📚 Magnetic Flux Density | 磁通量密度
Magnetic flux density, symbol B, is the fundamental vector quantity used to describe a magnetic field. It tells us how strong the field is and in which direction it points at any location. In A-Level physics, understanding B is essential for analysing forces on currents and charges, induction phenomena, and many practical applications such as motors and generators.
磁通量密度，符号 B，是描述磁场的基本矢量。它告诉我们空间中任意一点的磁场有多强以及指向什么方向。在 A-Level 物理中，理解 B 对于分析电流与电荷受力、电磁感应现象以及电动机、发电机等实际应用至关重要。
1. Understanding Magnetic Flux Density | 理解磁通量密度
Magnetic flux density is often simply called the magnetic field, but it is more precisely the measure of how many magnetic field lines pass through a unit area taken perpendicular to the field. The concept merges the idea of field strength with the density of flux lines. A stronger B means field lines are packed more tightly together.
磁通量密度常被简单地称为磁场，但更精确地说，它度量的是在垂直于磁场的方向上穿过单位面积的磁感线数目。这个概念将场强与磁感线密度结合起来。B 越大，表示磁感线越密集。
The direction of B at a point is the direction in which a free north pole would move if placed there. By convention, magnetic field lines emerge from north poles and enter south poles, and the tangent to a field line gives the direction of B.
空间中某点 B 的方向，就是放在该点的一个自由 N 极将会移动的方向。按照惯例，磁感线从 N 极出发进入 S 极，磁感线上某点的切线方向即为 B 的方向。
2. Defining Magnetic Flux Density: B = F/IL sinθ | 定义磁通量密度: B = F/IL sinθ
The formal definition of magnetic flux density comes from the force experienced by a current-carrying conductor in a magnetic field. If a straight wire of length L carrying a current I is placed in a uniform magnetic field, the magnitude of the force F is measured. The magnetic flux density B is then defined as:
磁通量密度的正式定义建立在载流导体在磁场中受力这一基础上。如果将一段长度为 L、通有电流 I 的直导线置于匀强磁场中，测出所受磁力 F 的大小，那么磁通量密度 B 的定义为:
B = F / (I L sinθ)
rent per unit length on a conductor placed perpendicular to the field (θ = 90°, sinθ = 1).
其中 θ 是电流方向与磁场方向之间的夹角。此定义表明，当导体与磁场垂直时（θ = 90°, sinθ = 1），B 就等于单位电流、单位长度导体所受的力。
3. The Tesla: SI Unit of B | 特斯拉: B 的国际单位
The SI unit of magnetic flux density is the tesla (T). From the defining equation, one tesla is equivalent to one newton per ampere per metre:
磁通量密度的国际单位是特斯拉（T）。由定义式可知，1 特斯拉等于 1 牛顿每安培每米:
1 T = 1 N A⁻¹ m⁻¹
common equivalent expression comes from the relationship with magnetic flux (Φ): B = Φ / A⊥, giving 1 T = 1 Wb m⁻² (weber per square metre). Familiar field strengths: Earth' is roughly 50 μT, a fridge magnet about 5 mT, and an MRI scanner can reach several tesla.
另一个常见的等价表述来自磁通量关系：B = Φ / A⊥，因此 1 T = 1 Wb m⁻²（韦伯每平方米）。常见的磁场强度对比：地磁场约为 50 μT，冰箱贴约 5 mT，而 MRI 扫描仪可达数特斯拉。
4. Measuring B with a Current Balance | 用电流天平测量 B
A classic laboratory method for measuring magnetic flux density is the current balance. A short straight wire of known length L is placed between the poles of a magnet, perpendicular to the field. The wire is part of a circuit that allows a variable current I to pass through it. The magnetic force on the wire is measured by placing the magnet assembly on a sensitive digital balance.
