📚 Magnetic Force | 磁力
A magnetic force is the attraction or repulsion that arises between electrically charged particles because of their motion. It is one of the fundamental forces of nature and underpins everything from the navigation of a compass needle to the operation of electric motors and particle accelerators. In A‑Level Physics, we learn to quantify this force on current‑carrying conductors and on individual moving charges, using vector equations and Fleming’s left‑hand rule.
磁力是由于带电粒子的运动而产生的吸引或排斥力。它是自然界的基本力之一，从指南针的指向到电动机和粒子加速器的运转，背后都有磁力的身影。在 A‑Level 物理中，我们学习如何利用矢量方程和弗莱明左手定则，量化载流导体以及单个运动电荷所受的磁力。
1. Introduction to Magnetic Force | 磁力简介
A magnetic field exerts a force on any moving charged particle that cuts through the field lines. This force is called the magnetic Lorentz force. Unlike gravitational or electrostatic forces, the magnetic force depends on the velocity of the particle and always acts perpendicular to both the magnetic field and the direction of motion. As a result, a pure magnetic force does no work on a free charge – it can change the direction of motion but not the speed.
磁场会对任何切割磁感线运动的带电粒子施加一个力，这个力称为磁洛伦兹力。与引力或静电力不同，磁力依赖于粒子的速度，并且始终垂直于磁场和运动方向。因此，纯磁力对自由电荷不做功——它只改变运动方向而不改变速率。
2. Magnetic Force on a Current-Carrying Conductor | 载流导体所受的磁力
When a straight conductor of length L carrying a current I is placed at an angle θ to a uniform magnetic field of flux density B, the magnitude of the magnetic force is given by:
F = B I L sin θ
The force is perpendicular to both the direction of the current and the magnetic field. Its direction can be determined by Fleming’s left‑hand rule. The equation assumes the entire length of the conductor lies in the uniform field.
当长度为 L 的直导体中通有电流 I，并与磁通量密度为 B 的匀强磁场成 θ 角放置时，磁力的大小为：
F = B I L sin θ
力的方向同时垂直于电流方向和磁场方向，可用弗莱明左手定则判断。该方程假定导线的整个长度都处于匀强磁场中。
3. Fleming’s Left‑Hand Rule | 弗莱明左手定则
Fleming’s left‑hand rule is a simple mnemonic used to find the direction of the force on a current‑carrying conductor in a magnetic field. Hold your left hand so that the thumb, first finger and second finger are mutually perpendicular:
- First finger – direction of the external magnetic field (North to South)
- Second finger – direction of conventional current (positive to negative)
- Thumb – resulting direction of the magnetic force (motion)
This rule is essential for predicting motor rotation, loudspeaker cone movement and many other electromagnetic devices.
弗莱明左手定则是一种简单的记忆方法，用来判断磁场中载流导体的受力方向。伸出左手，使拇指、食指和中指两两互相垂直：
- 食指 – 外磁场方向（N 极到 S 极）
- 中指 – 常规电流方向（正极到负极）
- 拇指 – 产生的磁力方向（运动方向）
该定则对于预测电动机旋转、扬声器锥盆运动以及其他众多电磁装置至关重要。
4. Magnetic Flux Density B | 磁通量密度 B
The quantity B represents the magnetic flux density, often simply called the magnetic field strength. It is measured in teslas (T). One tesla is defined as the flux density that produces a force of 1 newton on 1 metre of wire carrying 1 ampere perpendicular to the field:
1 T = 1 N A⁻¹ m⁻¹
From the force equation we can rearrange to obtain B = F / (I L sin θ), which allows us to measure B directly using a current balance – a simple apparatus involving a wire, a magnet and a balance.
物理量 B 表示磁通量密度，通常简称为磁场强度，单位是特斯拉（T）。1 特斯拉定义为：当 1 米长的导线通有 1 安培的电流且与磁场垂直时，若受到的力为 1 牛顿，则该磁场的磁通量密度为 1 特斯拉：
1 T = 1 N A⁻¹ m⁻¹
从力公式变形可得 B = F / (I L sin θ)，这使得我们可以利用电流天平——一种由导线、磁铁和天平组成的简单装置——直接测量 B。
5. Force on a Moving Charge | 运动电荷所受的力
At the microscopic level, the magnetic force on a current‑carrying wire originates from the force on each moving charge inside it. For a single particle with charge q moving at velocity v at an angle θ to the magnetic field:
F = q v B sin θ
If the particle is moving perpendicular to the field (θ = 90°), the force simplifies to F = q v B. The direction is again given by Fleming’s left‑hand rule, but with the second finger now representing the direction of motion of a positive charge. For a negative charge (such as an electron), the force direction is reversed.
在微观层面，载流导线所受的磁力源自导线内每个运动电荷所受的力。对于电荷量为 q、以速度 v 运动且与磁场成 θ 角的单个粒子：
F = q v B sin θ
若粒子垂直于磁场运动（θ = 90°），力简化为 F = q v B。方向仍用弗莱明左手定则判断，但此时中指代表 正电荷 的运动方向。对于负电荷（如电子），力的方向需要反向。
6. Motion of Charged Particles in a Magnetic Field | 带电粒子在磁场中的运动
When a charged particle enters a uniform magnetic field perpendicularly, the magnetic force provides a centripetal force, causing the particle to follow a circular path. Equating magnetic force and centripetal force:
B q v = m v² / r
From this we obtain the radius of the circular path:
r = m v / (B q)
The period T of the circular motion is independent of speed:
T = 2π m / (B q)
If the particle’s velocity has a component parallel to the field, the path becomes a helix because the parallel component is unaffected by the magnetic force.
