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AS Physics Unit 2 (June 2019) – Essential Formula Derivations | AS物理单元2（2019年6月）必备公式推导

2026-07-13 03:21 · 本文部分内容由AI辅助创作

📚 AS Physics Unit 2 (June 2019) – Essential Formula Derivations | AS物理单元2（2019年6月）必备公式推导

The June 2019 AS Physics Unit 2 examination paper required students not only to recall standard equations but also to demonstrate a thorough understanding of their origins. Many questions tested the ability to derive key formulas from first principles, linking concepts in electricity, waves, and quantum physics. This article walks through the essential derivations that underpinned the paper, breaking down each logical step so you can tackle similar challenges with confidence.

2019年6月的AS物理单元2试卷不仅要求学生记住标准公式，更要求对公式的来源有透彻理解。许多题目考查了从基本原理推导核心公式的能力，涵盖电学、波动和量子物理等概念。本文将逐一剖析这些关键推导，拆解每一步逻辑，帮助你从容应对类似问题。

1. Why Unit 2 Rewards Derivations | 为什么Unit 2如此重视公式推导

In Unit 2, examiners frequently award marks for the logical chain, not just the final boxed answer. A derivation question typically carries three to five marks, each tied to a specific step: stating the starting relationship, substituting correctly, rearranging symbolically, and simplifying. If you jump straight from the question to the answer, you can lose most of those marks even when the final expression is right.

在Unit 2中，考官给分往往看重整条逻辑链，而不仅是最后框出的答案。一道推导题通常有3到5分，每一分都对应一个具体步骤：写出起始关系式、正确代入、用符号整理、化简。如果你从题目直接跳到答案，即使最终表达式正确，也可能丢掉大部分分数。

hile R depends on shape as well, helps you answer explain questions about why a thick short wire has a lower resistance than a thin long one of the same metal.

推导题还能暴露你是否真正理解每个符号的物理意义。例如，知道 R = ρL / A 中的 ρ 是材料属性，而 R 还取决于形状，就能回答"为什么同样金属制成的粗短导线比细长导线电阻更小"这类解释题。

Quantity 物理量Symbol 符号Unit 单位
Resistance 电阻RΩ
Resistivity 电阻率ρΩ m
Current 电流IA
Drift velocity 漂移速度vm s⁻¹
Number density 数密度nm⁻³
Planck constant 普朗克常量hJ s

2. Deriving the Resistivity Formula | 电阻率公式的推导

For a uniform conductor, resistance depends on its geometry. Experiments show that R is directly proportional to the length L and inversely proportional to the cross-sectional area A. This can be written as R ∝ L / A, which becomes an equation once a proportionality constant is introduced.

对于均匀导体，电阻取决于其几何形状。实验表明，R与长度L成正比，与横截面积A成反比，即 R ∝ L / A。引入比例常数后即可写成等式。

The constant of proportionality is the resistivity ρ, a property that depends only on the material and its temperature, not on the shape of the sample. Therefore we obtain the defining equation:

该比例常数就是电阻率 ρ，它只与材料及其温度有关，而与样品的形状无关。于是得到定义式：

R = ρ L / A

Rearranging for ρ gives the form most often used in practical work, where you measure R, L and the diameter of a wire to find the resistivity of its metal.

移项求 ρ 得到实验中最常用的形式：通过测量导线的 R、L 和直径，即可求出该金属的电阻率。

ρ = R A / L

ivity increases with temperature for a metal, because lattice vibrations scatter the charge carriers more strongly.

由于 A = π r² = π d² / 4，直径为 d 的导线的工作公式为 ρ = R π d² / (4L)。请记住：金属的电阻率随温度升高而增大，因为晶格振动对载流子的散射更强。

3. Deriving the Drift Velocity Equation | 漂移速度方程的推导

Imagine a metallic conductor of with cross-sectional area A containing n free electrons per unit volume, each carrying charge e. When a voltage is applied, the electrons acquire an average drift velocity v along the wire. In a time interval t, each electron travels a distance v t, so the volume of the region swept out is A v t.

