Coulomb's Law | 库仑定律

📚 Coulomb's Law | 库仑定律

In A-Level physics, one of the fundamental principles governing electric interactions is Coulomb's law. It describes the force between two stationary point charges and is the cornerstone of electrostatics. Understanding this law is essential for grasping more advanced topics such as electric fields, potential, and capacitance. This article provides a detailed yet exam-focused explanation of Coulomb's law, carefully aligned with the Cambridge International (CIE) A-Level Physics syllabus.

在 A-Level 物理中，库仑定律是支配电相互作用的基本原理之一。它描述了两个静止点电荷之间的作用力，是静电学的基石。理解这一定律对于掌握电场、电势和电容等更深入的主题至关重要。本文以考试为重点，对库仑定律进行了详细的解释，并严格对标剑桥国际 (CIE) A-Level 物理课程大纲。

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1. Introduction to Coulomb's Law | 库仑定律简介

Electric charges exert forces on one another. The quantitative description of this force was first published by the French physicist Charles-Augustin de Coulomb in 1785. Using a sensitive torsion balance, he established that the force between two small charged spheres follows a simple inverse-square relationship with distance and depends on the product of the charges.

电荷会相互施加力的作用。法国物理学家查尔斯-奥古斯丁·德·库仑于 1785 年首次发表了这一作用力的定量描述。他利用灵敏的扭秤实验，证明了两个带电小球之间的力与距离的平方成反比，并与电荷量的乘积成正比。

In the CIE syllabus, Coulomb's law is introduced as the force law for point charges in a vacuum. A 'point charge' means a charged object whose size is negligible compared with the distances involved. The law is entirely analogous to Newton's law of universal gravitation in its mathematical form, but with charge replacing mass and the constant being much larger.

在 CIE 课程大纲中，库仑定律被介绍为真空中点电荷之间的力定律。'点电荷'指的是其尺寸相对所涉及的距离可以忽略不计的带电物体。该定律在数学形式上与牛顿万有引力定律完全类似，只是用电荷代替了质量，并且常数要大得多。

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2. Statement and Mathematical Form | 表述与数学形式

Coulomb's law states that the magnitude of the electrostatic force F between two point charges q₁ and q₂ separated by a distance r in a vacuum is directly proportional to the product of the absolute values of the charges and inversely proportional to the square of the distance between their centres:

库仑定律指出，真空中相距 r 的两个点电荷 q₁ 和 q₂ 之间的静电力大小 F，与两电荷绝对值的乘积成正比，与它们中心间距离的平方成反比：

F = k_e |q₁ q₂| / r²

Here k_e is the Coulomb constant (also written simply as k). The absolute value is used in this expression because the magnitude of the force is always positive. The direction of the force is determined separately: like charges repel and unlike charges attract.

这里 k_e 是库仑常数（通常也简写为 k）。该表达式中使用绝对值是因为力的大小总为正。力的方向单独确定：同种电荷相互排斥，异种电荷相互吸引。

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3. The Coulomb Constant k and Permittivity | 库仑常数 k 与介电常数

In SI units, the Coulomb constant k is related to the permittivity of free space ε₀ by:

在国际单位制中，库仑常数 k 与真空介电常数 ε₀ 有如下关系：

k = 1 / (4πε₀)

The value of ε₀ is 8.85 × 10⁻¹² F m⁻¹ (farads per metre). Consequently, k has a value of approximately 8.99 × 10⁹ N m² C⁻². For most A‑Level calculations, using k = 9.0 × 10⁹ N m² C⁻² is acceptable unless a more precise answer is required.

ε₀ 的值为 8.85 × 10⁻¹² F m⁻¹（法拉每米）。因此，k 的值约为 8.99 × 10⁹ N m² C⁻²。在大多数 A‑Level 计算中，除非要求更精确的答案，否则使用 k = 9.0 × 10⁹ N m² C⁻² 是可以的。

Some textbooks write Coulomb's law directly using ε₀:

F = (1 / (4πε₀)) × |q₁ q₂| / r²

This form is particularly useful when studying electric fields in dielectric materials, as ε₀ can be substituted with ε = ε₀ε_r, where ε_r is the relative permittivity of the medium.

