Coulomb's Law | 库仑定律

📚 Coulomb’s Law | 库仑定律

Coulomb’s law is the fundamental principle of electrostatics. It describes the electric force that acts between two point charges, providing the mathematical basis for understanding electric fields, potential, and many practical applications.

库仑定律是静电学的基本原理。它描述了两个点电荷之间的静电力，为理解电场、电势以及许多实际应用提供了数学基础。

1. Introduction to Electrostatics and Coulomb’s Law | 静电学与库仑定律简介

Electrostatics is the study of electric charges at rest. The central question is: how does one stationary charge exert a force on another stationary charge? The answer was first expressed quantitatively by Charles-Augustin de Coulomb in the 1780s.

静电学研究处于静止状态的电荷。核心问题是：一个静止电荷如何对另一个静止电荷施力？查尔斯-奥古斯丁·德·库仑在18世纪80年代首次定量给出了答案。

Coulomb’s law states that the magnitude of the electrostatic force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them.

库仑定律指出：两个点电荷之间静电力的大小与两电荷电量的乘积成正比，与它们之间距离的平方成反比。

2. The Point Charge Idealisation | 点电荷理想模型

A point charge is an idealised model in which all of an object’s electric charge is considered to be concentrated at a single point. This requires that the physical dimensions of the charged object are much smaller than the distances involved.

点电荷是一种理想模型，即认为物体的全部电荷集中在一个几何点上。这要求带电体本身的尺寸远小于所涉及的距离。

In A-Level Physics, small charged spheres are often treated as point charges. This approximation is valid when the sphere radius is negligible compared with the separation distance, allowing the force to be calculated using Coulomb’s law without considering charge distribution.

在A-Level物理中，小的带电球体通常被视为点电荷。当球体半径与球间距相比可以忽略时，这种近似成立，从而可以用库仑定律计算力，而无需考虑电荷分布。

3. Coulomb’s Law Equation | 库仑定律方程

For two point charges q₁ and q₂ separated by a distance r in a vacuum, the magnitude of the electrostatic force F is given by the equation below.

对于真空中相距为 r 的两个点电荷 q₁ 和 q₂，静电力 F 的大小由下式给出。

F = (1 / 4πε₀) × (q₁q₂ / r²)

Here ε₀ is the permittivity of free space. The constant k = 1 / 4πε₀ is often called Coulomb’s constant and has an approximate value of 8.99 × 10⁹ N m² C⁻². When answering numerical questions, both forms of the equation are acceptable.

其中 ε₀ 是真空介电常数。常数 k = 1 / 4πε₀ 通常称为库仑常数，其近似值为 8.99 × 10⁹ N m² C⁻²。在计算题中，这两种表达形式均可使用。

4. Units and the Constant k | 单位与常数 k

Before using Coulomb’s law, it is essential to understand each symbol and its SI unit. The charge unit is the coulomb (C), distance is measured in metres (m), and force is measured in newtons (N).

在使用库仑定律之前，必须弄清每个符号及其SI单位。电荷单位是库仑（C），距离单位为米（m），力的单位为牛顿（N）。

Symbol	Meaning	SI Unit
F	electrostatic force	N
q₁, q₂	electric charges	C
r	separation distance	m
ε₀	permittivity of free space	F m⁻¹

The value of ε₀ is 8.85 × 10⁻¹² F m⁻¹. In calculations, using k = 9 × 10⁹ N m² C⁻² is often accurate enough for A-Level questions.

ε₀ 的值为 8.85 × 10⁻¹² F m⁻¹。在A-Level题目中，使用 k = 9 × 10⁹ N m² C⁻² 通常已经足够精确。

5. Vector Nature of Electrostatic Force | 静电力的矢量性

Coulomb’s law gives only the magnitude of the force. The direction depends on the signs of the charges. Like charges repel, while opposite charges attract.

库仑定律只给出力的大小。力的方向取决于电荷的正负。同种电荷相互排斥，异种电荷相互吸引。

Because force is a vector quantity, each electrostatic force must be described with both magnitude and direction. When two charges interact, the force on q₁ due to q₂ is equal in magnitude and opposite in direction to the force on q₂ due to q₁, in agreement with Newton’s third law.

由于力是矢量，每个静电力都必须同时说明大小和方向。两个电荷相互作用时，q₂ 对 q₁ 的力与 q₁ 对 q₂ 的力大小相等、方向相反，符合牛顿第三定律。

When solving problems, it is helpful to draw a free-body diagram. Show each charge separately and draw arrows representing the forces acting on each charge, making sure the relative directions are consistent.

解题时最好画出受力分析图。分别画出每个电荷，并用箭头表示作用在每个电荷上的力，确保方向关系一致。

6. Permittivity and the Effect of a Dielectric Medium | 介电常数与介质的影响

The value ε₀ applies to a vacuum. If the charges are placed in a dielectric material, the electric force is reduced. This reduction is described by the relative permittivity εᵣ of the medium.

ε₀ 适用于真空。如果电荷置于电介质材料中，静电力会减弱。这种减弱用介质的相对介电常数 εᵣ 来描述。

F = q₁q₂ / (4πε₀εᵣr²)

Air has εᵣ very close to 1, so in most A-Level problems the vacuum formula can be used without significant error. For other materials, such as water, εᵣ is much larger and the force becomes considerably weaker.

空气的 εᵣ 非常接近 1，因此在大多数A-Level问题中可以直接使用真空公式而不产生明显误差。对于其他材料，例如水，εᵣ 大得多，静电力会被显著削弱。

7. Comparison with Newton’s Law of Gravitation | 与万有引力定律的比较

Coulomb’s law and Newton’s law of gravitation have a strikingly similar mathematical form: both are inverse-square laws. This similarity allows us to transfer problem-solving techniques between the two topics.