设想一个横截面积为A的金属导体，每单位体积含n个自由电子，每个电子带电荷e。施加电压后，电子沿导线获得平均漂移速度v。在时间t内，每个电子移动距离v t，因此扫过的体积为A v t。

passing a given cross-section is Q = n A v t e. Dividing by the time t gives the current.

该体积内的电子数为 n A v t，每个电子带电荷e，因此通过某横截面的总电荷为 Q = n A v t e。再除以时间t即得电流。

I = Q / t = n A v e

Rearranged for the drift velocity:

移项得到漂移速度：

v = I / (n A e)

order of 10⁻⁴ m s⁻¹. This is very slow compared with the signal speed, because the electric field is established along the wire almost instantly.

典型数值有助于检验答案。对铜而言 n ≈ 8 × 10²⁸ m⁻³，e = 1.6 × 10⁻¹⁹ C，若截面积 1 × 10⁻⁶ m² 的导线中通有1 A电流，则 v 约为 10⁻⁴ m s⁻¹ 量级。这比信号传播速度慢得多，因为电场几乎瞬间建立于整条导线中。

4. From Drift Velocity to Ohm's Law | 从漂移速度推导欧姆定律

charge e and mass m experiences a force eE from the applied field E and accelerates at a = eE / m. If the mean time between collisions is τ, the average drift velocity just before a collision is v = a τ = e E τ / m.

漂移速度方程的一个有力延伸，把微观图像与宏观电阻联系起来。在两次碰撞之间，电荷为e、质量为m的电子受到电场E的作用力eE，加速度为 a = eE / m。若平均碰撞间隔时间为τ，则碰撞前的平均漂移速度为 v = a τ = e E τ / m。

t, the field is E = V / L, so v = e V τ / (m L). Substituting this into I = n A v e gives:

对于长度为L、两端电势差为V的均匀导线，场强 E = V / L，故 v = e V τ / (m L)。代入 I = n A v e 得：

I = n A e² V τ / (m L)

Rearranging so that V is divided by I gives the resistance:

整理成 V 除以 I 的形式，即得电阻：

R = V / I = m L / (n A e² τ)

Because m, n, e and τ are properties of the material that do not change with V (for a metal at constant temperature), R is constant. That is exactly Ohm's law, and it also explains why R ∝ L and R ∝ 1 / A, matching the experimental result in Section 2.

由于m、n、e和τ对给定材料（恒温金属）不随V变化，故R为常量。这正是欧姆定律，同时也解释了 R ∝ L 与 R ∝ 1 / A，与第2节的实验结果一致。

5. Deriving the Wave Equation | 波速方程的推导

ave. By definition, one wavelength λ is the distance the wave advances in one period. Therefore the speed of the wave is distance divided by time.

频率f是每秒完成的完整振动次数，因此周期 T = 1 / f 是完成一次完整波动所需的时间。按定义，一个波长λ就是波在一个周期内前进的距离。因此波速等于距离除以时间。

v = λ / T = λ f

travelling in a vacuum, the speed is the speed of light c = 3.00 × 10⁸ m s⁻¹, so c = f λ.

该关系适用于所有行波，包括声波、水波和电磁波。真空中传播的电磁波速度即为光速 c = 3.00 × 10⁸ m s⁻¹，故 c = f λ。

. This single rearrangement appears repeatedly in quantum questions, so it is worth memorising both forms.

在2019年6月的试卷中，该方程常与其他公式结合使用。例如光子能量 E = h f，代入 f = c / λ 得 E = h c / λ。这一变形在量子题目中反复出现，两种形式都值得记住。

6. Deriving the Diffraction Grating Equation | 光栅方程的推导

/ N and N is the number of lines per metre. Waves from adjacent slits travel different distances to reach the same point on the screen. Constructive interference occurs when this path difference equals a whole number of wavelengths.

