📚 Relating SI Units | 关联国际单位制
In physics, quantities are measured and communicated using a coherent system of units. The International System of Units (SI) provides a set of seven base units from which all other physical quantities can be expressed. Being able to relate derived units back to these base units is an essential skill at A‑Level, as it strengthens conceptual understanding and allows you to verify the consistency of equations through dimensional analysis. This article systematically explains how common mechanical and electrical units are built from mass, length, time, electric current and other base quantities.
在物理学中，量的测量和交流依赖于一套协调的单位制。国际单位制（SI）给出了七个基本单位，所有其他物理量都可以用它们表示。能够将导出单位与基本单位联系起来是 A‑Level 阶段的核心技能，因为它不仅能加深概念理解，还能通过量纲分析检验方程的合理性。本文系统讲解如何用质量、长度、时间、电流等基本量构建常见的力学和电学单位。
1. The Seven Base Units | 七个基本单位
The SI is founded on seven carefully defined base quantities, each with its own base unit. Every other unit in physics can be expressed as a product of powers of these base units.
国际单位制建立在七个被精确定义的基本量之上，每个基本量都有自己的基本单位。物理学中其他所有单位都可以表示为这些基本单位的幂次乘积。
| Base quantity | 基本量 | Base unit | 基本单位 | Symbol | 符号 |
|---|---|---|
| Mass | 质量 | kilogram | 千克 | kg |
| Length | 长度 | metre | 米 | m |
| Time | 时间 | second | 秒 | s |
| Electric current | 电流 | ampere | 安培 | A |
| Thermodynamic temperature | 热力学温度 | kelvin | 开尔文 | K |
| Amount of substance | 物质的量 | mole | 摩尔 | mol |
| Luminous intensity | 发光强度 | candela | 坎德拉 | cd |
Memorising these seven is the starting point for relating all other units. At A‑Level, most derivations only need the first five (kg, m, s, A, K) because the mole and candela are used in specialised topics.
记住这七个单位是关联其他所有单位的起点。在 A‑Level 阶段，多数推导只需要前五个（kg、m、s、A、K），因为摩尔和坎德拉仅在特定专题中使用。
2. How Derived Units Are Formed | 导出单位如何形成
A derived unit is any unit that is not one of the seven base units. It is obtained by combining base units according to the defining equation of the physical quantity. The resulting combination can involve multiplication, division and powers. For example, speed is defined as distance divided by time, so its unit is metre per second (m s⁻¹).
导出单位是指七个基本单位之外的任何单位。它根据物理量的定义方程，通过基本单位的组合得到。这种组合可以包含乘、除和幂运算。例如，速率定义为距离除以时间，因此其单位是米每秒（m s⁻¹）。
The process is straightforward: replace each quantity in the defining equation with its unit in terms of base units, and simplify. This systematic approach ensures that every derived unit is dimensionally consistent.
方法很简单：用基于基本单位的单位替换定义方程中的每个量，然后化简。这种系统的方法能确保每个导出单位在量纲上都是协调的。
3. Expressing Force: The Newton | 表示力：牛顿
ocity per unit time, so its unit is m s⁻². Therefore the unit of force, the newton (N), can be expressed as kg m s⁻².
牛顿第二定律指出合力等于质量乘以加速度：F = m a。加速度是单位时间内的速度变化，其单位为 m s⁻²。因此力的单位牛顿（N）可表示为 kg m s⁻²。
1 N = 1 kg m s⁻²
helps when working with spring constant (N m⁻¹ = kg s⁻²) or pressure (N m⁻² = kg m⁻¹ s⁻²).
当检查涉及重力、张力、摩擦力或任何力项的方程量纲时，这一关系会频繁用到。它也有助于处理弹簧常数（N m⁻¹ = kg s⁻²）或压强（N m⁻² = kg m⁻¹ s⁻²）。
4. Work and Energy: The Joule | 功和能量：焦耳
Work done is the product of force and displacement in the direction of the force. Since force is measured in newtons and displacement in metres, the unit of work, the joule (J), is N m. Substituting the base unit expression for the newton gives:
功等于力与沿力方向位移的乘积。既然力的单位是牛顿，位移的单位是米，功的单位焦耳（J）就是 N m。代入牛顿的基本单位表达式可得：
1 J = 1 N m = 1 kg m² s⁻²
The same base unit combination applies to all forms of energy – kinetic energy (½ m v²), gravitational potential energy (m g h) and internal energy. Recognising this shared pattern reinforces the idea that energy is a unified concept.
