📚 A-Level Physics: Topic Test – Oxford AQA Int A-Level Physics – Magnetic Fields Formula Derivation | A-Level 物理：专题测试 – 牛津 AQA 国际 A-Level 物理磁场公式推导
Magnetic fields are fundamental to understanding the behaviour of moving charges and current-carrying conductors. In the Oxford AQA International A-Level Physics specification, this topic tests your ability not only to recall key formulas but also to derive them from first principles. This article walks through the essential derivations for magnetic forces, circular motion of charges, mass spectrometry, the cyclotron, and the force between two parallel wires, as well as magnetic flux. Each section presents the reasoning in clear steps, linking the underlying physics to the final equation.
磁场是理解运动电荷与载流导体行为的基础。在牛津 AQA 国际 A-Level 物理考纲中，本专题不仅考查对关键公式的记忆，还要求考生能从头推导这些公式。本文逐步讲解磁场力、带电粒子的圆周运动、质谱仪、回旋加速器、两根平行导线间的力以及磁通量的基本推导。每一节都以清晰的步骤展示推理过程，将物理本质与最终方程联系起来。
1. Magnetic Field and Flux Density B | 磁场与磁通量密度 B
A magnetic field is a region where a moving charge or a current-carrying conductor experiences a force. Its strength is described by the magnetic flux density B, measured in tesla (T). One tesla is defined as the flux density that produces a force of 1 newton per metre on a straight conductor carrying 1 ampere perpendicular to the field.
磁场是一种能使运动电荷或载流导体受力的区域。其强弱用磁通量密度 B 描述，单位为特斯拉 (T)。1 特斯拉的定义是：当磁场与电流方向垂直时，对每米长度通有 1 安培电流的直导体产生 1 牛顿的作用力。
The direction of the magnetic field is taken from north to south outside a magnet and can be represented by field lines. The spacing of these lines indicates the field strength – closer lines mean a stronger field.
磁场的方向在磁铁外部从北极指向南极，可用磁感线表示。磁感线的疏密反映场强——越密表示磁场越强。
2. Force on a Current-Carrying Conductor: F = BIL sinθ | 载流导体所受的力：F = BIL sinθ
When a straight wire of length L carries a current I and is placed in a uniform magnetic field of flux density B, it experiences a force. The magnitude is given by:
当一根长度为 L 的直导线通有电流 I，并置于磁通量密度为 B 的匀强磁场中时，它会受到力的作用。大小为：
F = B I L sinθ
where θ is the angle between the current direction and the magnetic field. The force is maximum when the conductor is perpendicular to the field (θ = 90°) and zero when parallel (θ = 0°).
其中 θ 为电流方向与磁场方向的夹角。当导体与磁场垂直 (θ = 90°) 时力最大，平行 (θ = 0°) 时不受力。
This equation can be remembered using Fleming’s left-hand rule: thumb – force, first finger – field, second finger – current, all mutually perpendicular. This rule gives the direction of the force on a conventional current.
该公式可结合弗莱明左手定则记忆：拇指——力，食指——磁场，中指——电流，三者相互垂直。此定则给出传统电流所受力的方向。
3. Deriving F = BIL from the Lorentz Force | 从洛伦兹力推导 F = BIL
The force on a current-carrying wire is actually the resultant of the Lorentz forces acting on each individual charge carrier moving inside the conductor. The Lorentz force on a single charge q moving with velocity v at right angles to a magnetic field is:
载流导线所受的力实际上是作用在导体内部每个运动电荷上的洛伦兹力的合力。单个电荷 q 以速度 v 垂直于磁场运动时受到的洛伦兹力为：
Fsingle = q v B
Consider a straight conductor of length L and cross-sectional area A, with n charge carriers per unit volume, each of charge q. The total number N of carriers in the wire that contribute to the force is n A L. The drift velocity v relates to the current by I = n A q v. Therefore, the total force on the wire is:
考虑一根长度为 L、截面积为 A 的直导体，单位体积内有 n 个电荷载流子，每个载流子电荷量为 q。导线中参与受力的载流子总数 N = n A L。漂移速度 v 与电流的关系为 I = n A q v。因此导线所受总力为：
F = N q v B = (n A L) q v B = (n A q v) L B = I L B
When the current is not perpendicular to the field, we replace B with its perpendicular component B sinθ, yielding F = B I L sinθ. This derivation elegantly connects the microscopic Lorentz force to the macroscopic behaviour of a current-carrying wire.
当电流与磁场不垂直时，用垂直分量 B sinθ 替代 B，即得 F = B I L sinθ。这个推导将微观的洛伦兹力与宏观载流导线的行为优雅地联系起来。
4. Motion of a Charged Particle in a Uniform Magnetic Field | 带电粒子在匀强磁场中的运动
When a charged particle enters a uniform magnetic field with its velocity perpendicular to the field, the Lorentz force acts as a centripetal force, causing circular motion. The force is always perpendicular to the velocity, so it does no work and the speed remains constant.
