📚 A-Level Physics: Interaction Forces Between Parallel Currents | A-Level 物理：平行电流间的相互作用力
When two straight conductors carrying electric currents are placed parallel to each other, they exert a force on each other. This force is not due to direct contact but arises from the magnetic fields each current creates and the interaction of those fields with the other current.
当两根通有电流的直导线平行放置时，它们之间会相互施力。这种力并非来自直接接触，而是源于每根导线中电流所产生的磁场，以及该磁场与另一根导线中电流的相互作用。
1. Magnetic Field Around a Long Straight Wire | 长直导线周围的磁场
A steady current I flowing through a long straight wire generates a magnetic field with circular field lines centred on the wire. At a perpendicular distance r from the wire, the magnetic flux density (magnetic field strength) B is given by:
恒定电流 I 流过一根长直导线时，会在导线周围产生以导线为中心的圆形磁场线。在距导线垂直距离 r 处，磁通密度（磁感应强度）B 的表达式为：
B = μ₀I / (2πr)
Here μ₀ is the permeability of free space, μ₀ = 4π × 10⁻⁷ T·m·A⁻¹. The direction of the field is given by the right-hand grip rule: if the thumb points along the current, the curled fingers show the direction of the magnetic field lines.
其中 μ₀ 是真空磁导率，μ₀ = 4π × 10⁻⁷ T·m·A⁻¹。磁场方向由右手螺旋定则判断：拇指指向电流方向，弯曲的四指即表示磁场线方向。
2. Force on a Current-Carrying Conductor in a Magnetic Field | 磁场中载流导线所受的力
A conductor of length L carrying current I placed in a uniform magnetic field B experiences a magnetic force given by:
长度为 L、通有电流 I 的导体置于匀强磁场 B 中时，受到的磁场力为：
F = BIL sinθ
where θ is the angle between the direction of the current and the magnetic field. When the current is perpendicular to the magnetic field (θ = 90°), the force is maximum: F = BIL. The direction of the force is determined by Fleming’s left-hand rule.
其中 θ 是电流方向与磁场方向之间的夹角。当电流垂直于磁场时（θ = 90°），力最大：F = BIL。力的方向用弗莱明左手定则判断。
3. Magnetic Field at One Wire Due to the Other | 一根导线在另一根导线处产生的磁场
Consider two long parallel wires separated by distance d. Wire 1 carries current I₁ and wire 2 carries current I₂. At the location of wire 2, wire 1 produces a magnetic field B₁ perpendicular to the plane containing the two wires, with magnitude:
考虑两根相距为 d 的长平行导线。导线 1 通有电流 I₁，导线 2 通有电流 I₂。在导线 2 所在位置，导线 1 产生的磁场 B₁ 垂直于两根导线所在的平面，其大小为：
B₁ = μ₀I₁ / (2πd)
This field is perpendicular to the current I₂ in wire 2, so the force on a length L of wire 2 is:
该磁场垂直于导线 2 中的电流 I₂，因此长度为 L 的导线 2 所受的力为：
F = B₁ I₂ L = (μ₀ I₁ I₂ L) / (2πd)
4. Force per Unit Length Between Two Parallel Wires | 两平行导线间单位长度的作用力
Rearranging the expression above gives the force per unit length between the two parallel wires:
整理上式可得两平行导线间单位长度的作用力：
F / L = μ₀ I₁ I₂ / (2πd)
This is the standard formula for the magnetic force between two long, straight, parallel currents. The force is attractive when the currents are in the same direction and repulsive when they are in opposite directions.
这就是两根长直平行电流之间磁力的标准公式。当电流方向相同时，力表现为吸引力；当电流方向相反时，力表现为排斥力。
5. Direction of the Force: Same Direction Attracts | 力的方向：同向相吸
Use the right-hand grip rule to find the magnetic field due to wire 1 at wire 2. Then use Fleming’s left-hand rule on wire 2: point the first finger along the field and the middle finger along the current I₂; the thumb gives the direction of the magnetic force. For parallel currents in the same direction, the force on wire 2 points toward wire 1. Similarly, the force on wire 1 points toward wire 2, so the wires attract.
用右手螺旋定则找出导线 1 在导线 2 处产生的磁场方向，然后对导线 2 应用弗莱明左手定则：食指指向磁场方向，中指指向电流 I₂ 方向，拇指即为磁场力方向。当两电流同向时，导线 2 所受的力指向导线 1；同理，导线 1 所受的力指向导线 2，因此两根导线相互吸引。
6. Direction of the Force: Opposite Directions Repel | 力的方向：反向相斥
If the currents flow in opposite directions, the same analysis shows that the magnetic field due to wire 1 at wire 2 is reversed, and the force on wire 2 points away from wire 1. The force on wire 1 also points away from wire 2, so the wires repel each other. This is consistent with Newton’s third law: the magnitudes of the forces on the two wires are always equal.
