Currents Crossing Fields | 电流穿越电场与磁场

📚 Currents Crossing Fields | 电流穿越电场与磁场

In CIE A-Level Physics, a current or a moving charge crossing a magnetic field experiences a magnetic force. This idea links straight-wire forces, forces between parallel currents, circular motion of charged particles, the Hall effect, and crossed electric and magnetic fields in a velocity selector.

在 CIE A-Level 物理中，电流或运动电荷穿越磁场时会受到磁力作用。这个知识点连接了直导线受力、平行电流之间的力、带电粒子圆周运动、霍尔效应，以及速度选择器中的正交电场与磁场。

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1. Magnetic Force on a Current-Carrying Conductor | 载流导体在磁场中的受力

A straight conductor carrying conventional current I in a uniform magnetic field of flux density B experiences a force when the current has a component perpendicular to the field. For a wire of length L making an angle θ with B, the magnitude is:

当载有常规电流 I 的直导体处于磁通量密度为 B 的匀强磁场中，且电流有垂直于磁场的分量时，导体会受到力的作用。长度为 L 的导线与 B 成 θ 角时，力的大小为：

F = BIL sin θ

Here F is the magnetic force in newtons (N), B is magnetic flux density in tesla (T), I is current in amperes (A), and L is the length of conductor inside the field in metres (m). The force is maximum when θ = 90° and zero when θ = 0° or 180°.

其中 F 是磁力，单位为牛顿（N）；B 是磁通量密度，单位为特斯拉（T）；I 是电流，单位为安培（A）；L 是导体处于磁场中的长度，单位为米（m）。当 θ = 90° 时力最大；当 θ = 0° 或 180° 时力为零。

This equation also defines magnetic flux density: 1 T is the flux density that produces a force of 1 N on 1 m of wire carrying 1 A perpendicular to the field.

该方程也定义了磁通量密度：1 T 表示 1 m 导线载有 1 A 电流且与磁场垂直时受到 1 N 的力。

Symbol	Quantity	SI unit
F	magnetic force	N
B	magnetic flux density	T
I	conventional current	A
L	length in field	m

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2. Direction: Fleming’s Left-Hand Rule | 方向判断：弗莱明左手定则

Fleming’s left-hand rule gives the direction of the force on a current-carrying conductor in a magnetic field. Point the first finger in the direction of the magnetic field B, the second finger in the direction of conventional current I, and the thumb then gives the direction of the force F.

弗莱明左手定则用于判断载流导体在磁场中的受力方向。将食指指向磁场 B 的方向，中指指向常规电流 I 的方向，拇指所指即为力 F 的方向。

Always use conventional current, which is opposite to the direction of electron flow. If the current is parallel to the magnetic field, no force is produced.

必须始终使用常规电流，其方向与电子流动方向相反。如果电流与磁场平行，则不会产生磁力。

For example, if B points into the page and the current flows from left to right, the force on the wire is upward.

例如，如果 B 垂直指向纸面内，电流从左向右流动，则导线受到的力方向向上。

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3. Force Between Parallel Currents | 平行电流之间的作用力

Two long, straight, parallel conductors exert magnetic forces on each other because each wire sits in the magnetic field produced by the other. A long wire carrying current I produces a field at distance d given by:

两根长直平行导体会相互施加磁力，因为每根导线都处于另一根导线产生的磁场中。载流 I 的长直导线在距离 d 处产生的磁场为：

B = μ₀I / (2πd)

The force per unit length between two wires carrying currents I₁ and I₂ separated by distance d is:

两根分别载有 I₁ 和 I₂ 电流、相距 d 的导线之间单位长度的力为：

F / L = μ₀I₁I₂ / (2πd)

Here μ₀ is the permeability of free space, approximately 4π × 10⁻⁷ H m⁻¹. Parallel currents in the same direction attract each other, while parallel currents in opposite directions repel each other.

其中 μ₀ 是真空磁导率，约为 4π × 10⁻⁷ H m⁻¹。同向平行电流相互吸引，反向平行电流相互排斥。

The ampere is defined using this force: 1 A is the constant current that, in two straight parallel conductors of infinite length and negligible cross-section placed 1 m apart in vacuum, produces a force of 2 × 10⁻⁷ N per metre of length.

安培就是利用这个力定义的：1 A 是指两根无限长、截面可忽略的平行直导线在真空中相距 1 m 放置时，每米长度上产生 2 × 10⁻⁷ N 力的恒定电流。

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4. Force on a Moving Charge | 运动电荷在磁场中的受力

A single charge Q moving with velocity v at angle θ to a magnetic field B experiences a force given by:

单个电荷 Q 以速度 v 运动，并与磁场 B 成 θ 角时，受到的磁力为：

F = BQv sin θ

When the velocity is perpendicular to the field, θ = 90° and the force has magnitude F = BQv. The direction for a positive charge is given by Fleming’s left-hand rule, with the second finger pointing in the direction of velocity.

当速度与磁场垂直时，θ = 90°，力的大小为 F = BQv。正电荷的受力方向由弗莱明左手定则判断，此时中指指向速度方向。

Because the magnetic force is always perpendicular to velocity, it does no work on the charge. It cannot change the kinetic energy or speed, only the direction of motion.

由于磁力始终与速度垂直，它不对电荷做功。它不能改变动能或速率，只能改变运动方向。

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5. Circular Motion in Magnetic Fields | 磁场中的圆周运动

When a charged particle moves perpendicular to a uniform magnetic field, the magnetic force acts as a centripetal force, causing circular motion. Equating magnetic force and centripetal force gives:

当带电粒子垂直于匀强磁场运动时，磁力充当向心力，使其做圆周运动。令磁力等于向心力可得：

BQv = mv² / r

Therefore the radius of the circular path is:

因此圆周运动的半径为：

r = mv / (BQ)

The time period of the circular motion is independent of speed:

圆周运动的周期与速度无关：

T = 2πm / (BQ)

A faster particle has a larger radius but completes one circle in the same time as a slower particle. This fact is used in cyclotron design and in mass spectrometers.

较快的粒子半径更大，但完成一圈所需的时间与较慢粒子相同。这一事实用于回旋加速器设计和质谱仪中。

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6. 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. Charge carriers are deflected to one side, creating a transverse voltage called the Hall voltage.

当载流导体或半导体置于与电流垂直的磁场中时，就会发生霍尔效应。电荷载流子被偏转到一侧，产生横向电压，称为霍尔电压。

At equilibrium, the electric force from the Hall field balances the magnetic force on the carriers. For a slab of thickness t in the direction of B, carrier density n, and carrier charge q, the Hall voltage is:

平衡时，霍尔电场产生的电场力与载流子受到的磁力平衡。对于沿 B 方向厚度为 t、载流子密度为 n、载流子电荷为 q 的薄片，霍尔电压为：

Vₕ = BI / (nqt)

Here B is the magnetic flux density, I is the current, and t is the thickness of the slab measured along the magnetic field. The polarity of the Hall voltage reveals whether the charge carriers are positive or negative.

其中 B 是磁通量密度，I 是电流，t 是薄片沿磁场方向的厚度。霍尔电压的极性可以揭示电荷载流子是正电荷还是负电荷。

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