当带电粒子以垂直于磁场的方向进入匀强磁场时，洛伦兹力充当向心力，使其做圆周运动。由于力始终与速度垂直，不做功，因此速率保持不变。
The centripetal force required for circular motion of mass m, speed v and radius r is mv²/r. Equating this to the magnetic force q v B:
质量为 m、速率为 v、半径为 r 的圆周运动所需向心力为 mv²/r。令其等于磁场力 q v B：
q v B = m v² / r
Cancelling one v and rearranging gives the radius of the circular path:
消去一个 v 并整理，得到轨道半径：
r = m v / (q B)
This equation shows that the radius increases with momentum m v and decreases with charge and field strength. Heavier or faster particles curve less sharply.
此式表明半径随动量 m v 增大而增大，随电荷量和场强增大而减小。较重或较快的粒子偏转幅度较小。
5. Period and Frequency of Circular Motion (Cyclotron) | 圆周运动的周期与频率（回旋加速器）
The period T of one complete revolution is the circumference divided by speed:
运动一周的周期 T 为圆周长除以速率：
T = 2πr / v
Substitute r = m v / (q B) into this expression:
将 r = m v / (q B) 代入：
T = 2π (m v / (q B)) / v = 2π m / (q B)
Crucially, the speed v cancels out. The period depends only on the particle’s charge and mass and the magnetic field strength, not on its speed. The frequency, known as the cyclotron frequency, is:
关键之处在于速度 v 消去了。周期只取决于粒子的比荷以及磁场的强弱，与速率无关。频率即为回旋加速器频率：
f = 1/T = q B / (2π m)
This constancy of T is the principle that allows a cyclotron to accelerate particles. The electric field alternates with this fixed frequency, so the particle always gains energy each time it crosses the gap between the dees, regardless of its increasing speed (until relativistic effects become significant).
周期 T 的恒定性是回旋加速器能够加速粒子的原理。电场以这一固定频率交替变化，使得粒子每次穿越 D 形盒间隙时都能获得能量，而不受其速度增加的影响（直到相对论效应变得明显）。
6. Velocity Selector: v = E/B | 速度选择器：v = E/B
A velocity selector uses perpendicular electric and magnetic fields to filter particles moving at a specific speed. Charged particles pass through a region where an electric field E exerts a force qE in one direction and a magnetic field B exerts a force q v B in the opposite direction.
速度选择器利用相互垂直的电场和磁场来筛选具有特定速度的粒子。带电粒子穿过一个区域，其中电场 E 施加一个方向的力 qE，磁场 B 施加相反方向的力 q v B。
When these two forces balance, the net force is zero and the particle continues undeflected. Therefore:
当两力平衡时，合力为零，粒子不发生偏转。因此：
q E = q v B → v = E / B
Only particles with speed exactly v = E/B emerge through a narrow slit. This is typically the first stage of a mass spectrometer, producing a monoenergetic beam of ions.
只有速度精确为 v = E/B 的粒子才能穿过狭窄的狭缝。这通常是质谱仪的第一级，用以产生单能离子束。
7. Mass Spectrometer: Determining m/q | 质谱仪：确定质荷比 m/q
After passing through a velocity selector, ions of known speed v = E/B₁ enter a uniform magnetic field B₂ perpendicular to their velocity, where they move in semicircular paths. The radius is:
在通过速度选择器后，速度已知且为 v = E/B₁ 的离子进入与其速度垂直的匀强磁场 B₂ 中，做半圆形运动。半径为：
r = m v / (q B₂)
Rearranging for the mass-to-charge ratio:
整理得到质荷比：
m/q = r B₂ / v = r B₂ B₁ / E
By measuring the radius r of the ion’s path (usually by the position where it hits a detector), both m/q and, if the charge is known, the mass m can be determined. This technique is widely used in chemical analysis and isotope separation. The derivation combines the velocity selector and circular motion formulas, showing how fundamental physics enables precise measurements.
