📚 AS Physics Unit 2 Mark Scheme Jun19: Formula Derivation | AS 物理 Unit 2 2019年6月评分标准与公式推导
This article focuses on the core formula derivations that frequently appear in the Edexcel AS Physics Unit 2 (WPH02) June 2019 mark scheme. Understanding these step-by-step derivations not only helps you secure method marks but also deepens your grasp of fundamental physics principles. We will walk through the key equations in mechanics, electricity, waves, and quantum physics, connecting them with typical exam requirements.
本文将重点讲解 Edexcel AS 物理 Unit 2 (WPH02) 2019年6月评分标准中经常出现的核心公式推导。逐步理解这些推导过程不仅有助于你在考试中获得方法分，还能加深你对基本物理原理的掌握。我们将逐一梳理力学、电学、波动和量子物理中的关键方程，并与典型的考试要求联系起来。
1. Deriving Kinetic Energy | 动能公式的推导
Kinetic energy (KE) is the energy an object possesses due to its motion. The standard expression can be derived from the work–energy principle and Newton’s second law.
动能 (KE) 是物体由于运动而具有的能量。它的标准表达式可以通过功能原理和牛顿第二定律推导出来。
Consider a constant resultant force F acting on an object of mass m, producing acceleration a over a displacement s. The work done by the force is:
假设一个恒定的合力 F 作用于质量为 m 的物体，产生加速度 a，并在位移 s 上做功。力所做的功为：
W = F s
Using Newton's second law, F = m a, we get:
使用牛顿第二定律 F = m a，可得：
W = m a s
Now apply the equation of motion linking initial velocity u, final velocity v, acceleration, and displacement:
现在应用将初速度 u、末速度 v、加速度和位移联系起来的运动学方程：
v² = u² + 2 a s
Rearrange to obtain the product a s:
重新整理以求得乘积 a s：
a s = (v² − u²) / 2
Substitute this into the work equation:
将 a s 代入功的表达式：
W = m × (v² − u²) / 2 = ½m v² − ½m u²
its kinetic energy. If the object starts from rest (u = 0), the kinetic energy gained is:
对物体做的功等于其动能的变化量。如果物体从静止开始 (u = 0)，那么获得的动能就是：
Eₖ = ½ m v²
This derivation is frequently assessed in the mark scheme; always state that work done equals the change in kinetic energy.
该推导经常在评分标准中被考查；请务必说明做功等于动能的变化量。
2. Proving v² = u² + 2 a s | 运动方程 v² = u² + 2 a s 的证明
This suvat equation can be derived from the definitions of velocity and acceleration without calculus, using average velocity.
这个匀加速运动方程可以在不借助微积分的情况下，利用平均速度的定义推导出来。
For uniform acceleration, displacement s is given by average velocity multiplied by time:
对于匀加速运动，位移 s 等于平均速度乘以时间：
s = ((u + v)/2) × t
Acceleration a is the rate of change of velocity:
加速度 a 是速度的变化率：
a = (v − u) / t ⇒ t = (v − u) / a
Substituting for t in the displacement equation eliminates time:
将时间 t 代入位移公式，消去时间：
s = (u + v)/2 × (v − u)/a
s = (v² − u²) / (2 a)
Multiplying both sides by 2a yields the required equation:
两边同时乘以 2a，即可得到所需方程：
v² = u² + 2 a s
In the Jun19 mark scheme, showing clear substitution steps is essential to gain full marks for a derivation question.
在2019年6月的评分标准中，清晰地展示代入步骤是获得推导题满分的关键。
3. Resistivity Formula R = ρ L / A | 电阻率公式 R = ρ L / A
Resistivity links the resistance of a uniform wire to its length and cross-sectional area. The mark scheme often expects a proportionality argument leading to the constant ρ.
电阻率将均匀导线的电阻与其长度和横截面积联系起来。评分标准通常要求先进行比例推理，再引入常数 ρ。
For a given material at constant temperature, resistance R is directly proportional to length L and inversely proportional to cross-sectional area A:
对于给定材料并在恒温条件下，电阻 R 与长度 L 成正比，与横截面积 A 成反比：
R ∝ L / A
Introducing a constant of proportionality ρ, which depends on the material, gives:
引入取决于材料的比例常数 ρ，可得：
R = ρ L / A
The unit of resistivity can be derived as Ω m. In the mark scheme, writing 'constant = ρ' often enough to secure the mark, but you should also state the proportional relationships.
