Forces between Currents | 电流之间的作用力

📚 Forces between Currents | 电流之间的作用力

When electric currents flow in parallel wires, the wires exert magnetic forces on each other. This interaction is not a direct electric force between charges in the wires; it arises because each current produces a magnetic field, and that field then applies a force to the other current-carrying wire. Understanding this effect is essential for explaining electromagnets, electric motors, and the formal definition of the ampere.

当电流在平行导线中流动时，导线之间会产生磁力作用。这种相互作用并不是导线中电荷之间的直接电场力，而是因为每个电流都会产生磁场，该磁场又对另一根载流导线施加力的作用。理解这一效应对于解释电磁铁、电动机以及安培的正式定义都至关重要。

1. The Origin of the Force | 力的来源

A current-carrying wire generates a magnetic field in the space around it. If a second current-carrying wire is placed in this field, the moving charges in the second wire experience a magnetic force. Therefore, two currents interact through their magnetic fields, even though the wires themselves are electrically neutral.

载流导线会在其周围空间产生磁场。如果将第二根载流导线放入该磁场中，第二根导线中的运动电荷就会受到磁力作用。因此，两个电流通过它们产生的磁场发生相互作用，即使导线本身是电中性的。

The force between currents is a consequence of the Lorentz force law, which states that a charge moving in a magnetic field experiences a force perpendicular to both its velocity and the magnetic field.

电流之间的力是洛伦兹力定律的结果。该定律指出，电荷在磁场中运动时会受到一个垂直于其速度方向和磁场方向的力。

2. Magnetic Field Around a Long Straight Current | 长直电流周围的磁场

For a long straight wire carrying current I, the magnetic flux density at a perpendicular distance r from the wire is given by:

对于载有电流 I 的长直导线，在距导线垂直距离为 r 处的磁感应强度由下式给出：

B = μ₀ I / (2π r)

Here μ₀ is the permeability of free space, with the value μ₀ = 4π × 10⁻⁷ T m A⁻¹. The field lines are concentric circles centred on the wire.

式中 μ₀ 是真空磁导率，其值为 μ₀ = 4π × 10⁻⁷ T m A⁻¹。磁感线是以导线为中心的同心圆。

The direction of the magnetic field is found using the right-hand grip rule: point the thumb of your right hand in the direction of the conventional current, and your fingers curl in the direction of the field lines.

磁场方向可以用右手螺旋定则判断：让右手拇指指向常规电流的方向，四指弯曲的方向就是磁感线的方向。

3. Force on a Current in a Magnetic Field | 磁场对电流的作用力

A straight conductor of length L carrying current I in a uniform magnetic field B experiences a force given by:

长度为 L 的直导体在均匀磁场 B 中载有电流 I 时，受到的力为：

F = B I L sin θ

where θ is the angle between the current direction and the magnetic field. If the current is perpendicular to the field, sin θ = 1 and the formula simplifies to F = B I L.

式中 θ 是电流方向与磁场方向之间的夹角。如果电流垂直于磁场，则 sin θ = 1，公式简化为 F = B I L。

The direction of this force is given by Fleming’s left-hand rule: first finger in the field direction, second finger in the current direction, and thumb shows the force direction.

该力的方向由弗莱明左手定则判断：食指指向磁场方向，中指指向电流方向，拇指所指即为力的方向。

4. Combining the Two Ideas | 综合两个概念

To calculate the force between two parallel currents, we combine the previous two results. Wire 1 produces a magnetic field at the position of wire 2. Wire 2 carries a current and therefore experiences a force in that field. Wire 2 produces a field at wire 1 as well, so wire 1 experiences an equal and opposite force.

要计算两个平行电流之间的力，我们需要综合前面两个结论。导线 1 在导线 2 所在位置产生磁场。导线 2 载有电流，因此在该磁场中受到力的作用。导线 2 也会在导线 1 处产生磁场，因此导线 1 受到大小相等、方向相反的力。

This is consistent with Newton’s third law: the force exerted by wire 1 on wire 2 is equal in magnitude and opposite in direction to the force exerted by wire 2 on wire 1.

这符合牛顿第三定律：导线 1 对导线 2 的力与导线 2 对导线 1 的力大小相等、方向相反。

5. Force per Unit Length Between Parallel Currents | 平行电流间单位长度作用力

Consider two long, straight, parallel wires separated by a distance d, carrying currents I₁ and I₂. The magnetic field produced by wire 1 at the position of wire 2 is:

考虑两根相距为 d 的长直平行导线，分别载有电流 I₁ 和 I₂。导线 1 在导线 2 处产生的磁场为：

B₁ = μ₀ I₁ / (2π d)

Since wire 2 is perpendicular to this field, the force on a length L of wire 2 is F = B₁ I₂ L. Substituting B₁ gives:

由于导线 2 与该磁场方向垂直，长度为 L 的导线 2 受到的力为 F = B₁ I₂ L。代入 B₁ 可得：

F / L = μ₀ I₁ I₂ / (2π d)

This is the force per unit length between two parallel current-carrying wires. It is directly proportional to the product of the currents and inversely proportional to their separation.

这就是两根平行载流导线之间单位长度上的力。该力与两电流的乘积成正比，与导线间的距离成反比。

This result applies exactly only for infinitely long wires in a vacuum, but it is an excellent approximation for long wires with separation much smaller than their length.

