📚 Force on a Charge | 电荷受力
In the study of electromagnetism, understanding how charges experience forces in electric and magnetic fields is essential. These forces govern phenomena from the deflection of an electron beam to the operation of particle accelerators. This article explores the principles of electric and magnetic forces on a charge, their directions, and key applications including circular motion, the velocity selector, and the Hall effect.
在电磁学中，理解电荷如何在电场和磁场中受力至关重要。这些力支配着从电子束偏转到粒子加速器运行的众多现象。本文探讨电荷所受电场力和磁场力的原理、方向判定以及在圆周运动、速度选择器和霍尔效应等重点应用中的表现。
1. The Two Fundamental Forces on a Charge | 电荷受到的两种基本作用力
A stationary charge experiences a force only in an electric field. However, a moving charge can be acted upon by both electric and magnetic fields simultaneously. The total electromagnetic force is the vector sum of the electric force and the magnetic force, commonly expressed by the Lorentz force law.
静止电荷只有在电场中才会受力。然而，运动的电荷可以同时受到电场和磁场的作用。总的电磁力是电场力与磁场力的矢量和，通常由洛伦兹力定律描述。
For a charge q moving with velocity v in an electric field E and a magnetic flux density B, the total force F is:
对于以速度v运动的电荷q，在电场E和磁通量密度B中受到的总力F为：
F = qE + q(v × B)
This fundamental equation underpins everything from cathode ray tubes to the aurora borealis.
这个基本方程从阴极射线管到北极光现象中无处不在。
2. Electric Force: F = qE | 电场力：F = qE
In any electric field, a charge q experiences a force directly proportional to the field strength. In a uniform electric field between parallel plates, the force is constant in magnitude and direction.
在任何电场中，电荷q受到的力与电场强度成正比。在平行板之间的匀强电场中，该力的大小和方向保持恒定。
F = qE
The direction of the electric force on a positive charge is the same as the direction of the electric field; for a negative charge, the force direction is opposite to the field. This simple relationship allows precise control of charged particle motion, for instance in electrostatic precipitators or ink-jet printers.
正电荷所受电场力的方向与电场方向相同；负电荷所受力的方向则与电场方向相反。这一简单关系使得对带电粒子运动的精确控制成为可能，例如在静电除尘器或喷墨打印机中。
done by the electric force is W = qV, which converts into kinetic energy in accelerators.
当电荷在电势差为V的两点之间移动时，电场力做的功为W = qV，在加速器中这部分功转化为电荷的动能。
3. Magnetic Force on a Moving Charge: F = qvB sinθ | 磁场对运动电荷的力
A magnetic field exerts a force only on a moving charge, and the force is always perpendicular to both the velocity of the charge and the magnetic field. The magnitude depends on the component of velocity perpendicular to the field.
磁场只对运动的电荷施加力，且该力始终同时垂直于电荷的速度和磁场方向。其大小取决于速度垂直于磁场方向的分量。
F = qvB sinθ
0°), the magnetic force is zero. The maximum force occurs when the charge moves perpendicular to the field (θ = 90°).
其中θ是速度矢量v与磁通量密度B之间的夹角。如果电荷平行或反平行于磁场方向运动（θ = 0° 或180°），磁场力为零。最大磁场力出现在电荷垂直于磁场运动时（θ = 90°）。
This force does no work on the charge because it is always perpendicular to the velocity; it only changes the direction of motion, not the speed. This property is exploited in magnetic confinement and circular accelerators.
由于此力总是垂直于速度，它不对电荷做功；它只改变运动方向，不改变速率。这一特性被用于磁约束和环形加速器中。
4. Direction of Forces: Fleming’s Left-Hand Rule | 力的方向：弗莱明左手定则
To determine the direction of the magnetic force on a positive charge, use Fleming’s left-hand rule: hold your left hand with the thumb, index finger and middle finger mutually perpendicular. The First finger points in the direction of the magnetic field (B), the seCond finger in the direction of conventional current (motion of a positive charge, v), then the thuMb gives the direction of the force (F).
要确定正电荷所受磁场力的方向，可以使用弗莱明左手定则：伸出左手，使拇指、食指和中指相互垂直。食指指向磁场方向（B），中指指向常规电流方向（正电荷的运动方向v），则拇指指向受力方向（F）。
For a negative charge such as an electron, the force direction is opposite to that given by the left-hand rule, because conventional current is opposite to the electron’s motion. An alternative right-hand palm rule can be used for positive charges: fingers along B, thumb along v, palm push gives F.
对于电子等负电荷，受力方向与左手定则判断的方向相反，因为常规电流方向与电子运动方向相反。另一种右掌定则可用于正电荷：四指沿B方向，拇指沿v方向，掌心推力方向即为F。
Direction: F ⊥ v and F ⊥ B
These directional rules are essential for analysing particle tracks in cloud chambers or for designing beam deflection systems.
这些方向判定规则对于分析云室中的粒子径迹或设计束流偏转系统至关重要。
5. Motion in a Uniform Magnetic Field | 匀强磁场中的运动
When a charged particle enters a uniform magnetic field perpendicularly, the magnetic force provides the centripetal force required for circular motion. The particle moves in a circular path with constant speed.
当带电粒子垂直进入匀强磁场时，磁场力提供圆周运动所需的向心力。粒子将以恒定速率做圆周运动。
The magnetic force Fmag = qvB acts as the centripetal force:
磁场力 Fmag = qvB 充当向心力：
qvB = mv² / r
If the velocity has a component parallel to the field, the path becomes a helix because the parallel component remains unchanged while the perpendicular component causes circular motion.
