📚 Currents Crossing Fields | 电流穿越场域
When electric currents and moving charges cross electric and magnetic fields, they experience forces that are fundamental to both pure physics and everyday technology. From the simple motor effect in a current-carrying wire to the precision mass separation inside a spectrometer, the interplay between currents and fields shapes the modern world. This article explores these interactions, with a strong focus on the principles required for the CIE A-Level Physics syllabus.
当电流和运动电荷穿越电场与磁场时，它们会受到各种力的作用，这些力不仅是基础物理的核心内容，也是许多现代技术的根基。从载流导线简单的电动机效应到质谱仪中精密的粒子分离，电流与场域之间的相互作用塑造了我们周围的世界。本文将深入探讨这些交互现象，紧扣 CIE A-Level 物理课程的要求。
1. Force on a Current-Carrying Conductor | 载流导体在磁场中所受的力
A straight conductor of length L carrying a current I and placed in a uniform magnetic field of flux density B experiences a magnetic force. The magnitude of this force is given by F = B I L sin θ 90°), the force is at its maximum: F = B I L. When the wire is parallel to the field, no force acts on it.
一根长度为 L 的直导体通有电流 I，置于磁通密度为 B 的匀强磁场中时，会受到磁力作用。力的大小由公式 F = B I L sin θ 决定，其中 θ 是常规电流方向与磁场线方向之间的夹角。当电流与磁场垂直（θ = 90°）时，力最大：F = B I L；当导线与磁场平行时，不受力。
| Quantity | 物理量 | Symbol | 符号 | Unit | 单位 |
|---|---|---|
| Force | F | N |
| Magnetic flux density | B | T or Wb m⁻² |
| Current | I | A |
| Length of conductor in field | L | m |
2. Fleming's Left-Hand Rule | 弗莱明左手定则
The direction of the force on a current-carrying conductor is given by Fleming's left-hand rule. Hold the thumb, forefinger and second finger of the left hand mutually at right angles. The forefinger points in the direction of the magnetic field (B), the second finger points in the direction of conventional current (I), and the thumb then points in the direction of the force (F) – also the direction of motion if the wire is free to move. This rule is crucial for solving problems involving the motor effect.
载流导体所受力的方向可由弗莱明左手定则来判断。将左手的拇指、食指和中指互相垂直伸直：食指指向磁场（B）的方向，中指指向常规电流（I）的方向，则拇指所指即为安培力（F）的方向，若导线可以自由移动，也是其运动方向。这一规则在解决电动机效应问题时至关重要。
3. Torque on a Rectangular Coil | 矩形线圈所受的力矩
When a current-carrying rectangular coil is placed in a uniform magnetic field, the forces on opposite sides produce a couple that tends to rotate the coil. The torque τ acting on a coil of n turns, each of area A, is τ = n B I A sin φ. Here φ is the angle between the magnetic field and the normal to the plane of the coil. This torque is the foundation of the electric motor: as the coil rotates, a split-ring commutator reverses the current every half-turn, maintaining rotation in the same direction.
当通电矩形线圈置于匀强磁场中时，对边上方向相反的力形成一个力偶，使线圈发生转动。作用在 n 匝、每匝面积为 A 的线圈上的力矩为 τ = n B I A sin φ，其中 φ 是磁场方向与线圈平面法线之间的夹角。这一力矩正是电动机的基础：线圈转动时，换向器每半圈反转电流方向，从而保持沿同一方向的持续旋转。
4. Lorentz Force on a Moving Charge | 运动电荷所受的洛伦兹力
A single charged particle moving through a magnetic field also experiences a force, often called the Lorentz force. For a particle of charge q moving with velocity v at angle θ to a uniform field B, the magnitude is F = B q v sin θ. The direction for a positive charge is found using Fleming's left-hand rule, treating the velocity direction as the "current" direction. For a negative charge (such as an electron), the force is in the opposite direction, or you can use the right hand for convenience.
