📚 A2 Physics: Magnetic Fields Key Points | A2 物理：磁场 考点精讲
In A2 Physics, magnetic fields are a core topic that bridges the understanding of forces on moving charges and currents, the motion of charged particles in circular paths, and electromagnetic induction. Mastering these concepts enables you to explain devices such as mass spectrometers, velocity selectors, Hall probes, generators and transformers, all of which frequently appear in exam questions.
在A2物理中，磁场是连接运动电荷与电流受力、带电粒子圆周运动以及电磁感应的核心主题。掌握这些概念能让你解释质谱仪、速度选择器、霍尔探头、发电机和变压器等设备的工作原理，它们都是考试中的常见考点。
1. Magnetic Force on a Current-Carrying Conductor | 电流在磁场中的受力
A straight conductor of length L carrying current I in a uniform magnetic field B experiences a force whose magnitude is F = B I L sin θ c field. When the conductor is perpendicular to the field (θ = 90°), the force is maximum: F = B I L.
长度为 L、通有电流 I 的直导体置于匀强磁场 B 中，会受到大小为 F = B I L sin θ 的力，其中 θ 为电流方向与磁场方向的夹角。当导体与磁场垂直时（θ = 90°），力为最大值 F = B I L。
The direction of the force is given by Fleming's left-hand rule: thumb – force, forefinger – field, second finger – conventional current. All three are mutually perpendicular.
力的方向由弗莱明左手定则确定：拇指表示力，食指表示磁场，中指表示电流（常规电流方向），三者互相垂直。
2. Force on a Moving Charge (Lorentz Force) | 运动电荷所受洛伦兹力
A charged particle moving with velocity v in a magnetic field B experiences the Lorentz force F = q v B sin θ, where q is the charge and θ is the angle between v and B. The direction of the force on a positive charge can be found by Fleming's left-hand rule, with the second finger representing the direction of conventional current (i.e. the direction of motion of the positive charge).
以速度 v 在磁场 B 中运动的带电粒子受到洛伦兹力 F = q v B sin θ，其中 q 为电荷量，θ 为 v 与 B 的夹角。正电荷受力方向可用弗莱明左手定则判断，此时中指指向正电荷运动方向（相当于常规电流方向）。
If the velocity is perpendicular to the magnetic field, the force becomes F = q v B and is always perpendicular to both v and B. This force changes the direction of the velocity without altering the speed, leading to circular motion for a charged particle in a uniform magnetic field.
若速度与磁场垂直，则力为 F = q v B，且始终垂直于速度与磁场。该力只改变速度方向，不改变大小，因此会使带电粒子在匀强磁场中做圆周运动。
3. Circular Motion of Charged Particles in Magnetic Fields | 带电粒子在磁场中的圆周运动
When a charged particle moves perpendicularly to a uniform magnetic field, the magnetic force provides the centripetal force: q v B = m v² / r. Solving for the radius gives r = m v / (B q). The radius depends on the particle's momentum and is larger for particles with higher mass or velocity, and smaller in stronger fields.
当带电粒子垂直于匀强磁场运动时，磁力提供向心力：q v B = m v² / r。解得半径 r = m v / (B q)。半径与粒子的动量相关：质量越大、速度越高，半径越大；磁场越强，半径越小。
The period of the circular motion is T = 2π m / (B q), which is independent of the particle's speed. This property is exploited in devices like the cyclotron.
圆周运动的周期为 T = 2π m / (B q)，与粒子的速率无关。这一性质被应用在回旋加速器等设备中。
4. Velocity Selector | 速度选择器
A velocity selector uses perpendicular electric and magnetic fields to allow only particles with a specific speed to pass through undeflected. The electric force FE = q E and magnetic force FB = q v B act in opposite directions. When the two forces balance, q E = q v B, giving v = E / B.
速度选择器利用相互垂直的电场和磁场，仅让特定速度的粒子不偏转地通过。电场力 FE = q E 与磁力 FB = q v B 方向相反。二者平衡时 q E = q v B，可得 v = E / B。
Particles with speed greater than E/B deflect towards the magnetic force side, while slower particles deflect towards the electric force side. This principle is essential in mass spectrometry to select ions of a known velocity.
速度大于 E/B 的粒子会偏向磁力方向，较慢的粒子则偏向电场力方向。这一原理在质谱仪中用于筛选出已知速度的离子。
5. Mass Spectrometer | 质谱仪
In a mass spectrometer, ions are first accelerated through a potential difference and then passed through a velocity selector to ensure a single speed. They then enter a region of uniform magnetic field and move in semicircular paths. The radius is r = m v / (B q), allowing the mass-to-charge ratio to be determined.
在质谱仪中，离子先被电势差加速，再经过速度选择器获得单一速度，随后进入匀强磁场区域做半圆运动。轨迹半径为 r = m v / (B q)，由此可以测定离子的荷质比。
Ions with different masses strike the detector at different positions; a larger mass gives a larger radius. By measuring the radius or the position of impact, the mass of the ion can be calculated if the charge is known.
不同质量的离子会打在不同位置上——质量越大，半径越大。通过测量半径或撞击位置，若已知电荷量，即可计算出离子质量。
6. Hall Effect | 霍尔效应
The Hall effect arises when a current-carrying conductor or semiconductor is placed in a perpendicular magnetic field. Charge carriers are deflected, creating a potential difference across the material, known as the Hall voltage VH. For a thin conducting strip, 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.