如果速度具有平行于磁场的分量，轨迹将变成螺旋线，因为平行分量保持不变，而垂直分量导致圆周运动。
6. Radius and Period of Circular Motion | 圆周运动半径与周期
From the centripetal force equation, the radius r of the circular path can be derived:
由向心力方程可推导出圆周运动的半径r：
r = mv / (qB)
The radius is directly proportional to the particle’s momentum mv and inversely proportional to both the charge and the magnetic flux density. This means heavier or faster particles have larger radii, while stronger magnetic fields produce tighter curves.
半径与粒子的动量mv成正比，与电荷量和磁通量密度成反比。这意味着更重或更快的粒子半径更大，而更强的磁场会使曲线更紧密。
The period T (time for one complete revolution) is independent of the speed:
周期T（完成一次完整圆周运动的时间）与速率无关：
T = 2πm / (qB)
This remarkable result is the basis of the cyclotron’s constant revolution time, regardless of the particle’s energy as long as the speed is non-relativistic.
这一显著结果是回旋加速器具有恒定回旋时间的基础，只要速率处于非相对论范围，粒子能量变化不影响周期。
7. The Velocity Selector | 速度选择器
A velocity selector uses perpendicular electric and magnetic fields to allow only particles of a specific velocity to pass through undeflected. The electric force qE and the magnetic force qvB on a charge oppose each other.
速度选择器利用相互垂直的电场和磁场，只允许具有特定速率的粒子无偏转通过。作用在电荷上的电场力qE与磁场力qvB彼此反向。
For a particle to travel straight through, the net force must be zero:
粒子若要直线通过，净力必须为零：
qE = qvB → v = E / B
Particles with speeds greater or smaller than E/B will be deflected and collide with the plates or miss the exit slit. Velocity selectors are crucial components in mass spectrometers for ensuring a monochromatic beam of charged particles.
速率大于或小于E/B的粒子将被偏转，撞击极板或错过出口狭缝。速度选择器是质谱仪中确保带电粒子束单色性的关键组件。
8. The Hall Effect | 霍尔效应
When a current-carrying conductor or semiconductor is placed in a perpendicular magnetic field, the moving charge carriers experience a magnetic force. This pushes them to one side, creating a transverse electric field and thus a Hall voltage across the material.
当载流导体或半导体置于垂直磁场中时，运动的载流子受到磁场力，被推向一侧，产生横向电场，从而在材料两端形成霍尔电压。
At equilibrium, the magnetic force on the charge carriers is balanced by the electric force from the built-up charge separation:
达到平衡时，作用在载流子上的磁场力与由电荷分离产生的电场力相平衡：
qEH = qvdB
where vd is the drift velocity of the carriers. The Hall voltage VH is given by VH ier density, t the thickness. The sign of the Hall voltage reveals whether the majority carriers are positive (holes) or negative (electrons), making it a powerful tool in semiconductor characterisation.
其中vd是载流子的漂移速度。霍尔电压VH由VH = B I / (n q t)给出，n是载流子密度，t是厚度。霍尔电压的符号揭示了多数载流子是正电（空穴）还是负电（电子），使其成为半导体表征中的有力工具。
9. Mass Spectrometry and the Cyclotron | 质谱分析与回旋加速器
Mass spectrometer: d enter a uniform magnetic field, moving in a semicircle of radius r = mv/(qB). By measuring r, the mass-to-charge ratio m/q can be determined, allowing isotope identification.
质谱仪：正离子经过电势差V加速，获得动能½mv² = qV。随后通过速度选择器，进入匀强磁场，以半径r = mv/(qB)做半圆运动。测量r即可确定质荷比m/q，从而实现同位素识别。
Cyclotron: , forcing them into a spiral path. Each time the particles cross the gap between dees, the electric field accelerates them, and the orbital radius increases. The constant revolution period T = 2πm/(qB) ensures synchronised acceleration regardless of radius.
回旋加速器：两个中空D形电极（D形盒）加上交变电压来加速带电粒子。垂直方向的匀强磁场强迫粒子做螺旋运动。每当粒子穿过D形盒之间的间隙，电场就对它们加速，轨道半径随之增大。恒定的回旋周期T = 2πm/(qB)保证了无论半径如何变化都能同步加速。
10. Summary and Key Equations | 总结与关键方程
The force on a charge in electric and magnetic fields is summarised below. Mastering these relationships and their vector directions is vital for tackling A-Level physics problems on electromagnetism.
以下总结了电荷在电场和磁场中的受力情况。掌握这些关系式及其矢量方向对解决A-Level物理电磁学问题至关重要。
| Quantity | Equation | Notes |
|---|---|---|
| Electric force | F = qE | Direction same as E for +q |
| Magnetic force | F = qvB sinθ | Perpendicular to v and B |
| Radius in B field | r = mv / (qB) | For v ⊥ B |
| Period in B field | T = 2πm / (qB) | Independent of speed |
| Velocity selector | v = E / B | For zero net force |
Always consider the sign of the charge when determining the direction of the electric force, and remember that magnetic force never does work on a free charge – it changes direction only.
在确定电场力方向时始终考虑电荷的符号，并牢记磁场力从不对自由电荷做功——它仅改变运动方向。
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