单个带电粒子穿过磁场时，也会受到力，这被称为洛伦兹力。对于电荷量为 q、速度为 v 且与匀强磁场 B 成 θ 角的粒子，力的大小为 F = B q v sin θ。正电荷的受力方向可用弗莱明左手定则判断，将速度方向视为“电流”方向即可。对于负电荷（如电子），受力方向与此相反，也可方便地使用右手。
5. Circular Motion in a Uniform Magnetic Field | 匀强磁场中的匀速圆周运动
When a charged particle enters a uniform magnetic field at right angles to the field lines, the Lorentz force is always perpendicular to the velocity. This provides the centripetal force required for circular motion. Equating the two gives B q v = m v² / r, from which the radius of the path is r = m v / (B q). The time taken for one complete revolution (the period) is T = 2π r / v = 2π m / (B q), which is remarkably independent of the particle's speed. This principle is exploited in particle accelerators such as the cyclotron.
当带电粒子垂直射入匀强磁场时，洛伦兹力始终与速度方向垂直，恰好提供圆周运动所需的向心力。由 B q v = m v² / r 可得轨迹半径 r = m v / (B q)。粒子运动一周所需的时间（周期）为 T = 2π m / (B q)，它与粒子的速率无关。这一特性被应用在回旋加速器等粒子加速器中。
r = m v / (B q) | T = 2π m / (B q)
6. Crossed Fields: The Velocity Selector | 正交场：速度选择器
A velocity selector uses perpendicular electric and magnetic fields to filter charged particles moving at a specific speed. Inside the device, positively charged particles experience an electric force qE in one direction and a magnetic force B q v in the opposite direction. Only particles for which these two forces exactly balance will travel in a straight line and exit through a narrow slit. The condition for balance is q E = B q v, which simplifies to v = E / B. Particles with a different speed are deflected and do not pass through.
速度选择器利用相互垂直的电场和磁场，筛选出具有特定速度的带电粒子。在仪器内部，正电荷粒子受到一个方向的电场力 qE，以及相反方向的磁场力 B q v。只有当这两个力大小恰好相等时，粒子才能沿直线运动并通过狭缝。平衡条件为 q E = B q v，即 v = E / B。速度不同的粒子会被偏转，无法通过。
v = E / B
7. The Mass Spectrometer | 质谱仪
A mass spectrometer combines a velocity selector with a region of uniform magnetic field to measure the mass-to-charge ratio of ions. First, ions are accelerated and passed through the velocity selector, so they all emerge with the same speed v = E / Bs. Then they enter a second magnetic field Bm where they follow circular paths. Measuring the radius r of the semicircle allows the calculation of m/q: from r = m v / (Bm q), we obtain m/q = Bm r / v. Mass spectrometers are widely used to identify isotopes, analyse chemical samples, and determine atomic masses.
质谱仪将速度选择器与另一匀强磁场区域结合在一起，用来测量离子的荷质比。离子先被加速并通过速度选择器，使它们以相同的速率 v = E / Bs 射出。随后进入第二个磁场 Bm，在其中做圆弧运动。测量半圆轨迹的半径 r 后，可由 r = m v / (Bm q) 导出 m/q = Bm r / v。质谱仪广泛用于鉴定同位素、分析化学样品以及测定原子质量。
8. The Hall Effect | 霍尔效应
The Hall effect occurs when a current-carrying conductor or semiconductor is placed in a magnetic field perpendicular to the current. The charge carriers (electrons or holes) are deflected by the Lorentz force, accumulating on one side and creating a transverse electric field. At equilibrium, the electric force balances the magnetic force, giving a steady Hall voltage VH = B vd d. Using the current formula I = n A vd q and thickness t for area A = t d, the Hall voltage becomes VH = B I / (n q t), where n is the number density of charge carriers and q the charge on each. The sign of VH reveals whether the majority carriers are positive or negative.