霍尔效应是通电导体或半导体置于垂直磁场中时，载流子发生偏转，从而在材料两侧形成电势差，即霍尔电压 VH。对于薄片导体，VH = B I / (n q t)，式中 n 为载流子数密度，q 为每个载流子的电荷量，t 为厚度。
The sign of the Hall voltage reveals the sign of the charge carriers. Hall probes exploit this effect to measure magnetic field strength by calibrating VH against a known B.
霍尔电压的符号可揭示载流子的正负。霍尔探头利用这一效应，通过标定 VH 与已知 B 的关系来测量磁场强度。
7. Magnetic Fields due to Currents | 电流产生的磁场
A long straight wire produces circular magnetic field lines concentric with the wire. The magnetic flux density at a distance r is B = μ₀ I / (2π r), where μ₀ is the permeability of free space. The direction is given by the right-hand grip rule: thumb points in current direction, fingers curl in the field direction.
长直导线产生以导线为中心的同心圆形磁感线。距离导线 r 处的磁通量密度为 B = μ₀ I / (2π r)，其中 μ₀ 为真空磁导率。方向由右手螺旋定则确定：拇指指向电流方向，四指弯曲方向即为磁场方向。
Inside a long solenoid, the field is uniform and parallel to the axis: B = μ₀ n I, where n is the number of turns per unit length. This strong, uniform field is used in electromagnets and transformers.
长直螺线管内部的磁场是匀强且平行于轴线的：B = μ₀ n I，n 为单位长度上的匝数。这种强而均匀的磁场被用于电磁铁和变压器中。
8. Magnetic Flux and Flux Linkage | 磁通量与磁链
Magnetic flux Φ through a surface is defined as Φ = B A cos θ, where B is the magnetic flux density, A is the area, and θ is the angle between the field lines and the normal to the surface. It is measured in webers (Wb).
磁通量 Φ 定义为穿过某个面的磁感线总数：Φ = B A cos θ，B 为磁通量密度，A 为面积，θ 为磁场方向与面法线间的夹角。单位是韦伯（Wb）。
When a coil has N turns, the flux linkage is Φ_link = N Φ. Changes in flux linkage induce an electromotive force (emf), which is the foundation of electromagnetic induction.
当线圈有 N 匝时，磁链为 Φ_link = N Φ。磁链的变化会感应出电动势，这是电磁感应的基础。
9. Electromagnetic Induction – Faraday's Law | 电磁感应——法拉第定律
Faraday's law of electromagnetic induction states that the induced emf in a circuit is equal to the negative rate of change of magnetic flux linkage: ε = - N ΔΦ / Δt. The negative sign indicates the direction of the induced emf as given by Lenz's law.
法拉第电磁感应定律指出，回路中感生电动势等于磁链变化率的负值：ε = - N ΔΦ / Δt。负号表示感生电动势的方向，由楞次定律决定。
The emf can be generated by changing B, changing the area A, or changing the angle θ between the coil and the field. An instantaneous emf can be written as ε = - d(NΦ)/dt, but at A-level we usually work with average changes.
电动势可通过改变磁场 B、改变面积 A 或改变线圈与磁场间的夹角 θ 来产生。瞬时电动势可写为 ε = - d(NΦ)/dt，但在 A Level 阶段我们通常处理平均变化量。
10. Lenz's Law | 楞次定律
Lenz's law gives the direction of the induced current: the induced current always flows in a direction that opposes the change in magnetic flux that produced it. This is a consequence of the conservation of energy.
楞次定律决定了感应电流的方向：感应电流总是沿这样一个方向，即它自身产生的磁场阻碍引起感应电流的磁通量变化。这是能量守恒的结果。
For example, when a magnet's north pole approaches a coil, the coil generates a north pole on the side facing the magnet to repel the approach. The mechanical work done against this repulsion is converted into electrical energy.
例如，当磁铁 N 极靠近线圈时，线圈面对磁铁的一侧产生 N 极，以排斥趋近的磁铁。克服这种排斥力所做的机械功转化为电能。
11. AC Generator and Transformers | 交流发电机与变压器
An AC generator consists of a coil rotating in a uniform magnetic field. The induced emf varies sinusoidally: ε = N B A ω sin ω t, where ω is the angular frequency. Slip rings and brushes ensure the alternating voltage is taken out.
交流发电机由在匀强磁场中转动的线圈构成。感生电动势随时间按正弦变化：ε = N B A ω sin ω t，ω 为角频率。滑环与电刷保证输出交变电压。
A transformer changes the voltage of an AC supply. For an ideal transformer, the ratio of voltages equals the turns ratio: Vp / Vs = Np / Ns. Power is conserved, so Ip Vp = Is Vs.
变压器用于改变交流电压。对于理想变压器，电压比等于匝数比：Vp / Vs = Np / Ns，且功率守恒，即 Ip Vp = Is Vs。
12. Eddy Currents | 涡流
Eddy currents are circulating currents induced in a conductor when it is exposed to a changing magnetic field. They flow in closed loops within the plane of the conductor and produce heating as well as magnetic effects that oppose the changing flux, in line with Lenz's law.
涡流是导体处于变化磁场中时，其内部感应出的环行电流。它们在导体平面内形成闭合回路，并产生热量以及反抗磁通变化的磁效应，符合楞次定律。
Eddy currents cause energy losses in transformer cores, so the cores are laminated to restrict the paths of the eddy currents. However, eddy currents are also used advantageously in electromagnetic braking and induction heating.
涡流会在变压器铁心中造成能量损失，因此铁心采用叠片结构来限制涡流路径。不过，涡流也被有利地应用于电磁制动和感应加热中。
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