📚 Magnetic Fields: Key Concepts for IB and CCEA Physics | 磁场：IB与CCEA物理考点精讲
Magnetic fields are fundamental to understanding how moving charges and currents interact. In both the IB and CCEA Physics syllabuses, the topic of magnetic fields includes field patterns, magnetic force on moving charges and current-carrying conductors, and applications such as particle accelerators and electric motors. This article covers the essential key points to help you master the concepts and solve typical exam problems.
磁场是理解运动电荷与电流如何相互作用的基础。在IB和CCEA物理课程中，磁场主题涵盖磁感线分布、运动电荷与载流导体所受的磁力，以及粒子加速器、电动机等应用。本文梳理核心考点，助你掌握概念并解决典型考题。
1. Magnetic Fields and Field Lines | 磁场与磁感线
A magnetic field is a region in which a magnetic force is experienced by a moving charge, a current-carrying wire, or a magnetic material. It is a vector field, denoted by B. Magnetic field lines (lines of flux) represent the direction and strength of the field: they point from north to south outside a magnet, are continuous, and never cross. The closer the lines, the stronger the field.
磁场是运动电荷、载流导线或磁性材料会受到磁力的区域。它是一个矢量场，用B表示。磁感线（磁通线）表示磁场的方向和强弱：在磁体外从北极指向南极，连续且永不相交。磁感线越密集，磁场越强。
2. Magnetic Flux Density B and the Tesla | 磁通量密度 B 与特斯拉
The magnetic flux density B is defined by the force on a current element: for a straight conductor of length L carrying current I perpendicular to the field, the magnetic force F is F = BIL. Thus, B = F/(IL). The SI unit is the tesla (T), where 1 T = 1 N A⁻¹ m⁻¹. It is also equivalent to Wb m⁻² because magnetic flux Φ = BA.
磁通量密度B由电流元所受的力定义：对于长度为L、与磁场垂直的通电直导线，磁力F = BIL，因此B = F/(IL)。国际单位是特斯拉(T)，1 T = 1 N A⁻¹ m⁻¹。因磁通量Φ = BA，故B的单位也等价于Wb m⁻²。
3. Magnetic Fields Produced by Currents | 电流产生的磁场
A straight wire: The magnetic field lines form concentric circles around the wire. The direction is given by the right-hand grip rule: point your thumb in the direction of conventional current, and your fingers curl in the direction of the field.
直导线：磁感线是以导线为圆心的同心圆。方向由右手螺旋定则判断：大拇指指向电流方向，弯曲的四指指向磁场方向。
The magnitude at a distance r from a long straight wire is B = μ₀I/(2πr), where μ₀ = 4π×10⁻⁷ T m A⁻¹.
距长直导线r处的磁通量密度大小为 B = μ₀I/(2πr)，其中μ₀ = 4π×10⁻⁷ T m A⁻¹。
B = μ₀ I / (2π r)
A flat circular coil: At its centre, the field is perpendicular to the plane of the coil and its magnitude is B = μ₀NI/(2R), where N is the number of turns and R is the radius.
平面圆形线圈：圆心处磁场垂直于线圈平面，大小 B = μ₀NI/(2R)，N为匝数，R为半径。
B = μ₀ N I / (2 R)
A solenoid: Inside a long solenoid, the field is uniform and parallel to the axis, with B = μ₀nI, where n = N/L is the number of turns per unit length. Outside it is very weak.
螺线管：长直螺线管内部磁场均匀且平行于轴线，B = μ₀nI，其中n = N/L为单位长度匝数。外部磁场非常弱。
B = μ₀ n I
4. Magnetic Force on a Moving Charge (Lorentz Force) | 运动电荷在磁场中的力（洛伦兹力）
A charge q moving with velocity v in a magnetic field B experiences a force given by F = qvB sinθ, where θ is the angle between v and B. In vector form: F = q(v × B).
电荷q以速度v在磁场B中运动时，所受磁力为F = qvB sinθ，θ为v与B的夹角。矢量形式：F = q(v × B)。
The direction is perpendicular to both v and B, determined by Fleming’s left-hand rule for a positive charge: thumb – force (for + charge), first finger – field (N to S), second finger – velocity (conventional current direction). For a negative charge, reverse the direction of force.