电阻率的单位可推导出 Ω·m。在评分标准中，写出“常数 = ρ”通常就能得分，但你还需要清晰表述比例关系。
4. Power as Force × Velocity | 功率 P = F v
moves its point of application at a constant speed, the useful expression P = F v emerges.
机械功率是做功的快慢。当恒力以恒定速度移动其作用点时，便能得到有用的表达式 P = F v。
Work done by a force over a small displacement Δs is:
力在一小段位移 Δs 上做的功为：
W = F Δs
Power is work done divided by time taken:
功率是功除以所用时间：
P = W / Δt = F Δs / Δt
Since velocity v = Δs / Δt, we obtain:
因为速度 v = Δs / Δt，所以得到：
P = F v
This derivation is especially relevant for questions involving vehicles moving at constant speed against resistive forces.
该推导对于涉及车辆以恒定速度行驶克服阻力的问题尤其重要。
5. Electrical Power P = I V, P = I² R, and P = V² / R | 电功率的三种形式
Starting from the definition of potential difference and current, we can derive the basic power formula and then combine it with Ohm’s law.
从电势差和电流的定义出发，我们可以推导出基本的功率公式，再结合欧姆定律进行变形。
per unit charge: W = Q V. Current I is the rate of flow of charge: I = Q / t, so Q = I t.
电势差 V 是每单位电荷所做的功：W = Q V。电流 I 是电荷流动的速率：I = Q / t，所以 Q = I t。
Substituting Q into the work equation and dividing by time gives power:
将 Q 代入功的表达式并除以时间，得到功率：
P = W / t = (Q V) / t = (I t V) / t = I V
For ohmic conductors, V = I R. Substituting V or I yields the alternative forms:
对于欧姆导体，V = I R。代入 V 或 I 可得出其他形式：
P = I × (I R) = I² R
P = (V / R) × V = V² / R
tement P = I V and then the correct substitution from V = I R to obtain the required form.
评分标准通常认可先写出 P = I V，然后正确代入 V = I R 来得到所需形式。
6. EMF and Internal Resistance: ε = I (R + r) | 电动势与内阻公式的推导
A real source of emf has internal resistance r, causing the terminal potential difference to be less than the emf when current flows. The energy conservation approach leads to the key equation.
实际的电动势源具有内阻 r，当有电流流过时，端电压会低于电动势。从能量守恒出发可推导出关键方程。
The emf ε is the energy supplied per unit charge. This energy is partly used in the external resistance R and partly dissipated in the internal resistance r:
电动势 ε 是每单位电荷提供的能量。这部分能量一部分在外电阻 R 上被利用，另一部分在内阻 r 上消耗：
ε = VR + Vr
Using Ohm’s law, VR = I R and Vr = I r. Therefore:
根据欧姆定律，VR = I R，Vr = I r。因此：
ε = I R + I r = I (R + r)
lost-volts experiments. The Jun19 paper specifically tested the rearrangement V = ε − I r.
该公式是电位差计和“损失电压”实验的核心。2019年6月的试卷专门考查了其变形 V = ε − I r。
7. Drift Velocity I = n A v e | 漂移速度方程的推导
The drift velocity of charge carriers in a conductor is derived by considering the volume of charge passing a cross-section per unit time.
导体中载流子的漂移速度可以通过考虑单位时间内通过横截面的电荷量来推导。
In a time Δt, the charges travel a distance v Δt. The volume of charge carriers passing a cross-sectional area A is A v Δt.
在时间 Δt 内，电荷移动的距离为 v Δt。通过横截面积 A 的载流子体积为 A v Δt。
The number of charge carriers in this volume is n A v Δt, where n is the number density. The total charge ΔQ is the number of carriers multiplied by the elementary charge e:
该体积内的载流子数目为 n A v Δt，其中 n 是数密度。总电荷量 ΔQ 等于载流子数目乘以元电荷 e：
ΔQ = n A v Δt e
Current I is charge per unit time:
电流 I 是单位时间内的电荷量：
I = ΔQ / Δt = n A v e
In the mark scheme, defining n clearly and showing the volume argument are crucial steps.
在评分标准中，清楚地定义 n 并展示体积推理是关键步骤。
8. Photon Energy and Laser Power | 光子能量与激光功率的推导
The wave–particle duality of light is expressed through the photon model. The mark scheme often links photon energy to the power of a monochromatic source.