相同的基本单位组合适用于所有形式的能量——动能（½ m v²）、重力势能（m g h）和内能。认识到这一共同模式有助于强化“能量是一个统一概念”的观念。
5. Power: The Watt | 功率：瓦特
has the unit J and time s, the unit of power, the watt (W), is simply a joule per second.
功率是做功的速率，P = W / t 或 P = ΔE / t。既然功的单位是 J，时间的单位是 s，功率的单位瓦特（W）就是焦耳每秒。
1 W = 1 J s⁻¹ = 1 kg m² s⁻³
er see that this reduces to the same base units, confirming the consistency between mechanical and electrical power.
在电学情境中，功率也可写成 P = I V，后面你会看到它可化简为相同的基本单位，这印证了机械功率与电功率之间的一致性。
6. Electric Charge and Current: The Coulomb | 电荷与电流：库仑
rearranges to Q = I t. Therefore the unit of charge, the coulomb (C), is an ampere second.
电流是一个基本量，但电荷是从它导出的。其定义关系为 I = Q / t，整理后得 Q = I t。因此电荷的单位库仑（C）是安培秒。
1 C = 1 A s
ce, the elementary charge e = 1.60 × 10⁻¹⁹ C, linking micro- and macroscopic descriptions.
这个简单关系对于连接电学单位和力学单位至关重要。例如，元电荷 e = 1.60 × 10⁻¹⁹ C，将微观描述与宏观描述联系起来。
7. Potential Difference: The Volt | 电势差：伏特
pressions for work (J = kg m² s⁻²) and charge (C = A s), we obtain the volt in base units:
电势差（电压）定义为单位电荷转移的能量：V = W / Q。利用功（J = kg m² s⁻²）和电荷（C = A s）的基本单位表达式，可得伏特用基本单位表示为：
1 V = 1 J C⁻¹ = 1 kg m² s⁻³ A⁻¹
d in the definition of the electronvolt (eV = 1.60 × 10⁻¹⁹ J), a convenient energy unit in particle physics.
这个表达式会出现在许多变换后的电学方程中，例如推导电阻或电容的单位时。它也用于电子伏特（eV = 1.60 × 10⁻¹⁹ J）的定义，后者是粒子物理中常用的能量单位。
8. Resistance: The Ohm | 电阻：欧姆
g the base unit representation of the volt and the ampere gives the ohm (Ω) in base units:
欧姆定律将电阻定义为电势差与电流的比值：R = V / I。代入伏特和安培的基本单位表示，可得欧姆（Ω）用基本单位表示：
1 Ω = 1 V A⁻¹ = 1 kg m² s⁻³ A⁻²
You can test your understanding by deriving the base units of resistivity (Ω m) or conductivity (Ω⁻¹ m⁻¹). Such exercises strengthen fluency with unit transformations and are often part of A‑Level exam questions.
你可以通过推导电阻率（Ω m）或电导率（Ω⁻¹ m⁻¹）的基本单位来检验理解。这种练习能提高单位转换的熟练度，也是 A‑Level 考试中常见的题型。
9. Other Important Derived Units | 其他重要的导出单位
is force per unit area, so 1 Pa = 1 N m⁻² = 1 kg m⁻¹ s⁻². Density is mass per volume, kg m⁻³. Frequency (hertz, Hz) is cycles per second, s⁻¹.
力学和材料学中出现许多其他量，每一个都可用 kg、m、s 表示。以下是几个关键例子：压强（帕斯卡，Pa）是单位面积上的力，因此 1 Pa = 1 N m⁻² = 1 kg m⁻¹ s⁻²。密度是质量除以体积，kg m⁻³。频率（赫兹，Hz）是周期每秒，s⁻¹。
In thermodynamics, temperature is a base quantity, but the Boltzmann constant kB connects it to energy: kB is underlines how SI base units provide a unified language across all branches of physics.
