📚 Mastering Key Formula Derivations from Edexcel IAS Physics Unit 2 (PH02) Specimen Paper | 精通Edexcel IAS物理Unit 2(PH02)样卷关键公式推导
The Edexcel International AS Physics Unit 2 (PH02) specification covers waves, electricity and the nature of light. In the 2016 specimen paper, candidates are tested on their ability to derive and apply key equations. This article walks you through the essential derivations, helping you understand the logic behind each formula rather than just memorising it. Mastering these derivations will boost your problem-solving skills and exam confidence.
Edexcel International AS物理Unit 2（PH02）课程涵盖波、电和光的本质。在2016年的样卷中，考生需要能够推导并应用关键公式。本文将带你梳理这些重要推导过程，帮助你理解每个公式背后的逻辑，而不仅是死记硬背。掌握这些推导将提升你的解题能力和考试信心。
1. Wave Speed v = fλ | 波速公式推导
The speed of a wave is the distance travelled per unit time. One full wavelength λ is the distance between two consecutive crests. The time taken for a wave to travel one wavelength is the period T. Therefore speed v = λ / T. Since frequency f = 1 / T, we substitute to obtain v = fλ.
波速是单位时间内传播的距离。一个完整波长λ是两个相邻波峰之间的距离。波传播一个波长所需的时间是周期T。因此速度v = λ / T。由于频率f = 1 / T，代入后得到v = fλ。
v = f λ
This derivation holds for all waves – mechanical or electromagnetic – provided the medium is nondispersive so that v is constant.
该推导适用于所有波，无论是机械波还是电磁波，只要介质没有频散，即波速恒定。
2. Snell’s Law of Refraction n1 sin θ1 = n2 sin θ2 | 斯涅尔折射定律推导
Consider a wavefront AB incident on a boundary between two media. In medium 1, speed v1; in medium 2, speed v2. During the time t the edge A travels distance v1t to reach point C, while edge B travels v2t to D. From geometry, sin θ1 = v1t / AC and sin θ2 = v2t / AC. Dividing gives sin θ1 / sin θ2 = v1 / v2. Defining refractive index n = c / v, we get n1 sin θ1 = n2 sin θ2.
考虑入射到两种介质界面的波前AB。在介质1中速度为v1，在介质2中速度为v2。在时间t内，边缘A移动v1t到达点C，而边缘B移动v2t到达D。由几何关系，sin θ1 = v1t / AC，sin θ2 = v2t / AC。两式相除得sin θ1 / sin θ2 = v1 / v2。定义折射率n = c / v，可得n1 sin θ1 = n2 sin θ2。
n1 sin θ1 = n2 sin θ2
This derivation based on Huygens’ principle explains why light changes direction when crossing a boundary between materials of different optical densities.
这一基于惠更斯原理的推导解释了光在穿越不同光密度介质界面时为何改变方向。
3. Diffraction Grating Equation d sin θ = nλ | 衍射光栅方程推导
For a transmission grating with slit separation d, each slit acts as a coherent source. Constructive interference occurs when the path difference between adjacent slits equals an integer multiple of the wavelength. From the diagram, the extra distance travelled by the wave from the lower slit is d sin θ. Therefore, the condition for bright fringes is d sin θ = nλ, where n = 0, ±1, ±2, …
对于缝间距为d的透射光栅，每个狭缝作为相干的次波源。当相邻狭缝的光程差等于波长的整数倍时发生相长干涉。由几何图可知，下方狭缝的波多走的距离为d sin θ。因此，亮纹条件为d sin θ = nλ，其中n = 0, ±1, ±2, …
d sin θ = nλ
This equation allows us to determine the wavelength of light or the slit spacing by measuring the angle θ for a given order n. It is a core tool in spectroscopy.
通过测量某一级次n对应的角度θ，该方程可用于测定光波长或狭缝间距。它是光谱学中的核心工具。
4. Photon Energy E = hf | 光子能量公式
Max Planck proposed that electromagnetic radiation is emitted and absorbed in discrete quanta, each with energy E = hf, where h is Planck’s constant (6.63 × 10-34 J s) and f is the frequency. Albert Einstein extended this idea to explain the photoelectric effect, treating light as a stream of photons. The energy of a single photon is therefore directly proportional to its frequency.
普朗克提出电磁辐射以分立的量子形式发射和吸收，每个量子能量E = hf，其中h为普朗克常量（6.63 × 10-34 J s），f为频率。爱因斯坦推广这一思想以解释光电效应，将光视为光子流。因此单个光子的能量与其频率成正比。
Ephoton = h f
Using the wave equation c = fλ, we can also write E = hc / λ. This formula underpins the calculation of photon energies across the electromagnetic spectrum.
结合波速公式c = fλ，还可写为E = hc / λ。该公式是计算整个电磁波谱光子能量的基础。
5. Electronvolt and Kinetic Energy Conversion | 电子伏特与动能转换
An electronvolt (eV) is the energy gained by an electron when it is accelerated through a potential difference of 1 volt. Energy = charge × voltage, so 1 eV = 1.60 × 10-19 C × 1 V = 1.60 × 10-19 J. For an electron accelerated through a p.d. V, its kinetic energy Ek = eV. Thus we can express photon energies or work functions conveniently in eV.
电子伏特（eV）是一个电子经过1伏特电势差加速后所获得的能量。能量 = 电荷量 × 电压，因此1 eV = 1.60 × 10-19 C × 1 V = 1.60 × 10-19 J。对于经电压V加速的电子，其动能Ek = eV。这样我们可以方便地用eV表示光子能量或功函数。
1 eV = 1.60 × 10-19 J
In the PH02 specimen, you may need to convert between joules and eV when working with photoelectricity or energy levels. Always remember to multiply or divide by the elementary charge e.