这种形式在研究介质中的电场时特别有用，因为 ε₀ 可被 ε = ε₀ε_r 替代，其中 ε_r 是介质的相对介电常数。

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4. Vector Nature of Electric Force | 电场力的矢量性质

Force is a vector quantity, so Coulomb's law must include direction information. The force on a charge q₁ due to q₂ acts along the line joining the two charges. The vector form of Coulomb’s law is:

力是矢量，因此库仑定律必须包含方向信息。电荷 q₁ 受到来自 q₂ 的力沿着两电荷的连线方向。库仑定律的矢量形式为：

F₁₂ = k q₁q₂ / r² r̂₁₂

where r̂₁₂ is a unit vector pointing from q₁ to q₂. If the product q₁q₂ is positive (like charges), F₁₂ points away from q₂, indicating repulsion; if negative, it points towards q₂, indicating attraction. The force on q₂ is equal in magnitude and opposite in direction (Newton's third law).

其中 r̂₁₂ 是从 q₁ 指向 q₂ 的单位矢量。若乘积 q₁q₂ 为正（同号电荷），则 F₁₂ 指向远离 q₂ 的方向，代表排斥力；若为负，则指向 q₂，代表吸引力。q₂ 所受的力大小相等、方向相反（牛顿第三定律）。

Students should be comfortable working with the vector form when dealing with more than two charges in a line or in a plane. The sign of the charges automatically gives the correct direction when using the vector equation, without needing separate sign rules.

学生应能熟练运用矢量形式来处理直线上或平面内多于两个电荷的情况。使用矢量方程时，电荷的符号会自动给出正确的方向，无需额外的符号规则。

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5. Superposition Principle | 叠加原理

When a charge experiences forces due to several other charges, the net electrostatic force is the vector sum of the individual forces. This is the principle of superposition. Mathematically, the net force on charge q₀ due to charges q₁, q₂, …, q_n is:

当一个电荷受到来自多个其他电荷的作用力时，净静电力是各个力的矢量之和，这就是叠加原理。在数学上，电荷 q₀ 受到来自 q₁、q₂、…、q_n 的净力为：

F_net = F₀₁ + F₀₂ + … + F_0n

Each force is calculated using Coulomb's law independently, as if the other charges were absent. The vector addition can be performed using components, which is the standard approach for A‑Level problems.

每个力用库仑定律单独计算，就像其他电荷不存在一样。矢量叠加可以通过分量法进行，这是解决 A‑Level 问题的标准方法。

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6. Forces Between Multiple Point Charges | 多个点电荷之间的力

In examination questions, you will often encounter three point charges placed along a straight line or at the vertices of a triangle. The most common task is to calculate the resultant force on one of the charges. The procedure is always the same:

在考试题目中，你经常会遇到三个点电荷排列在一条直线上或位于三角形顶点的情况。最常见的任务是计算其中一个电荷所受的合力。步骤总是相同的：

• Identify all pairs of interacting charges and determine the direction of each force (attraction or repulsion).

  确定所有相互作用的电荷对，并确定每个力的方向（吸引还是排斥）。

• Calculate the magnitude of each force using F = k |q_a q_b| / r².

  使用 F = k |q_a q_b| / r² 计算每个力的大小。

• Sketch clear free-body diagrams showing all force vectors.

  绘制清晰的受力图显示所有力矢量。

• Resolve the forces into perpendicular components if they are not collinear.

  如果力不共线，则将其分解为相互垂直的分量。

• Find the resultant force, giving both magnitude and direction.

  求出合力，并给出大小和方向。

For charges in a right‑angled triangle, using Pythagoras’ theorem and trigonometry is often sufficient. Always remember to state the direction relative to a specified axis or charge.

