Magnetic Force | 磁力

📚 Magnetic Force | 磁力

In A-Level Physics, magnetic force is the force experienced by a moving charge or a current-carrying conductor when placed in a magnetic field. It is a non-contact force that arises from the interaction between moving charges and magnetic fields. Understanding magnetic force is essential for explaining the motion of charged particles, the operation of electric motors, and many modern instruments such as mass spectrometers and cyclotrons.

在 A-Level 物理中，磁力是运动电荷或载流导体置于磁场中时所受到的力。它是一种非接触力，源于运动电荷与磁场之间的相互作用。理解磁力对于解释带电粒子运动、电动机工作原理以及质谱仪、回旋加速器等现代仪器至关重要。

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1. Magnetic Fields and Flux Density | 磁场与磁通量密度

A magnetic field is a region of space in which a magnetic force can be detected. The strength of a magnetic field is described by the magnetic flux density B, which is measured in tesla (T). One tesla is defined as the flux density that produces a force of one newton on a one-metre length of wire carrying a current of one ampere perpendicular to the field.

磁场是能够检测到磁力的空间区域。磁场强度用磁通量密度 B 来描述，单位是特斯拉（T）。1 特斯拉的定义是：当 1 米长的导线载有 1 安培电流且与磁场垂直时，受到 1 牛顿的力，此时磁通量密度为 1 特斯拉。

For a straight conductor placed perpendicular to a uniform magnetic field, the flux density can be written as:

B = F ÷ (IL)

对于垂直于均匀磁场放置的直导线，磁通量密度可写为：

B = F ÷ (IL)

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2. Force on a Moving Charge | 运动电荷受到的磁力

A charged particle moving through a magnetic field experiences a magnetic force only if its velocity has a component perpendicular to the magnetic field. The magnitude of this force is given by:

带电粒子在磁场中运动时，只有当其速度具有垂直于磁场的分量时，才会受到磁力。磁力大小由下式给出：

F = Bqv sin θ

where q is the charge, v is the speed of the particle, B is the magnetic flux density, and θ is the angle between the velocity vector and the magnetic field direction.

其中 q 是电荷量，v 是粒子速度，B 是磁通量密度，θ 是速度矢量与磁场方向之间的夹角。

If the charge moves parallel to the field (θ = 0° or 180°), sin θ = 0, so the magnetic force is zero. If the charge moves perpendicular to the field (θ = 90°), sin θ = 1 and the force reaches its maximum value F = Bqv.

如果电荷沿磁场方向运动（θ = 0° 或 180°），sin θ = 0，因此磁力为零。如果电荷垂直于磁场运动（θ = 90°），sin θ = 1，磁力达到最大值 F = Bqv。

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3. Direction of the Magnetic Force | 磁力的方向

The direction of the magnetic force on a moving positive charge is determined by Fleming’s left-hand rule. Point the first finger in the direction of the magnetic field, the second finger in the direction of conventional current (or velocity of a positive charge), then the thumb points in the direction of the force. For a negative charge, the force direction is reversed.

运动正电荷所受磁力的方向由弗莱明左手定则确定。将食指指向磁场方向，中指指向常规电流（或正电荷速度）方向，则拇指指向力的方向。对于负电荷，力的方向相反。

Fleming’s left-hand rule can be summarised as follows:

• First finger: magnetic field direction
• Second finger: conventional current direction (positive charge motion)
• Thumb: magnetic force direction

弗莱明左手定则可总结如下：

• 食指：磁场方向
• 中指：常规电流方向（正电荷运动方向）
• 拇指：磁力方向

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4. Magnetic Force on a Current-Carrying Conductor | 载流导体受到的磁力

A current in a wire is a flow of charge, so a current-carrying conductor also experiences a magnetic force when placed in a magnetic field. For a straight wire of length L carrying current I, the force is:

导线中的电流是电荷的流动，因此载流导体置于磁场中时也会受到磁力。对于长度为 L、载有电流 I 的直导线，力为：

F = BIL sin θ

where θ is the angle between the current direction and the magnetic field.

其中 θ 是电流方向与磁场方向之间的夹角。

This equation is consistent with F = Bqv sin θ because the total charge passing through the wire in time t is q = It and the drift distance is L = vt, so F = B(It)v sin θ = BIL sin θ.

该方程与 F = Bqv sin θ 一致，因为在时间 t 内通过导线的总电荷为 q = It，漂移距离为 L = vt，所以 F = B(It)v sin θ = BIL sin θ。

When the wire is perpendicular to the field, θ = 90° and F = BIL. When the wire is parallel to the field, θ = 0° and the force is zero.

当导线垂直于磁场时，θ = 90°，F = BIL。当导线平行于磁场时，θ = 0°，力为零。

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5. Motion of a Charged Particle in a Uniform Magnetic Field | 带电粒子在均匀磁场中的运动

When a charged particle enters a uniform magnetic field at right angles to the field, the magnetic force is always perpendicular to the velocity. This force changes the direction of the velocity but not its speed, so the particle undergoes uniform circular motion. The magnetic force provides the centripetal force:

当带电粒子以垂直于磁场的方向进入均匀磁场时，磁力始终垂直于速度。该力改变速度的方向而不改变其大小，因此粒子做匀速圆周运动。磁力提供向心力：

F = Bqv = mv² ÷ r

Here m is the mass of the particle, v is its speed, and r is the radius of the circular path. Rearranging gives the radius:

其中 m 是粒子质量，v 是其速度，r 是圆周路径的半径。重新整理可得半径：

r = mv ÷ (Bq)

The time period T for one full revolution is independent of the speed and is given by:

完整旋转一圈的周期 T 与速度无关，由下式给出：

T = 2πm ÷ (Bq)

The angular frequency ω = 2π ÷ T = Bq ÷ m is sometimes called the cyclotron frequency.

角频率 ω = 2π ÷ T = Bq ÷ m

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