测量磁通量密度的经典实验方法是电流天平。将一段已知长度 L 的短直导线置于磁铁两极之间，并与磁场方向垂直。导线连接到一个可调电流的电路中。将磁铁组件放在灵敏的数字天平上，通过天平读数变化来测量导线所受磁力。
current I, the wire experiences a magnetic force F = B I L (since θ = 90°), and by Newton' obtain B = Δmg / (I L). Repeating for different currents and averaging improves accuracy.
未通电流时，天平读数为磁铁的重量。通以电流 I 后，导线受到磁力 F = B I L（因 θ = 90°），根据牛顿第三定律，磁铁受到大小相等、方向相反的力，使天平读数改变。由 F = Δmg 可得 B = Δmg / (I L)。改变电流多次测量取平均，可提高精度。
5. Force on a Current-Carrying Conductor | 磁场对载流导体的作用力
The force on a straight current-carrying conductor in a uniform magnetic field is given by Fleming's left-hand rule for direction and the equation for magnitude:
匀强磁场中直载流导体所受的力，其方向由弗莱明左手定则判定，大小由下式给出:
F = B I L sinθ
magnetic field lines. The force is maximum when the current is perpendicular to the field (θ = 90°) and zero when parallel (θ = 0°).
式中 L 是导体处于磁场中的长度，I 为电流，B 为磁通量密度，θ 是导体与磁感线之间的夹角。当电流垂直于磁场时力最大（θ = 90°），当平行时力为零（θ = 0°）。
Fleming's left-hand rule: hold the thumb, first finger and second finger mutually at right angles. The first finger points in the direction of the field (B), the second finger in the direction of the current (I), and the thumb then indicates the direction of the force (F).
弗莱明左手定则：将拇指、食指和中指互相垂直伸开，食指指向磁场（B）方向，中指指向电流（I）方向，则拇指所指即为力（F）的方向。
6. Force on a Moving Charge: F = Bqv sinθ | 磁场对运动电荷的作用力: F = Bqv sinθ
ow of charged particles. By substituting I = n A v q into F = B I L sinθ (where n is the number of charge carriers per unit volume, A is the cross-sectional area, v is the drift velocity, and q is the charge of each carrier), the total force on N = n A L carriers becomes F = B q v N sinθ. Hence the force on a single charge is:
磁场不仅对载流导线有力的作用，对单个运动电荷也同样有力。这是因为电流本质上是电荷的定向运动。将 I = n A v q（n 为单位体积载流子数，A 为横截面积，v 为漂移速度，q 为每个载流子的电荷量）代入 F = B I L sinθ，可得 N = n A L 个载流子所受总力为 F = B q v N sinθ。因此单个运动电荷所受的磁力为:
F = B q v sinθ
This Lorentz force is always perpendicular to both the velocity of the charge and the magnetic field. For a positive charge, Fleming's left-hand rule can be adapted: point the first finger in the direction of B, the second finger in the direction of v, and the thumb gives the direction of F. For a negative charge (e.g. an electron), the force direction is reversed.
这种洛伦兹力始终垂直于电荷速度与磁场方向。对于正电荷，可沿用弗莱明左手定则：食指指向 B 方向，中指指向 v 方向，则拇指即指向力 F 的方向。对于负电荷（如电子），力的方向则相反。
7. Circular Motion of Charges in a Uniform B Field | 均匀磁场中电荷的圆周运动
motion. Since F = B q v and this force always acts at right angles to the velocity, the particle undergoes uniform circular motion. Equating B q v = m v² / r gives the radius of the path:
当带电粒子垂直于匀强磁场运动时，磁力恰好提供圆周运动所需的向心力。因为 F = B q v 且该力始终与速度垂直，粒子将做匀速圆周运动。由 B q v = m v² / r 可得轨道半径:
r = m v / (B q)
The time for one complete revolution (the period T) is independent of the speed and radius:
绕行一周的时间（周期 T）与速度及半径无关:
T = 2π r / v = 2π m / (B q)
This principle is exploited in particle accelerators like the cyclotron and in mass spectrometers to separate isotopes based on their mass-to-charge ratio. The equation also confirms that a stronger B results in tighter circular paths.