在衍射光栅中，相邻狭缝间距为d，其中 d = 1 / N，N 为每米刻线数。来自相邻狭缝的波到达屏幕上同一点时路程不同。当路程差等于波长的整数倍时发生相长干涉。

From the geometry of the triangle formed by the slit separation and the wavefronts, the path difference is d sin θ, where θ is the angle between the straight-through direction and the direction of the diffracted beam. Setting the path difference equal to n λ gives the grating equation.

由狭缝间距与波前构成的三角形几何关系可知，路程差为 d sin θ，其中θ为直射方向与衍射光束方向之间的夹角。令路程差等于 n λ，即得光栅方程。

d sin θ = n λ

le order is n(max) ≤ d / λ. If your calculated n exceeds d / λ, that order does not exist.

其中n为衍射级次，可取 0, 1, 2, 3 ... 一个有用的检验是 sin θ 不可能超过1，因此最大级次满足 n(max) ≤ d / λ。若计算出的n超过 d / λ，则该级次不存在。

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7. Deriving the Photoelectric Equation | 光电效应方程的推导

lectron near the metal surface, some of that energy is used to escape from the metal and the rest becomes kinetic energy. The minimum energy needed to escape is the work function φ.

光电效应表明，光以称为光子的离散能量包形式传递能量，每个光子能量 E = h f。当光子被金属表面附近的电子吸收时，一部分能量用于逸出金属，其余成为动能。逸出所需的最小能量称为逸出功 φ。

Applying conservation of energy to the most loosely bound surface electron gives the photoelectric equation:

对束缚最弱、最易逸出的表面电子应用能量守恒，即得光电效应方程：

h f = φ + E(k max)

n has zero kinetic energy. Substituting φ = h f₀ gives E(k max) = h f − h f₀ = h (f − f₀).

截止频率 f₀ 由 h f₀ = φ 定义，因为在该频率下逸出电子的动能为零。代入 φ = h f₀ 得 E(k max) = h f − h f₀ = h (f − f₀)。

(s) = E(k max). Combining these relations produces the practical form used in experiments:

若加上遏止电压 V(s) 恰好使最快电子无法到达阳极，则 e V(s) = E(k max)。联立可得实验常用的形式：

e V(s) = h f − φ

A graph of V(s) against f is therefore a straight line of gradient h / e and intercept − φ / e on the vertical axis. This is one of the most commonly examined graphs in Unit 2, so be ready to explain each feature.

因此 V(s) 对 f 的图像是一条直线，斜率 h / e，纵轴截距 − φ / e。这是Unit 2中最常考的图像之一，务必能解释每个特征。

8. Deriving the de Broglie Relation and Electron Acceleration | 德布罗意关系与电子加速的推导

two gives p = h / λ, or λ = h / p. De Broglie assumed the same relation holds for matter.

德布罗意提出，粒子与光子一样具有波长。对光子，E = h f = h c / λ，又 E = p c，两式相等得 p = h / λ，即 λ = h / p。德布罗意假设该关系对实物粒子同样成立。

λ = h / p = h / (m v)

the field equals the kinetic energy gained, so e V = ½ m v², which rearranges to give v = √2 e V / m.

当电子从静止经电势差V加速时，该关系尤为有用。电场做功等于获得的动能，即 e V = ½ m v²，整理得 v = √2 e V / m。

The momentum is then p = m v = √2 m e V, so the de Broglie wavelength becomes:

于是动量 p = m v = √2 m e V，德布罗意波长为：

λ = h / √2 m e V

Note that the wavelength depends on 1 / √V. Doubling the accelerating voltage therefore reduces the wavelength by a factor of √2, which is a favourite calculation in multiple-choice questions.

