 指向一个方向，磁场力 BQv 指向相反方向；只有满足 QE = BQv 的粒子才能不受偏转地通过。因此被筛选出的速度与电荷量、质量均无关。
v = E ÷ B
tive ions. The ions are then accelerated through a potential difference V, so QV = ½mv², and they enter a velocity selector or a uniform magnetic field where they travel in a circular arc of radius r = mv ÷ (BQ).
质谱仪综合了上述所有思想。首先样品被汽化并电离，通常通过电子轰击产生单电荷正离子。随后离子经电势差 V 加速，故 QV = ½mv²，然后进入速度选择器或匀强磁场，在其中沿半径 r = mv ÷ (BQ) 的圆弧运动。
Combining the accelerating equation with the circular-motion equation and eliminating v gives a direct expression for the mass of the ion, and the detector records the relative abundance of each value of r.
把加速方程与圆周运动方程联立并消去 v，便可直接得到离子质量的表达式，探测器则记录各 r 值所对应的相对丰度。
m = B²Q²r² ÷ (2V) = BQr ÷ v
Because the mass appears in the expression, ions of different isotopes arrive at slightly different positions and can be counted separately. This is how relative atomic masses and isotopic abundances are measured, and the same technique is used in carbon dating, in checking for banned substances in sport, and in forensic drug testing. When writing answers, always specify the charge state of the ion, because a doubly charged ion would curve with half the radius.
由于表达式中含有质量，不同同位素的离子会到达略微不同的位置，从而可以被分别计数。这就是相对原子质量与同位素丰度的测量方法，同一技术也用于碳定年、体育违禁药物检测以及法医学毒品检验。答题时务必说明离子的电荷状态，因为二价离子的偏转半径只有一半。
9. The Hall Effect | 霍尔效应
The Hall effect occurs when a current-carrying conductor or semiconductor is placed in a magnetic field perpendicular to the current. Charge carriers are pushed sideways by the magnetic force, creating a transverse potential difference known as the Hall voltage across the sample.
当载流导体或半导体置于与电流垂直的磁场中时，就会产生霍尔效应。载流子受磁场力作用向侧面偏转，在样品两侧形成横向电势差，称为霍尔电压。
ier is balanced by the electric force from the accumulating charge, Bqv = qE_H, and with EH = VH mber density of charge carriers, t the thickness in the field direction and I the current.
平衡时，载流子所受磁场力与积累电荷产生的电场力平衡，Bqv = qE_H；结合 EH = VH ÷ d 与 I = nAvq，可得稳态霍尔电压如下，其中 n 为载流子数密度，t 为沿磁场方向的厚度，I 为电流。
VH = BI ÷ (ntq)
Several important consequences follow. A large Hall voltage requires
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