这一原理被应用于回旋加速器、质谱仪等装置中，根据荷质比分离同位素。公式也表明，B 越强，粒子运动的圆周半径越小。
8. Magnetic Flux (Φ) and Flux Density | 磁通量 (Φ) 与磁通量密度
Magnetic flux Φ is a measure of the total magnetic field passing through a given area. For a uniform magnetic field, it is defined as:
磁通量 Φ 度量的是穿过某一给定面积的总磁场量。对于匀强磁场，其定义为:
Φ = B A sinθ
e plane (θ = 90°), Φ = B A. If the field lies parallel to the plane (θ = 0°), Φ = 0. This definition makes it clear why B is called flux density: B = Φ / (A sinθ), the flux per unit area measured perpendicular to the field.
式中 A 为面积，θ 为磁感线与平面之间的夹角。当磁场垂直于平面时（θ = 90°），Φ = B A；当磁场平行于平面时（θ = 0°），Φ = 0。这一定义清楚地表明 B 被称为“磁通量密度”的原因：B = Φ / (A sinθ)，即垂直于磁场方向单位面积上的磁通量。
The SI unit of magnetic flux is the weber (Wb), and 1 Wb = 1 T m².
磁通量的国际单位是韦伯（Wb），1 Wb = 1 T m²。
9. Flux Linkage and Electromagnetic Induction | 磁链与电磁感应
For a coil of N turns each of area A placed in a magnetic field, the total flux linkage is NΦ. According to Faraday's law of electromagnetic induction, the magnitude of the induced emf is equal to the rate of change of flux linkage:
对于面积为 A、匝数为 N 且置于磁场中的线圈，总磁链为 NΦ。根据法拉第电磁感应定律，感应电动势的大小等于磁链的变化率:
ε = − d(NΦ) / dt
by changing B, changing A, or changing the orientation θ. In many A-Level problems, the variation of B with time is a key focus, linking back directly to the concept of magnetic flux density.
由于 Φ = B A sinθ，改变 B、改变面积 A 或改变取向 θ 都可以产生感应电动势。在许多 A-Level 问题中，B 随时间的变化是核心考点之一，直接关联着磁通量密度这个概念。
Lenz's law gives the direction of the induced emf: the induced current flows in a direction that opposes the change in flux producing it. This opposition is a manifestation of energy conservation.
楞次定律给出了感应电动势的方向：感应电流的方向总是使其阻碍引起感应的磁通变化。这种阻碍作用是能量守恒的体现。
10. Hall Effect: A Tool to Measure B | 霍尔效应: 测量 B 的工具
ing slab of thickness t placed perpendicular to a magnetic field B, charge carriers experience a magnetic force q v B deflecting them to one side. This creates a transverse electric field, and at equilibrium the electric force balances the magnetic force: q E = q v B. The resulting Hall voltage VH across the slab is:
霍尔效应提供了一种利用半导体探头直接测量磁通量密度的方法。当电流 I 流过厚度为 t 的薄片，并将薄片垂直于磁场 B 放置时，载流子受到磁力 q v B 的作用向一侧偏转，从而建立起横向电场。平衡时电场力与磁力相等: q E = q v B。此时在薄片两侧产生的霍尔电压 VH 为:
VH = (B I) / (n q t)
charge carrier density and q is the charge of the carriers. Rearranging gives B = (n q t VH) / I. For a given Hall probe, n, q, and t are constants, so B is directly proportional to VH for a fixed current. This linear relationship makes the Hall probe a convenient and accurate instrument for measuring magnetic flux density in laboratories and industrial settings.
式中 n 是载流子密度，q 是载流子电荷量。变形可得 B = (n q t VH) / I。对于给定的霍尔探头，n、q 和 t 为常数，因此在电流固定时 B 与 VH 成正比。这一线性关系使霍尔探头成为实验室和工业环境中方便、精确的磁通量密度测量工具。
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