当通电导体或半导体置于与电流方向垂直的磁场中时，会发生霍尔效应。载流子（电子或空穴）受洛伦兹力偏转，聚集在一侧，从而建立起横向电场。达到平衡时，电场力与磁力等大反向，形成稳定的霍尔电压 VH = B vd d。利用电流公式 I = n A vd q 并将横截面积表示为 A = t d，可得 VH = B I / (n q t)，其中 n 为载流子数密度，q 为每个载流子的电荷量。霍尔电压的符号可以揭示多数载流子带正电还是负电。
9. Hall Probes and Electromagnetic Flowmeter | 霍尔探头与电磁流量计
Hall probes exploit the Hall effect to measure magnetic flux density. A small semiconductor slice with known n and t is placed in the field, and the measured Hall voltage gives B = VH n q t / I. These probes are compact, robust and widely used in laboratories and industry. A related application is the electromagnetic flowmeter, which measures the flow speed of a conducting liquid (like water or blood). When the liquid flows through a pipe of diameter D placed in a perpendicular magnetic field, a voltage V = B D v is induced, allowing the volume flow rate to be determined without moving parts.
霍尔探头利用霍尔效应测量磁通密度。将已知 n 和 t 的小块半导体置于待测磁场中，测出霍尔电压即可求出 B = VH n q t / I。这种探头小巧、坚固，在实验室和工业中被广泛应用。另一个相关应用是电磁流量计，用来测量导电流体（如水或血液）的流速。当液体流经处于垂直磁场中、直径为 D 的管道时，会感应出电压 V = B D v，从而在不使用运动部件的情况下测定体积流量。
10. Comparing Electric and Magnetic Forces | 电场力与磁场力的比较
| Feature | 特征 | Electric Force | 电场力 | Magnetic Force | 磁场力 |
|---|---|---|
| Formula | F = q E | F = B q v sin θ (on moving charge) |
| Condition for force | Charge must be in an electric field | Charge must be moving in a magnetic field |
| Direction | Parallel or antiparallel to E | Perpendicular to both v and B (left-hand rule) |
| Work done | Can change kinetic energy | Does no work; only changes direction |
| Speed dependence | Independent of speed | Proportional to speed |
11. Worked Examples | 典型计算示例
Example 1: A wire of length 0.50 m rests on a horizontal table and carries a current of 4.0 A to the east. The uniform magnetic field is 0.30 T directed vertically downwards. Calculate the magnetic force on the wire.
The current is east, field is down – the angle between them is 90°. Using F = B I L sin 90° = 0.30 × 4.0 × 0.50 = 0.60 N. By Fleming's left-hand rule, forefinger down, second finger east, thumb points south. So the force is 0.60 N towards the south.
示例1：一根长 0.50 m 的导线平放在水平桌面上，通有 4.0 A 电流方向向东。匀强磁场为 0.30 T，方向竖直向下。计算导线受到的磁力。
电流向东，磁场向下，夹角为 90°。F = B I L sin 90° = 0.30 × 4.0 × 0.50 = 0.60 N。由左手定则，食指向下，中指指向东，拇指指向南。因此力的大小为 0.60 N，方向向南。
Example 2: 2.0 × 10⁶ m s⁻¹ perpendicularly into a 0.20 T magnetic field. Find the radius of its path.
Using r = m v / (B q) = (1.67 × 10⁻²⁷ × 2.0 × 10⁶) / (0.20 × 1.60 × 10⁻¹⁹) ≈ 0.104 m. The period of revolution is T = 2π m / (B q) ≈ 3.28 × 10⁻⁷ s, independent of speed.
示例2：一个质子（质量 1.67 × 10⁻²⁷ kg，电荷量 1.60 × 10⁻¹⁹ C）以 2.0 × 10⁶ m s⁻¹ 的速度垂直射入 0.20 T 的磁场。求其运动轨迹的半径。
由 r = m v / (B q) = (1.67 × 10⁻²⁷ × 2.0 × 10⁶) / (0.20 × 1.60 × 10⁻¹⁹)
Published by TutorHao | A-Level Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图