库仑定律与牛顿万有引力定律在数学形式上非常相似：两者都是平方反比定律。这种相似性使我们能在两个主题间迁移解题技巧。

Feature	Coulomb’s law	Gravitation
Equation	F = kq₁q₂ / r²	F = Gm₁m₂ / r²
Source	electric charge	mass
Nature of force	attractive or repulsive	attractive only
Relative strength	very large	extremely small

Both forces obey the inverse-square relationship with distance, but electric forces can be either attractive or repulsive because charges have two signs. Gravitational forces are always attractive because mass is always positive.

两种力都与距离平方成反比，但静电力可以是引力或斥力，因为电荷有两种符号。万有引力则始终是吸引力，因为质量总是正的。

8. Electric Field Strength and Coulomb’s Law | 电场强度与库仑定律

The concept of an electric field is closely linked to Coulomb’s law. The electric field strength E at a point is defined as the force per unit positive charge experienced by a small test charge placed at that point.

电场的概念与库仑定律密切相关。电场强度 E 定义为放在某点的微小正试探电荷所受的力与电荷量之比。

E = F / q

Combining this with Coulomb’s law gives the electric field strength due to a point charge Q at a distance r.

将上式与库仑定律结合，可得点电荷 Q 在距离 r 处产生的电场强度。

E = Q / (4πε₀r²)

The field is directed radially away from a positive charge and radially towards a negative charge. This result is essential for solving many CIE A-Level questions on electric fields.

电场方向从正电荷沿径向向外，指向负电荷沿径向向内。这一结果对于解决许多CIE A-Level电场问题至关重要。

9. Superposition Principle for Multiple Charges | 多个电荷的叠加原理

When more than two charges are present, the net force on any one charge is the vector sum of the individual forces exerted by each of the other charges. This is called the principle of superposition.

当存在两个以上电荷时，任意一个电荷所受的合力等于其他各个电荷单独作用时对该电荷作用力的矢量和。这称为叠加原理。

To apply superposition, first calculate each pairwise force using Coulomb’s law. Then break the forces into components along x- and y-axes if needed, and add the components separately.

应用叠加原理时，先用库仑定律计算每一对电荷之间的力。如有必要，将各个力沿 x 轴和 y 轴分解，然后分别将各分量相加。

Do not simply add the magnitudes arithmetically unless all forces act along the same straight line. Pay careful attention to the directions of the forces, since sign errors are the most common mistakes in these problems.

除非所有力都沿同一直线，否则不能简单地将大小代数相加。必须仔细注意力的方向，因为符号错误是这类问题最常见的错误。

10. Worked Example: Two Charges | 例题：两个电荷

A charge q₁ = +2.0 μC and a charge q₂ = −3.0 μC are separated by 0.50 m in a vacuum. Calculate the magnitude of the force between them.

真空中有电荷 q₁ = +2.0 μC 和 q₂ = −3.0 μC，两者相距 0.50 m。计算它们之间力的大小。

Using k = 9 × 10⁹ N m² C⁻² and the relation F = kq₁q₂ / r²:

使用 k = 9 × 10⁹ N m² C⁻² 以及关系式 F = kq₁q₂ / r²：

F = (9 × 10⁹) × (2.0 × 10⁻⁶) × (3.0 × 10⁻⁶) / (0.50)²

The product of the charges is 6.0 × 10⁻¹² C². Dividing by 0.25 m² gives F = 0.216 N. Since the charges have opposite signs, the force is attractive.

两电荷乘积为 6.0 × 10⁻¹² C²。除以 0.25 m² 得 F = 0.216 N。由于两电荷异号，该力为吸引力。

11. Common Mistakes and Exam Pitfalls | 常见错误与考试陷阱

Students often forget to square the distance in the denominator. The term r² is not optional; using r instead of r² will give a force that is too large by a factor of r.

学生经常忘记对分母中的距离取平方。r² 不是可选项；若误用 r 代替 r²，所得力会偏大 r 倍。

Another common mistake is failing to convert units before substitution. Distances must be in metres and charges in coulombs. Values given in centimetres or microcoulombs must be converted first.

另一个常见错误是代入前未能进行单位换算。距离必须以米为单位，电荷必须以库仑为单位。如果题目给出厘米或微库仑，必须先换算。

It is also important to decide how to treat the sign of charges. When finding the magnitude of the force, use the absolute values of the charges. Use the signs only to determine whether the force is attractive or repulsive.

还要注意如何处理电荷的正负号。在求力的大小时，使用电荷的绝对值；仅用正负号判断力是引力还是斥力。

12. Summary and Key Formulae | 总结与关键公式

Coulomb’s law is one of the most important equations in electrostatics. For two point charges in a vacuum, the force is proportional to the product of the charges and inversely proportional to the square of their separation.

库仑定律是静电学中最重要的方程之一。对于真空中的两个点电荷，力与电荷量的乘积成正比，与距离的平方成反比。

F = q₁q₂ / (4πε₀r²)

Always remember that the force is a vector. Draw a diagram, convert all quantities to SI units, square the distance, and use superposition when multiple charges are present. With these habits, you will avoid the most common errors in CIE A-Level exams.

始终记住力是矢量。画图、将所有量换算为SI单位、对距离取平方，并在存在多个电荷时使用叠加原理。养成这些习惯，就能避免CIE A-Level考试中最常见的错误。

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Coulomb’s Law | 库仑定律