注意波长与 1 / √V 成正比。因此把加速电压加倍会使波长减小为原来的 1 / √2，这是选择题中常见的计算点。

9. Deriving Electrical Power Equations | 电功率公式的推导

o P = Q V / t. Since current I = Q / t, this simplifies to the fundamental power equation.

功率是能量传递的速率 P = W / t。在电路中，把电荷Q移动经过电势差V所做的功为 W = Q V，故 P = Q V / t。又因为电流 I = Q / t，于是化简为最基本的功率公式。

P = V I

Substituting Ohm' is most convenient. When voltage is the same across components in parallel, P = V² / R is easier to use.

代入欧姆定律 V = I R 可再得两个极为实用的形式。当串联元件中电流相同时，使用 P = I² R 最方便；当并联元件两端电压相同时，使用 P = V² / R 更简单。

P = I² R = V² / R

ys check whether a question asks for power or for energy, because they differ by the factor t.

时间t内传递的能量为 E = P t，生活用电常用千瓦时作单位，1 kW h = 3.6 × 10⁶ J。务必看清题目问的是功率还是能量，二者相差一个时间因子t。

10. Deriving the Potential Divider Equation | 分压器公式的推导

ries, the same current I flows through both, and the total resistance is R₁ + R₂. From Ohm's law for the whole circuit, I = V(in) / (R₁ + R₂).

分压器由两个串联电阻接在电动势为 V(in) 的电源两端组成。由于串联，两电阻中电流I相同，总电阻为 R₁ + R₂。对整个电路应用欧姆定律，I = V(in) / (R₁ + R₂)。

V(out) = I R₂, so substituting the expression for I gives the potential divider equation.

R₂ 两端的输出电压为 V(out) = I R₂，代入I的表达式即得分压器公式。

V(out) = V(in) × R₂ / (R₁ + R₂)

In practice, connecting a load across R₂ draws current and alters the division. The loaded output voltage is found by combining the load resistance with R₂ in parallel first, then applying the same equation. This is a common source of error in practical questions.

实际使用时，在R₂两端接上负载会分走电流，从而改变分压比。此时应先把负载电阻与R₂并联求出等效电阻，再套用同一公式。这是实验题中常见的错误来源。

11. Deriving the E.M.F. and Internal Resistance Equation | 电动势与内阻公式的推导

A real source of e.m.f. ε has internal resistance r. When current I flows, energy is dissipated inside the source as well as in the external circuit. The total power delivered by the source equals the power in the external resistor plus the power wasted inside.

真实的电动势源 ε 具有内阻 r。当电流I流动时，能量不仅在外部电路消耗，也在电源内部消耗。电源提供的总功率等于外电阻上的功率加上内部损耗的功率。

ε I = I² R + I² r

Dividing every term by I gives the standard circuit equation:

各项同除以I，即得标准电路方程：

ε = I (R + r)

falls as the current rises, and that the short-circuit current, when R = 0, is I = ε / r.

电源两端的端电压 V（即电压表在电源两端读到的值）为 V = I R = ε − I r。这说明端电压随电流增大而下降；当 R = 0 短路时，短路电流为 I = ε / r。

A graph of V against I has gradient − r and vertical intercept ε. This is the standard experimental method for measuring both quantities and is widely examined in Unit 2 practical work.

以V对I作图，斜率为 − r，纵轴截距为ε。这是测量这两个量的标准实验方法，也是Unit 2实验部分的高频考点。

12. Exam Technique and Common Pitfalls | 考试技巧与常见误区

Most lost marks in derivation questions come from presentation rather than physics. Always start with a named equation, define every new symbol the first time it appears, and keep the algebra symbolic until the final line. Substituting numbers early makes errors harder to spot and often breaks the logical chain the mark scheme rewards.

推导题失分大多源于表达而非物理本身。务必先写出所依据的公式名称，首次出现的每个符号都要定义，并一直用符号运算到最后一行为止。过早代入数字既难发现错误，也常常打断评分标准所看重的逻辑链。