光的波粒二象性通过光子模型得以体现。评分标准经常将光子能量与单色光源的功率联系起来。
The energy of a single photon is given by Planck’s equation:
单个光子的能量由普朗克方程给出：
E = h f = h c / λ
For a laser emitting N photons in time t, the total energy delivered is N h f, so the power is:
对于在时间 t 内发射 N 个光子的激光，其输出的总能量为 N h f，因此功率为：
P = (N h f) / t
If the number of photons per second (N/t) is known, the intensity or power can be related to wavelength. The Jun19 mark scheme rewarded explicit linking of these relationships.
如果每秒钟的光子数 (N/t) 已知，则可以将强度或功率与波长联系起来。2019年6月的评分标准对明确表述这些关系的作答给予了分数。
9. Refractive Index and Critical Angle | 折射率与临界角的公式推导
Total internal reflection occurs when light travels from a denser medium to a less dense one at an angle greater than the critical angle C. Snell’s law gives the link.
当光从光密介质射向光疏介质且入射角大于临界角 C 时，就会发生全内反射。斯涅尔定律给出了两者之间的联系。
Snell’s law at the boundary between medium of refractive index n and air (n ≈ 1) is:
在折射率为 n 的介质与空气 (n ≈ 1) 的交界面，斯涅尔定律为：
n sin θ₁ = 1 × sin θ₂
At the critical angle, the angle of refraction is 90°, so sin θ₂ = sin 90° = 1. Therefore:
在临界角处，折射角为 90°，所以 sin θ₂ = sin 90° = 1。因此：
n sin C = 1 ⇒ sin C = 1 / n or n = 1 / sin C
This simple derivation is often tested with a requirement to state Snell’s law first.
这个简单的推导经常被考查，并且通常要求先写出斯涅尔定律。
10. Wave Speed v = f λ | 波速公式的推导
The wave equation is derived directly from the definitions of frequency and wavelength, connecting the wave’s speed to its temporal and spatial periods.
波动方程直接从频率和波长的定义导出，将波速与时间周期和空间周期联系起来。
the time period T = 1 / f. In one period T, a wave advances by precisely one wavelength λ.
频率 f 是每秒完整振动的次数，因此周期 T = 1 / f。在一个周期 T 内，波刚好前进一个波长 λ。
Speed is distance divided by time:
速度等于距离除以时间：
v = λ / T = λ / (1 / f) = f λ
This derivation is a staple of Unit 2 mark schemes, where a simple substitution is sufficient.
这个推导是 Unit 2 评分标准中的常客，只需简单的代入即可得分。
11. Stress, Strain, and the Young Modulus | 应力、应变与杨氏模量
The Young modulus describes the stiffness of a material within its elastic limit. The defining equation is derived from the ratio of tensile stress to tensile strain.
杨氏模量描述了材料在弹性限度内的刚度。其定义方程来源于拉伸应力与拉伸应变的比值。
Tensile stress is the force applied per unit cross-sectional area:
拉伸应力是单位横截面积上施加的力：
Stress = F / A
Tensile strain is the extension per unit original length:
拉伸应变是单位原始长度的伸长量：
Strain = ΔL / L
The Young modulus E is defined as the stress divided by strain, provided Hooke’s law is obeyed:
杨氏模量 E 定义为应力与应变的比值，前提是满足胡克定律：
E = (F / A) / (ΔL / L) = (F L) / (A ΔL)
In the mark scheme, you must give the formula and state the quantities clearly, often with the unit Pa or N m⁻².
在评分标准中，必须写出公式并清楚地说明各物理量，通常会涉及单位 Pa 或 N m⁻²。
12. Deriving the Efficiency Equation | 效率公式的推导
Efficiency is a concept that frequently appears in energy-transfer contexts, and its derivation from power or energy ratios is straightforward but examiners pay attention to the use of correct forms.
效率是能量转化情境中经常出现的概念，它从功率比或能量比推导而来虽简单，但考官会关注所用公式是否正确。
Efficiency η is defined as the ratio of useful output power (or energy) to total input power (or energy):
效率 η 定义为有用输出功率（或能量）与总输入功率（或能量）的比值：
η = Useful output power / Total input power
Since power is energy per unit time, the same ratio holds for energy:
由于功率是单位时间的能量，因此该比值对能量同样适用：
η = Useful output energy / Total input energy
e Jun19 mark scheme often requires stating that total power = useful power + wasted power.
要将效率表示为百分比，需乘以 100。2019年6月的评分标准经常要求说明总功率 = 有用功率 + 浪费功率。
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