在热力学中，温度是基本量，但玻尔兹曼常数 kB 将它与能量联系起来：kB 的单位为 J K⁻¹ = kg m² s⁻² K⁻¹。这凸显了 SI 基本单位如何为物理学所有分支提供统一的语言。
10. Dimensional Analysis: A Powerful Checking Tool | 量纲分析：强大的检验工具
le is that both sides of a valid equation must have the same base unit dimensions. For example, consider the equation for the period of a simple pendulum: T = 2π √l/g side: length l has dimension L, g (acceleration) has dimension L T⁻², so √(L / (L T⁻²)) = √T² = T. Both sides match, confirming the equation is dimensionally correct.
量纲分析利用导出量的基本单位表达式来验证物理方程的内部一致性。原理是：一个有效方程的两边必须具有相同的基本单位量纲。例如，考虑单摆的周期方程：T = 2π √l/g。左边量纲为 T（时间）。右边：长度 l 量纲为 L，g（加速度）量纲为 L T⁻²，因此 √(L / (L T⁻²)) = √T² = T。两边一致，确认了方程在量纲上是正确的。
When you derive an unfamiliar formula, quickly checking its dimensions can catch algebraic slips. It also helps when choosing between similar‑looking equations in multiple‑choice questions.
当你推导一个不熟悉的公式时，快速检查量纲可以发现代数错误。在选择题中面对相似的方程时，这也有助于作出选择。
11. Prefixes and Scaling | 前缀与比例缩放
SI prefixes such as kilo (k, 10³), mega (M, 10⁶), milli (m, 10⁻³) and micro (µ of ten correctly. For instance, 1 km = 10³ m, so 1 km² = (10³ m)² = 10⁶ m². Similarly, 1 cm = 10⁻² m, so 1 cm³ = 10⁻⁶ m³.
SI 前缀如千（k, 10³）、兆（M, 10⁶）、毫（m, 10⁻³）和微（µ, 10⁻⁶）附加在基本单位或导出单位前，用于简洁地表示极大或极小的数值。在关联单位时，必须正确处理这些十的幂次。例如，1 km = 10³ m，因此 1 km² = (10³ m)² = 10⁶ m²。类似地，1 cm = 10⁻² m，因此 1 cm³ = 10⁻⁶ m³。
A common error is to forget squaring or cubing the prefix factor when converting areas or volumes. Practising with milliamperes (mA), kilohms (kΩ) and microcoulombs (µC) will reinforce the need to convert everything to base units before starting a calculation.
常犯的错误是在转换面积或体积时忘记对前缀因子进行平方或立方运算。多练习毫安（mA）、千欧（kΩ）和微库（µC）的单位转换，能强化“计算前先将所有单位转换为基本单位”的习惯。
12. Putting It All Together in Exam Scenarios | 在考试情境中综合运用
CIE A‑Level Physics papers often include questions that ask you to “express this unit in terms of SI base units” or to “show that the units on both sides of the equation are the same” onsider a question requiring the base units of capacitance (C = Q / V). Q = A s, V = kg m² s⁻³ A⁻¹, so capacitance = C = (A s) / (kg m² s⁻³ A⁻¹) = kg⁻¹ m⁻² s⁴ A². This matches the farad (F).
CIE A‑Level 物理试卷中常有这样的题目：要求“用 SI 基本单位表示该单位”或“证明方程两边的单位相同”。系统的步骤是关键：写出定义方程，代入各项的 SI 基本单位表达式，利用指数运算法则化简。以电容（C = Q / V）的基本单位为例：Q = A s，V = kg m² s⁻³ A⁻¹，因此电容 = C = (A s) / (kg m² s⁻³ A⁻¹) = kg⁻¹ m⁻² s⁴ A²。这符合法拉（F）的单位。
Build speed by memorising the core derived expressions (N, J, W, C, V, Ω) in base units. Then, when faced with a complex quantity such as magnetic flux density, you can break it down step by step. Ultimately, relating SI units transforms abstract definitions into concrete, testable relationships.
通过熟记核心导出表达式（N、J、W、C、V、Ω）的基本单位形式，可以提高解题速度。之后，当遇到磁通量密度等复杂量时，你就能逐步拆解。最终，关联国际单位制能将抽象的定义转化为具体、可检验的关系。
Published by TutorHao | Physics Revision Series | aleveler.com
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