在PH02样卷中，处理光电效应或能级问题时可能需要你在焦耳和电子伏特之间转换。请始终牢记乘以或除以基本电荷e。
6. Resistivity Formula R = ρL / A | 电阻率公式推导
For a uniform conductor, resistance R depends on length L, cross-sectional area A, and a material property called resistivity ρ. Experimentally, R ∝ L and R ∝ 1/A. Combining these gives R = ρ (L / A), where ρ is the constant of proportionality. The unit of resistivity is ohm-metre (Ω m).
对于均匀导体，电阻R取决于长度L、横截面积A以及材料的电阻率ρ。实验表明，R ∝ L 且 R ∝ 1/A。将两者结合得R = ρ (L / A)，其中ρ为比例常数。电阻率的单位是欧姆·米（Ω m）。
R = ρ L / A
This derivation assumes ohmic behaviour and uniform cross-section. Resistivity depends on temperature and material structure.
该推导假设欧姆行为和均匀截面。电阻率受温度和材料微观结构影响。
7. Electrical Power Dissipation P = IV = I²R = V²/R | 电功率公式推导
Power P is the rate of energy transfer. In a circuit component, moving charge Q through a p.d. V does work W = QV. Current I = Q/t, so P = W/t = (QV)/t = IV. For a resistor, V = IR, so substituting gives P = I(IR) = I²R. Alternatively, I = V/R gives P = V(V/R) = V²/R. These three forms are used depending on which quantities are known.
功率P是能量转换的速率。在电路元件中，使电荷Q通过电势差V做功W = QV。电流I = Q/t，因此P = W/t = (QV)/t = IV。对于电阻，V = IR，代入得P = I(IR) = I²R。或者由I = V/R得P = V(V/R) = V²/R。这三个形式视已知量而选用。
P = I V = I² R = V² / R
These relationships are fundamental for analysing energy dissipation in resistors and designing safe circuits that do not overheat.
这些关系是分析电阻能耗以及设计不会过热的电路的基础。
8. Derivation of Equivalent Resistance: Series and Parallel | 串并联等效电阻推导
Series: The same current I flows through resistors R1 and R2. Total voltage V = V1 + V2 = IR1 + IR2 = I(R1+R2). Therefore total resistance Rtotal = V/I = R1+R2. For n resistors in series, Rtotal = ΣRi.
串联：相同电流I流过电阻R1和R2。总电压V = V1 + V2 = IR1 + IR2 = I(R1+R2)。因此总电阻Rtotal = V/I = R1+R2。对于n个串联电阻，Rtotal = ΣRi。
Parallel: The same voltage V is across each resistor. Total current I = I1 + I2 = V/R1 + V/R2 = V(1/R1+1/R2). From V = I Rtotal, we get 1/Rtotal = I/V = 1/R1 + 1/R2. In general, 1/Rtotal = Σ(1/Ri).
并联：各电阻两端电压相同为V。总电流I = I1 + I2 = V/R1 + V/R2 = V(1/R1+1/R2)。由V = I Rtotal可得1/Rtotal = I/V = 1/R1 + 1/R2。一般地，1/Rtotal = Σ(1/Ri)。
Series: Rtotal = R1 + R2 + …
Parallel: 1/Rtotal = 1/R1 + 1/R2 + …
These derivations stem directly from Kirchhoff’s laws and are essential for analysing any circuit.
这些推导直接来源于基尔霍夫定律，对于分析任何电路都至关重要。
9. EMF and Internal Resistance ε = I(R + r) | 电动势与内阻关系推导
A real source of emf (such as a battery) has an internal resistance r. The terminal voltage V across the source when delivering current I is less than the emf ε because some energy is dissipated internally. Applying Kirchhoff’s voltage law to the circuit: ε – Ir – IR = 0, hence ε = I(R + r), where R is the external load. Rearranging, V = ε – Ir.
真实的电动势源（如电池）具有内阻r。当输出电流I时，电源两端电压V小于电动势ε，因为一部分能量在内阻上耗散。对电路应用基尔霍夫电压定律：ε – Ir – IR = 0，从而ε = I(R + r)，其中R为外负载。移项得V = ε – Ir。
ε = I (R + r) and V = ε – I r
The terminal voltage V equals the emf only when no current flows (open circuit). This explains why a battery’s measured voltage drops under load.
端电压V仅在无电流（断路）时才等于电动势。这解释了负载下电池端电压下降的现象。
10. Einstein’s Photoelectric Equation hf = φ + Kmax | 爱因斯坦光电效应方程推导
In the photoelectric effect, a photon of energy hf strikes a metal surface. If hf exceeds the work function φ (the minimum energy to release an electron), the electron is emitted with maximum kinetic energy Kmax. Energy conservation yields: hf = φ + Kmax. Kmax can be measured as the stopping potential Vs times e: Kmax = eVs. Thus hf = φ + eVs.
在光电效应中，能量为hf的光子撞击金属表面。若hf大于功函数φ（释放电子所需的最小能量），电子将以最大动能Kmax逸出。根据能量守恒：hf = φ + Kmax。Kmax可通过遏止电势Vs测量：Kmax = eVs。因此hf = φ + eVs。
h f = φ + ½ m v2max
This equation beautifully explains the threshold frequency, the instantaneous emission, and the independence of Kmax on light intensity. It was a milestone in quantum physics.
该方程精妙地解释了截止频率、瞬时发射以及Kmax与光强无关。这是量子物理学的一个里程碑。
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