对于位于直角三角形中的电荷，使用勾股定理和三角函数通常就够了。务必牢记在表述方向时需指明相对于特定轴或电荷的方向。

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7. Comparison with Gravitational Force | 与万有引力的比较

There is a striking similarity between Coulomb's law and Newton's law of universal gravitation:

库仑定律与牛顿万有引力定律之间存在惊人的相似性：

Property / 性质	Electrostatic Force / 静电力	Gravitational Force / 引力
Law / 定律	F = k |q1q2| / r2	F = G m1m2 / r2
Constant / 常数	k ≈ 9.0 × 109 N m2 C−2	G ≈ 6.67 × 10−11 N m2 kg−2
Nature / 性质	Attractive or repulsive / 吸引或排斥	Always attractive / 总是吸引
Relative strength / 相对强度	Much stronger / 强得多	Extremely weak / 极弱

For example, the electrostatic force between two protons is about 10³⁶ times larger than the gravitational force between them. This comparison highlights why gravity is negligible in particle physics experiments but dominates on astronomical scales, where masses are huge and net charge is nearly zero.

例如，两个质子之间的静电力大约是它们之间引力的 10³⁶ 倍。这一比较说明了为什么在粒子物理实验中引力可以忽略，而在天体尺度上，由于质量巨大且净电荷几乎为零，引力才占主导地位。

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8. Electric Field of a Point Charge | 点电荷的电场

From Coulomb's law, we can derive the electric field strength E due to a point charge Q. By definition, E = F / q, where F is the force on a small positive test charge q placed at a distance r from Q. Substituting Coulomb's law gives:

由库仑定律，我们可以导出点电荷 Q 产生的电场强度 E。根据定义，E = F / q，其中 F 是放置在距离 Q 为 r 处的小正试探电荷 q 所受的力。代入库仑定律可得：

E = k |Q| / r²

The direction of E is radially outward from a positive Q and radially inward towards a negative Q. The electric field of a point charge is a radial field, and the field lines are straight lines emanating from (or converging on) the charge. This directly links Coulomb's law to the concept of electric field, which is a major topic in the CIE syllabus.

E 的方向由正 Q 径向向外，指向负 Q 径向向内。点电荷的电场是径向场，电场线是从电荷发出（或汇聚于电荷）的直线。这直接建立了库仑定律与电场概念之间的联系，而电场是 CIE 课程大纲中的一大主题。

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9. Coulomb’s Law in a Dielectric Medium | 介质中的库仑定律

When the charges are placed in an insulating material (a dielectric), the electrostatic force is reduced. The reduction is accounted for by introducing the relative permittivity ε_r (also called the dielectric constant) of the medium. The modified law is:

当电荷被置于绝缘材料（电介质）中时，静电力会减小。通过引入介质的相对介电常数 ε_r（也称为介电常数）来体现这种减小。修正后的定律为：

F_medium = (1 / (4πε₀ε_r)) × |q₁ q₂| / r²

For vacuum (and approximately for air), ε_r = 1. For other materials ε_r > 1; for example, water has ε_r tically weakens electrostatic interactions. This concept is essential for understanding capacitors with dielectrics.

对于真空（以及近似地对于空气），ε_r = 1。对于其他材料，ε_r > 1；例如，水在室温下的 ε_r ≈ 80，这极大地削弱了静电相互作用。这一概念对于理解带介质的电容器至关重要。

It is also worth noting that in a medium the constant k effectively becomes k / ε_r. Free‑body force calculations in a medium follow the same superposition rules, but with a reduced force magnitude.

同样值得注意的是，在介质中常数 k 实际上变为 k / ε_r。介质中的受力分析遵循相同的叠加规则，但力的大小有所减小。

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10. Experimental Verification: Coulomb’s Torsion Balance | 实验验证：库仑的扭秤

Coulomb verified the inverse‑square law using a torsion balance. A lightweight insulating rod with a small charged sphere at each end was suspended from a fine wire. A second charged sphere was brought close to one of the suspended spheres. The repulsive force twisted the wire, and the angle of twist was proportional to the force.

库仑通过扭秤验证了平方反比定律。一根轻质绝缘杆两端各带有一个小带电球，由一根细丝悬挂。另一个带电球被移近其中一个悬挂的小球。排斥力扭转了细丝，扭转角度与力的大小成正比。

By varying the distance between the spheres and the magnitude of the charges, Coulomb established that F ∝ 1/r² and F ∝ q₁q₂. Modern high‑precision experiments have confirmed the exponent of r to be −2 to within 1 part in 10¹⁶. A‑Level questions often ask

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