Producing and representing magnetic fields | 产生和表示磁场

📚 Producing and representing magnetic fields | 产生和表示磁场

Magnetic fields are fundamental to many areas of physics, from the behaviour of compass needles to the operation of electric motors and generators. In A-Level Physics, it is essential to understand how magnetic fields are produced, how they can be represented by field lines and magnetic flux, and how their strength — the magnetic flux density — is defined, calculated and measured. This article covers both permanent magnets and electromagnets, the rules for determining field directions, the concept of magnetic flux and flux linkage, and practical techniques such as using a Hall probe to measure magnetic flux density.

磁场是物理学许多领域的基础，从指南针的指向到电动机和发电机的运行都离不开它。在 A-Level 物理中，理解磁场如何产生、如何用磁感线和磁通量来表示，以及磁场强度——磁通密度——的定义、计算和测量至关重要。本文涵盖了永磁体和电磁体、判断磁场方向的定则、磁通量与磁链的概念，以及使用霍尔探头测量磁通密度的实际操作技术。

1. What is a magnetic field? | 什么是磁场？

A magnetic field is a region of space where a moving charged particle or a magnetic material experiences a force. It is a vector field, meaning it has both magnitude and direction at every point. The SI unit of magnetic field strength is the tesla (T).

磁场是一个空间区域，其中运动的带电粒子或磁性材料会受到力的作用。它是一个矢量场，意味着空间每一点既有大小也有方向。磁场强度的国际单位是特斯拉（T）。

Magnetic fields are produced by moving charges (electric currents) and by elementary particles with intrinsic magnetic moments, such as electrons in atoms. In permanent magnets, the combined effect of electron spin and orbital motion gives rise to a net magnetic field. In wires and coils, a magnetic field is generated whenever an electric current flows.

磁场由运动电荷（电流）以及具有内禀磁矩的基本粒子（如原子中的电子）产生。在永磁体中，电子自旋和轨道运动的综合效应产生了净磁场。在导线和线圈中，只要有电流流过，就会产生磁场。

2. Magnetic fields of permanent magnets | 永磁体的磁场

A permanent magnet has two poles: north (N) and south (S). By convention, magnetic field lines emerge from the north pole, curve through the surrounding space, and enter the south pole, continuing inside the magnet from south to north to form closed loops.

永磁体有两个磁极：北极（N）和南极（S）。按惯例，磁感线从北极发出，穿过周围空间弯曲后进入南极，并在磁体内部由南到北继续形成闭合曲线。

When two magnets are brought close together, like poles repel and unlike poles attract. The field lines between two unlike poles run directly from N to S, producing a more concentrated field; between two like poles, the lines bulge away from each other, and there is a neutral point where the net field is zero.

当两块磁铁靠近时，同性磁极相互排斥，异性磁极相互吸引。异极之间的磁感线直接从 N 指向 S，使磁场更集中；同极之间的磁感线相互隆起，并存在一个合场强为零的中性点。

The shape of a magnetic field can be visualised by sprinkling iron filings around a magnet. The filings align themselves along the field lines, providing a simple way to trace the field pattern.

通过在磁铁周围撒上铁屑可以直观地看到磁场的形状。铁屑会沿着磁感线排列，这提供了描绘磁场图案的简单方法。

3. Representing magnetic fields: field lines | 表示磁场：磁感线

Magnetic field lines (also called lines of magnetic flux) are a visual tool used to represent magnetic fields. The direction of a field line at any point gives the direction of the magnetic field, which is the direction of the force on a hypothetical north pole placed there. The spacing of the lines indicates the strength of the field: closely spaced lines mean a strong field; widely spaced lines mean a weak field.

磁感线（也称为磁通线）是用来表示磁场的可视化工具。线上任意点的切线方向代表该点的磁场方向，即放在该处的假想北极所受力的方向。线的疏密表示场的强弱：线越密场越强，线越疏场越弱。

Key properties of magnetic field lines: they always form closed loops (no magnetic monopoles exist); they never cross one another; they are continuous; and they tend to shorten and repel each other laterally if represented by a stretched elastic model.

磁感线的主要性质包括：始终形成闭合曲线（不存在磁单极子）；永不相交；连续不断；若用弹性模型来类比，磁感线会倾向于缩短，且彼此间存在着侧向排斥的趋势。

4. Magnetic flux density B | 磁通密度 B

Magnetic flux density, symbol B, measures the strength of a magnetic field. It is defined by the force experienced by a current-carrying conductor placed perpendicular to the field. For a straight wire of length L carrying a current I at right angles to a uniform magnetic field, the force F satisfies:

磁通密度，符号 B，衡量磁场的强弱。其定义基于放置于磁场中且与磁场垂直的载流导体所受的力。对于一段长度为 L、通有电流 I 且与均匀磁场方向垂直的直导线，所受的力 F 满足：

B = F / (I L)

e θ with the field, the component of B perpendicular to the current is B sin θ, and F = B I L sin θ.