这一结果仅在真空中无限长导线的条件下严格成立，但对于长度远大于间距的长导线来说，它是一个非常好的近似。

6. Direction Rules: Attraction and Repulsion | 方向判断：同向吸引、异向排斥

If the currents in the two parallel wires flow in the same direction, the wires attract each other. If the currents flow in opposite directions, the wires repel each other.

如果两根平行导线中的电流方向相同，导线会相互吸引；如果电流方向相反，导线会相互排斥。

To verify this, use the right-hand grip rule to find the field direction at wire 2 due to wire 1, and then use Fleming’s left-hand rule to find the force on wire 2.

要验证这一点，可以先用右手螺旋定则确定导线 1 在导线 2 处产生的磁场方向，再用弗莱明左手定则判断导线 2 所受力的方向。

Same direction currents: field from one wire pushes the other wire towards it, so attraction occurs.
同向电流：一根导线产生的磁场使另一根导线受到朝向它的力，因此表现为吸引。
Opposite direction currents: field from one wire pushes the other wire away, so repulsion occurs.
反向电流：一根导线产生的磁场使另一根导线受到远离它的力，因此表现为排斥。

This attraction and repulsion is the magnetic analogue of the forces between electric charges, but here the cause is the magnetic field produced by moving charges.

这种吸引和排斥是电荷之间电场力的磁学类比，但此处的原因是运动电荷产生的磁场。

7. Definition of the Ampere | 安培的定义

The force between parallel currents provides the formal SI definition of the ampere. One ampere is that constant current which, if maintained in two straight parallel conductors of infinite length and negligible circular cross-section, placed 1 metre apart in vacuum, would produce between these conductors a force equal to 2 × 10⁻⁷ newtons per metre of length.

平行电流之间的力为安培提供了正式的 SI 定义。一安培是指这样的恒定电流：若将其保持在两根相距 1 米、置于真空中、长度无限且圆截面可忽略的平行直导线中，则两导线之间每米长度上产生的力为 2 × 10⁻⁷ 牛顿。

This definition was adopted because it links the ampere to mechanical quantities such as force and length, which can be measured very accurately in a laboratory.

采用这一定义是因为它将安培与力和长度等力学量联系起来，而这些力学量可以在实验室中非常精确地测量。

8. Experimental Demonstration | 实验演示

In a school laboratory, the force between currents can be demonstrated using two parallel strips of aluminium foil connected to a power supply. When a large current flows in the same direction through both strips, they visibly move towards each other. When the current direction in one strip is reversed, they move apart.

在学校实验室中，可以用两根连接电源的平行铝箔条来演示电流之间的力。当大电流以相同方向通过两根铝箔条时，可以看到它们相互靠近。当其中一根铝箔条中的电流方向反向时，它们会相互远离。

A more accurate method uses a current balance, where the force per unit length between two coils or straight conductors is balanced against a known weight. This method can be used to measure μ₀ or to verify the inverse relationship with separation.

更精确的方法是使用电流天平，使两个线圈或直导线之间单位长度的力与已知砝码的重量相平衡。这种方法可用来测量 μ₀，或验证力与距离的反比关系。

9. Worked Example | 例题

Two long straight parallel wires are separated by 5.0 cm in air. They carry currents of 3.0 A and 4.0 A in the same direction. Calculate the force per metre between them and state whether it is attractive or repulsive.

两根长直平行导线在空气中相距 5.0 cm，分别载有 3.0 A 和 4.0 A 的同向电流。计算它们之间每米长度上的力，并说明该力是吸引力还是排斥力。

Using the formula F / L = μ₀ I₁ I₂ / (2π d):

使用公式 F / L = μ₀ I₁ I₂ / (2π d)：

F / L = (4π × 10⁻⁷ × 3.0 × 4.0) / (2π × 0.050)

F / L = (48π × 10⁻⁷) / (0.10π) = 4.8 × 10⁻⁵ N m⁻¹

Since the currents are in the same direction, the force is attractive.

由于电流方向相同，该力为吸引力。

10. Common Misconceptions and Exam Tips | 常见误区与应试提示

A common mistake is to think that a wire experiences a force due to its own magnetic field. A current-carrying wire does produce its own field, but its own field produces no net force on the wire itself. The force on a wire is caused by the field produced by the other current.

一个常见误区是认为导线会受到自身磁场的力。载流导线确实会产生自身的磁场，但自身的磁场不会对导线本身产生净力。导线所受的力是由另一电流产生的磁场引起的。

Another common confusion is between the right-hand grip rule and Fleming’s left-hand rule. The right-hand grip rule gives the field direction around a current, while Fleming’s left-hand rule gives the force direction on a current in an external field. Do not swap them.

另一个常见的混淆是右手螺旋定则和弗莱明左手定则。右手螺旋定则给出电流周围的磁场方向，而弗莱明左手定则给出电流在外部磁场中的受力方向。不要将它们互换。

In exam answers, always convert distances to metres and currents to amperes before substituting into the formula. If the question asks for direction, you must state whether the force is attractive or repulsive, or draw arrows showing the direction.

在考试答案中，代入公式前务必将距离转换为米、电流转换为安培。如果题目要求判断方向，必须说明力是吸引力还是排斥力，或者画出显示方向的箭头。

Published by TutorHao | Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Forces between Currents | 电流之间的作用力