通过测量离子轨道的半径 r（通常根据它打在探测器上的位置），既可以确定 m/q，若已知电荷，还可以求出质量 m。这项技术广泛用于化学分析与同位素分离。推导综合了速度选择器和圆周运动公式，展示了基础物理如何实现精密测量。
8. Force Between Two Parallel Current-Carrying Wires | 两根平行载流导线之间的力
Two long, straight, parallel wires carrying currents exert magnetic forces on each other. Consider wire 1 carrying current I₁, which creates a magnetic field at the location of wire 2. The flux density at a perpendicular distance d from a long straight wire is given by:
两根长直平行载流导线会相互施加磁场力。设导线 1 通有电流 I₁，它在导线 2 处产生磁场。距离长直导线垂直距离 d 处的磁通量密度为：
B₁ = μ₀ I₁ / (2π d)
where μ₀ is the permeability of free space. Wire 2, carrying current I₂, lies in this field over length L, so it experiences a force:
其中 μ₀ 为真空磁导率。载有电流 I₂ 的导线 2 长度为 L，处于该磁场中，受到的力为：
F = B₁ I₂ L = (μ₀ I₁ I₂ L) / (2π d)
The force per unit length on each wire is therefore:
因此，每单位长度导线所受的力为：
F/L = μ₀ I₁ I₂ / (2π d)
If the currents are in the same direction, the wires attract; if opposite, they repel. This result is the basis for the definition of the ampere: one ampere is the constant current that, when flowing in two infinitely long parallel conductors of negligible cross-section placed 1 metre apart in a vacuum, produces a force of 2 × 10⁻⁷ newtons per metre of length.
若电流同向，导线相互吸引；若反向，则相互排斥。这一结果是安培定义的基础：1 安培是指两条截面可忽略、相距 1 米的无限长平行直导线在真空中通以恒定电流，每米长度产生 2 × 10⁻⁷ 牛顿的力时的电流值。
9. Torque on a Current-Carrying Coil in a Magnetic Field | 磁场中载流线圈所受的力矩
A rectangular coil of N turns, each of area A, carrying a current I and placed in a uniform magnetic field B, experiences a torque. Consider a coil free to rotate about a central axis. The forces on opposite sides parallel to the axis produce a couple. For a coil whose plane makes an angle θ with the magnetic field, the perpendicular distance between the forces is w sinθ (where w is the width of the coil perpendicular to the axis).
一个匝数为 N、每匝面积为 A 的矩形线圈，通有电流 I 并置于匀强磁场 B 中，会受到力矩作用。考虑一个可绕中心轴自由旋转的线圈，平行于转轴的两条对边所受的力构成力偶。若线圈平面与磁场的夹角为 θ，力之间的垂直距离为 w sinθ（其中 w 为线圈垂直于转轴的宽度）。
The force on one side of length L (parallel to the axis) is F = N I L B (since N turns). The torque about the axis is then force times perpendicular distance from the axis for both sides:
长度为 L（平行于轴）的边所受的力为 F = N I L B（因为 N 匝）。对轴的力矩为两边力乘以其到轴的垂直距离：
τ = F (w sinθ) = N I L B w sinθ = N I A B sinθ
where A = L w is the area of the coil. In vector form, this is τ = N I (A × B). When the coil is perpendicular to the field (θ = 90°), torque is maximum; when parallel (θ = 0°), torque is zero. This principle is used in electric motors and moving-coil galvanometers.
其中 A = L w 为线圈面积。矢量形式为 τ = N I (A × B)。当线圈与磁场垂直 (θ = 90°) 时力矩最大；平行 (θ = 0°) 时力矩为零。此原理应用于电动机和动圈式检流计。
10. Magnetic Flux and Flux Linkage | 磁通量与磁链
Magnetic flux Φ is a measure of the number of magnetic field lines passing through a given area. For a uniform field B and a flat area A, with the normal to the area making an angle θ to the field direction:
磁通量 Φ 是衡量穿过某一面积的磁感线数量的物理量。对于匀强磁场 B 和平面面积 A，若面积法线与磁场方向夹角为 θ：
Φ = B A cosθ
When the field is perpendicular to the area (θ = 0°), flux is maximum Φ = B A. When parallel (θ = 90°), flux is zero. Flux is measured in webers (Wb).
当磁场垂直于面积 (θ = 0°) 时，磁通量最大，为 Φ = B A；当平行 (θ = 90°) 时磁通量为零。磁通量的单位是韦伯 (Wb)。
Flux linkage ψ through a coil of N turns is simply ψ = N Φ. It becomes important in the study of electromagnetic induction, where the induced e.m.f. is proportional to the rate of change of flux linkage (Faraday’s law). Understanding flux is essential before moving on to induction topics.
穿过 N 匝线圈的磁链 ψ 就是 ψ = N Φ。在研究电磁感应时，磁链至关重要，因为感应电动势与磁链的变化率成正比（法拉第定律）。在学习感应内容之前，理解磁通量是基本前提。
In a typical Oxford AQA topic test, you may be asked to derive any of the above formulas, perform calculations, or explain the meaning of flux and flux linkage. Memorising the derivations step by step will give you confidence and deepen your understanding of how magnetic fields interact with moving charges and currents.
在典型的牛津 AQA 专题测试中，你可能需要推导以上任一公式、进行计算，或者解释磁通量与磁链的含义。逐步记住这些推导过程将增强你的信心，并加深你对磁场如何与运动电荷和电流相互作用的理解。
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