B 的单位是特斯拉（T），1 T = 1 N A⁻¹ m⁻¹。该定义仅在导线与 B 垂直时成立。如果导线与磁场成角度 θ，则垂直于电流方向的 B 的分量为 B sin θ，此时 F = B I L sin θ。

Magnetic flux density can also be defined in terms of the number of flux lines per unit area taken perpendicular to the field. This is closely related to magnetic flux Φ.

磁通密度也可以定义为单位垂直面积上通过的磁通线数目。这与磁通量 Φ 密切相关。

5. Magnetic flux Φ and flux linkage | 磁通量 Φ 与磁链

Magnetic flux Φ through a surface is the product of the magnetic flux density B and the area A perpendicular to the field. If the field is uniform and perpendicular to the area, we have:

穿过某一面积 S 的磁通量 Φ 等于磁通密度 B 与垂直于磁场方向的面积 A 的乘积。若磁场均匀且与面积垂直，则有：

Φ = B A

s, each turn links the same flux, so the flux linkage is NΦ. This is measured in weber-turns (Wb-turns).

磁通量的国际单位是韦伯（Wb）。1 Wb = 1 T m²。对于一个 N 匝的线圈，每一匝连接相同的磁通量，因此磁链为 NΦ，其单位是韦伯·匝（Wb-turns）。

he area, Φ = B A cos θ. This concept is essential for understanding electromagnetic induction.

当磁场不与面积垂直时，只有 B 的垂直分量起作用。若均匀磁场与面积法线成 θ 角，则 Φ = B A cos θ。这一概念对于理解电磁感应至关重要。

6. Magnetic field around a straight current-carrying wire | 直载流导线周围的磁场

A current in a straight wire produces a circular magnetic field around the wire. The field lines form concentric circles centred on the wire, lying in planes perpendicular to it. The magnetic flux density B at a distance r from a long straight wire carrying current I is given by:

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直导线中的电流会在其周围产生环形的磁场。磁感线是以导线为圆心的同心圆，位于与导线垂直的平面内。与无限长直导线距离为 r 处的磁通密度 B 由下式给出：

B = μ₀ I / (2π r)

f the circular field can be found using the right-hand grip rule (see next section). The field strength decreases as 1/r, so a larger current or a smaller distance produces a stronger field.

其中 μ₀ 为真空磁导率（4π × 10⁻⁷ H m⁻¹）。环形磁场的方向可用右手螺旋定则判断（见下节）。磁场强度随 1/r 衰减，因此电流越大或距离越近，磁场越强。

7. Right-hand grip rule and direction of magnetic fields | 右手螺旋定则与磁场方向

To determine the direction of the magnetic field generated by a current, CIE A-Level uses the right-hand grip rule: if you grasp a straight current-carrying wire with your right hand such that the thumb points in the direction of conventional current (positive to negative), your curled fingers show the direction of the magnetic field lines encircling the wire.

为了判断电流产生的磁场方向，CIE A-Level 使用右手螺旋定则：用右手握住直导线，使拇指指向常规电流的方向（正到负），则弯曲的四指所指的方向为环绕导线的磁感线方向。

For a circular loop or a solenoid, the right-hand grip rule is adapted: if the fingers of the right hand curl in the direction of the current around the coil, the thumb points towards the north pole of the resulting magnetic field (the direction of the field lines inside the coil). This version is sometimes called the right-hand rule for coils.

对于圆形线圈或螺线管，可调整右手定则：让右手四指沿线圈中电流的方向弯曲，则拇指指向所产生的磁场的北极（即线圈内部磁感线的方向）。这一版本有时被称为线圈的右手定则。

These rules are crucial for predicting the polarity of electromagnets and for understanding the forces between current-carrying conductors.

这些定则对于判断电磁铁的极性和理解载流导体之间的相互作用力至关重要。

8. Magnetic field of a flat circular coil and a solenoid | 圆形线圈与螺线管的磁场

A single flat circular loop carrying a current produces a magnetic field pattern similar to that of a short bar magnet. At the centre of the loop, the field is perpendicular to the plane of the loop and nearly uniform over a small region. The field lines emerge from one face (north) and enter the opposite face (south), then curve around outside.