方向垂直于v和B，用弗莱明左手定则判断——对正电荷：拇指—力（正电荷），食指—磁场（N到S），中指—速度（电流方向）。若是负电荷，力的方向反向。
F = q v B sinθ
If the velocity is parallel (θ = 0° or 180°), the force is zero; if perpendicular (θ = 90°), the force is maximum: F = qvB.
若速度平行于磁场（θ = 0°或180°），力为零；垂直时（θ = 90°）力最大：F = qvB。
5. Magnetic Force on a Current-Carrying Conductor | 通电导体在磁场中的力（安培力）
A straight wire of length L carrying current I in a magnetic field B experiences a force F = BIL sinθ, where θ is the angle between the wire and the field. Direction is given by Fleming’s left-hand rule, with the second finger representing conventional current.
长度为L、通有电流I的直导线在磁场B中所受安培力F = BIL sinθ，θ为导线与磁场的夹角。方向用弗莱明左手定则，中指代表电流方向。
F = B I L sinθ
When the wire is perpendicular, F = BIL. This force is the basis of electric motor torque.
当导线垂直时，F = BIL。此力是电动机转矩的基础。
6. Force Between Two Parallel Current-Carrying Wires | 两平行载流导线间的力
Two long, parallel wires carrying currents I₁ and I₂ separated by distance r attract each other if the currents are in the same direction, and repel if opposite. The force per unit length on each wire is F/L = μ₀ I₁ I₂ / (2πr). This is used to define the ampere.
两根长直平行导线相距r，通有电流I₁和I₂。若电流同向则相互吸引，反向则排斥。每根导线单位长度受力为F/L = μ₀ I₁ I₂ / (2πr)。这是安培定义的基础。
F / L = μ₀ I₁ I₂ / (2π r)
7. Torque on a Rectangular Current-Carrying Coil | 矩形载流线圈所受的力矩
A rectangular coil of N turns, area A, carrying current I, placed in a uniform magnetic field B experiences a torque τ = NIAB sinφ, where φ is the angle between the normal to the coil and the field direction. The torque rotates the coil to align its plane perpendicular to the field (normal parallel). This principle is used in moving-coil galvanometers and electric motors.
一个N匝、面积A的矩形线圈通有电流I，置于匀强磁场B中，所受力矩τ = NIAB sinφ，其中φ为线圈法线与磁场的夹角。力矩使线圈转动至平面垂直于磁场（法线平行）。这一原理用于动圈式电流计和电动机。
τ = N I A B sinφ
8. Circular Motion of a Charged Particle in a Magnetic Field | 带电粒子在磁场中的圆周运动
When a charged particle moves perpendicular to a uniform magnetic field, the magnetic force provides the centripetal force: qvB = mv²/r. Hence the radius of the circular path is r = mv/(qB).
当带电粒子垂直于匀强磁场运动时，磁力提供向心力：qvB = mv²/r。因此圆周轨道半径r = mv/(qB)。
q v B = m v² / r → r = m v / (q B)
The period T (time for one revolution) is independent of speed: T = 2πm/(qB), and the frequency (cyclotron frequency) f = qB/(2πm).
周期T（回转一周的时间）与速度无关：T = 2πm/(qB)，频率（回旋频率）f = qB/(2πm)。
T = 2π m / (q B)
f = q B / (2π m)
These relations are essential for understanding mass spectrometers and cyclotrons.
这些关系式是理解质谱仪和回旋加速器的关键。
9. Velocity Selector | 速度选择器
A velocity selector uses perpendicular electric and magnetic fields. Only particles with velocity v = E/B pass through undeflected, because the electric force qE balances the magnetic force qvB. For other speeds, the net force deflects them.
速度选择器利用相互垂直的电场和磁场。只有速度v = E/B的粒子不受偏转，因为电场力qE与磁力qvB平衡。其他速度的粒子因净力而偏转。
q E = q v B → v = E / B
10. Mass Spectrometer | 质谱仪
In a Bainbridge mass spectrometer, ions are accelerated through a potential difference V, then pass through a velocity selector, and finally enter a uniform magnetic field where they move in a semicircle of radius r. The mass m can be found from q, B, r, and V.
在班布里奇质谱仪中，离子经电势差V加速后，通过速度选择器，再进入匀强磁场中作半径为r的半圆运动。质量m可由q、B、r和V求得。
After acceleration, ½ mv² = qV. With the selector giving v = E/B, and r = mv/(qB), we can eliminate v to get m = q B² r² / (2V).
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