当带电粒子垂直进入匀强磁场时，磁力充当向心力，使粒子做圆周运动。令磁力等于向心力：
B q v = m v² / r
由此可得圆周半径：
r = m v / (B q)
圆周运动的周期 T 与速度无关：
T = 2π m / (B q)
若粒子的速度具有平行于磁场的分量，则路径变为螺旋线，因为平行分量不受磁力影响。
7. Application: Hall Effect | 应用：霍尔效应
When a current‑carrying conductor is placed in a perpendicular magnetic field, charge carriers (electrons or holes) are deflected to one side, creating a transverse voltage known as the Hall voltage VH. The Hall voltage is given by:
VH = B I / (n q t)
where n is the number density of charge carriers, q is the charge on each carrier and t is the thickness of the conductor in the direction of the magnetic field. The Hall effect is used to measure magnetic fields, determine the type of charge carriers (positive or negative) and calculate carrier density.
当载流导体置于垂直磁场中时，电荷载流子（电子或空穴）会被推向一侧，产生一个横向电压，称为霍尔电压 VH。霍尔电压的表达式为：
VH = B I / (n q t)
其中 n 是载流子数密度，q 是每个载流子的电荷量，t 是导体在磁场方向上的厚度。霍尔效应可用于测量磁场、判断载流子类型（正电或负电）以及计算载流子密度。
8. Force between Two Parallel Current‑Carrying Wires | 两平行载流导线间的力
Two long, straight, parallel wires carrying currents I₁ and I₂ separated by a distance r exert a force on each other. The force per unit length on each wire is:
F / L = μ₀ I₁ I₂ / (2π r)
where μ₀ . This is the basis for the SI definition of the ampere: one ampere is that constant current which, if maintained in two straight parallel conductors of infinite length and negligible cross‑n placed 1 metre apart in vacuum, would produce a force of 2 × 10⁻⁷ N per metre of length.
两根平行的长直导线，分别通有电流 I₁ 和 I₂，相距 r，彼此会施加作用力。每根导线单位长度上的力为：
F / L = μ₀ I₁ I₂ / (2π r)
其中 μ₀ = 4π × 10⁻⁷ N A⁻² 是真空磁导率。若电流同向，力为吸引力；若反向，则为排斥力。这正是安培的国际单位制定义的基础：1 安培是指，在真空中相距 1 米的两根无限长、截面可忽略的平行直导线中，通以恒定电流，若每米长度上产生的力为 2 × 10⁻⁷ 牛，则每根导线中的电流为 1 安培。
9. Comparison with Electric Force | 与电力的比较
While both electric and magnetic forces act on charged particles, they differ in several crucial ways:
| Property | Electric Force | Magnetic Force |
|---|---|---|
| Acts on | Stationary or moving charges | Only moving charges |
| Direction | Parallel to field | Perpendicular to field and velocity |
| Work done | Can do work (changes speed) | Does no work (only changes direction) |
| Expression | FE = q E | FB = q v B sin θ |
These differences explain why magnetic fields deflect charged particles without altering their kinetic energy, making them ideal for applications such as mass spectrometry and circular particle accelerators.
虽然电力和磁力都作用于带电粒子，但它们在几个关键方面存在差异：
| 性质 | 电力 | 磁力 |
|---|---|---|
| 作用对象 | 静止或运动的电荷 | 仅运动的电荷 |
| 方向 | 平行于场 | 垂直于场和速度 |
| 做功 | 能做功（改变速率） | 不做功（仅改变方向） |
| 表达式 | FE = q E | FB = q v B sin θ |
这些差异解释了为何磁场能使带电粒子偏转却不改变其动能，这使其成为质谱分析和圆形粒子加速器等应用的理想选择。
10. Summary and Common Misconceptions | 总结与常见误区
Key points to remember:
- Magnetic force always acts perpendicular to both the magnetic field and the current/velocity.
- Use F = B I L sin θ for a conductor and F = q v B sin θ for a point charge.
- The constant magnetic flux density B is measured in teslas (T).
- A magnetic force does no work, so kinetic energy of a particle moving purely under magnetic force remains constant.
- Fleming’s left‑hand rule applies to motors (converting electrical energy to kinetic); Fleming’s right‑hand rule applies to generators and is not needed for magnetic force questions.
Common pitfalls: confusing conventional current direction with electron flow when using Fleming’s rule; forgetting the sin θ factor when the angle is not 90°; and thinking that a magnetic field can speed up a charged particle (it cannot – only an electric field can).
需要牢记的要点：
- 磁力始终垂直于磁场和电流（或速度）方向。
- 对导体使用 F = B I L sin θ，对点电荷使用 F = q v B sin θ。
- 恒定磁场的磁通量密度 B 以特斯拉（T）为单位。
- 磁力不做功，因此仅在磁力作用下运动的粒子动能保持不变。
- 弗莱明左手定则适用于电动机（电能转为动能）；弗莱明右手定则适用于发电机，磁力问题中不需要使用右手定则。
常见误区：在使用弗莱明定则时混淆常规电流方向与电子流方向；当角度并非 90° 时忘记乘 sin θ 因子；认为磁场能使带电粒子加速（磁场不能，只有电场可以）。
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