载流单匝圆形线圈产生的磁场图样类似于短条形磁铁。在线圈中心，磁场垂直于线圈平面，且在小区域内几乎是均匀的。磁感线从一面（北极）发出，进入另一面（南极），然后从外部弯曲连接。

A solenoid is a long coil consisting of many turns of wire. When current flows, it produces a strong, nearly uniform magnetic field inside the solenoid and a weak, diverging field outside. The field pattern resembles that of a bar magnet. The magnetic flux density inside a long air-core solenoid is given by:

螺线管是由许多匝导线构成的长线圈。当通电时，其内部产生较强的、几乎均匀的磁场，而外部磁场较弱且发散。其场图类似于条形磁铁。长直空心螺线管内部的磁通密度为：

B = μ₀ n I

where n is the number of turns per unit length (N / L). The direction of the field inside the solenoid is given by the right-hand rule for coils: if the current enters at one end and curls in the direction of the fingers, that end becomes the south pole, while the opposite end becomes the north pole.

其中 n 为单位长度的匝数（N / L）。内部磁场的方向由线圈右手定则给出：若电流从一端进入并按四指方向旋绕，则该端为南极，另一端为北极。

9. Uniform magnetic fields and how to produce them | 均匀磁场及其产生方法

A uniform magnetic field is one in which the magnetic flux density B has the same magnitude and direction at all points. In such a field, field lines are parallel and equally spaced. Uniform fields are important in many experiments and devices, such as velocity selectors and Hall effect measurements.

均匀磁场是指场中每一点的磁通密度 B 大小和方向都相同的磁场。在这种场中，磁感线平行且等间距。均匀磁场在速度选择器、霍尔效应测量等许多实验和器件中十分重要。

Uniform fields can be produced in several ways: the interior of a long solenoid provides a nearly uniform field; the region between two flat, opposite magnetic poles (for example, between the N and S poles of two permanent magnets with plane faces facing each other) produces a reasonably uniform field; Helmholtz coils (a pair of identical coaxial coils separated by a distance equal to their radius) give a highly uniform field near the midpoint.

产生均匀磁场的方法有几种：长螺线管内部可提供近乎均匀的场；两个扁平异名磁极之间的区域（例如两块平面相对的永磁体 N 极与 S 极之间）可产生相当均匀的场；亥姆霍兹线圈（一对间距等于其半径的同轴相同线圈）在其中点附近产生高度均匀的场。

In CIE practical work, the uniform field between two slab magnets is often used to investigate Fleming’s left-hand rule or to calibrate a Hall probe.

在 CIE 实验操作中，两块板状磁铁之间的均匀场常被用来探究弗莱明左手定则或标定霍尔探头。

10. Measuring magnetic flux density: the Hall probe | 测量磁通密度：霍尔探头

A Hall probe is a device that uses the Hall effect to measure magnetic flux density B. When a slab of semiconductor carries a current I and is placed in a magnetic field perpendicular to the current, charge carriers are deflected, creating a transverse voltage V_H — the Hall voltage — across the slab. For a given material, V_H is proportional to B and to I, and inversely proportional to the thickness t and the charge carrier density n:

霍尔探头是一种利用霍尔效应测量磁通密度 B 的装置。当一块半导体薄片通有电流 I 并置于与电流垂直的磁场中时，电荷载流子会发生偏转，从而在薄片两侧产生横向电压 V_H——即霍尔电压。对于给定的材料，V_H 正比于 B 和 I，反比于薄片厚度 t 和载流子浓度 n：

V_H = B I / (n q t)

where q is the charge of the carriers (e for electrons). In practice, the Hall probe is calibrated so that a known B gives a certain output voltage. To measure an unknown magnetic field, the probe is placed in the field with its plane perpendicular to the field lines, and the voltage reading is converted to a tesla value using the calibration.

其中 q 为载流子的电荷量（对于电子为 e）。实际应用中，霍尔探头已标定好，使已知的 B 值对应一定的输出电压。要测量未知磁场，只需将探头平面垂直于磁感线放置，读取电压并通过标定关系转换成特斯拉值。

Hall probes are versatile: they can measure both the strength and, by rotating the probe, the direction of a magnetic field. They are used to map field patterns around magnets, solenoids, and current-carrying conductors, and are standard equipment in advanced physics laboratories.

霍尔探头功能多样：它们不仅可以测量磁场强度，还可以通过旋转探头探测磁场方向。它们被用于描绘磁体、螺线管和载流导体周围的磁场图样，是高等物理实验室中的标准设备。

Published by TutorHao | Physics Revision Series | aleveler.com

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