如果电流方向相反，同样的分析表明，导线 1 在导线 2 处产生的磁场方向反转，导线 2 所受的力指向远离导线 1 的方向；导线 1 所受的力也指向远离导线 2 的方向，因此两根导线相互排斥。这与牛顿第三定律一致：两根导线所受的力大小总是相等。
7. Definition of the Ampere | 安培的定义
The force between parallel currents provides the operational definition of the ampere, the SI base unit of electric current. One ampere is defined as the constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, placed one metre apart in vacuum, would produce between these conductors a force equal to 2 × 10⁻⁷ newtons per metre of length.
平行电流之间的作用力为安培提供了操作定义。安培是国际单位制中电流的基本单位。安培定义为：真空中相距 1 米的两根无限长、圆截面可忽略的平行直导线内通以等量恒定电流时，若导线间相互作用力在每米长度上为 2 × 10⁻⁷ 牛顿，则此恒定电流的大小为 1 安培。
Substituting I₁ = I₂ = 1 A and d = 1 m into the formula gives F/L = (4π × 10⁻⁷ × 1 × 1) / (2π × 1) = 2 × 10⁻⁷ N·m⁻¹, confirming the definition.
将 I₁ = I₂ = 1 A、d = 1 m 代入公式，可得 F/L = (4π × 10⁻⁷ × 1 × 1) / (2π × 1) = 2 × 10⁻⁷ N·m⁻¹，与定义相符。
8. Worked Example: Numerical Calculation | 例题：数值计算
Two long parallel wires are separated by 5.0 cm. Wire A carries a current of 3.0 A and wire B carries a current of 4.0 A in the same direction. Calculate the force per unit length between them and state whether it is attractive or repulsive.
两根长平行导线相距 5.0 cm。导线 A 通有 3.0 A 电流，导线 B 通有 4.0 A 电流，方向相同。计算它们之间的单位长度作用力，并说明是吸引力还是排斥力。
Using the formula with d = 0.050 m:
将 d = 0.050 m 代入公式：
F / L = (4π × 10⁻⁷ × 3.0 × 4.0) / (2π × 0.050)
= (4.8π × 10⁻⁶) / (0.100π) = 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。
9. Common Pitfalls and Exam Tips | 常见易错点与考试提示
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Remember to use the perpendicular distance between the wires, not the length of the wire, in the denominator.
注意公式中分母使用的是导线间的垂直距离，而不是导线本身的长度。
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The force per unit length formula already includes L in the denominator; do not multiply by L again unless asked for the total force.
单位长度力的公式已经包含 L 在分母中；除非题目要求总力，否则不要再乘以 L。
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The force is magnetic, not electrostatic, even though the wires carry currents. Charges are not static.
这种力是磁力而非静电力，即使导线中载有电流——电荷并非静止。
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For non-parallel wires, the force is not given by this simple formula; the magnetic field varies along the wire and the angle between B and I must be considered.
对于不平行导线，此简单公式不适用；磁场沿导线变化，且必须考虑 B 与 I 之间的夹角。
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Always state the direction (attractive/repulsive) in your answer if the question asks for it.
若题目要求说明方向（吸引/排斥），务必在答案中写出。
10. Experimental Demonstration: Current Balance | 实验演示：电流天平
The force between parallel currents can be measured using a current balance. Two horizontal parallel conductors are arranged so that the upper one is balanced on a pivot. When currents flow in the same direction, the attractive force adds to the weight of the upper conductor; when opposite, the repulsive force reduces the apparent weight. By calibrating the balance, the value of F/L can be determined and compared with the theoretical formula.
平行电流间的作用力可通过电流天平进行测量。将两根水平平行导体布置成上导体可在支点上平衡的结构。当电流同向流动时，吸引力叠加到上导体的重量上；当电流反向时，排斥力减小表观重量。通过校准天平，可以测定 F/L 的值并与理论公式进行比较。
11. Relation to Magnetic Flux Density | 与磁通密度的关系
From the force per unit length expression, we can also define the magnetic flux density B produced by a long straight wire. Since F/L = B₁ I₂, and B₁ = μ₀ I₁ / (2πd), the formula is consistent. This shows that the concept of magnetic field strength is a convenient intermediate step in understanding the force between currents.
由单位长度力的表达式，我们也可以定义长直导线产生的磁通密度 B。因为 F/L = B₁ I₂，且 B₁ = μ₀ I₁ / (2πd)，公式自洽。这表明磁感应强度的概念是理解电流间作用力的一个便捷中间步骤。
12. Summary | 总结
Two parallel currents exert equal and opposite forces on each other. The magnitude of the force per unit length is given by F/L = μ₀ I₁ I₂ / (2πd). Parallel currents in the same direction attract; opposite currents repel. This principle is fundamental to the definition of the ampere and has numerous practical applications, including electromagnetic switches, railguns, and the forces between wires in electrical circuits.
两平行电流对彼此施加等大反向的作用力。单位长度力的大小为 F/L = μ₀ I₁ I₂ / (2πd)。同向平行电流相吸，反向平行电流相斥。这一原理是安培定义的基础，并具有众多实际应用，包括电磁开关、电磁轨道炮以及电路中导线之间的作用力。
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