Photoelectric Effect Explained | 光电效应考点精讲

📚 Photoelectric Effect Explained | 光电效应考点精讲

The photoelectric effect is one of the most important phenomena in modern physics, providing strong evidence for the particle nature of light. When electromagnetic radiation of a sufficiently high frequency shines on a metal surface, electrons are emitted. This emission cannot be explained by classical wave theory but is beautifully accounted for by Einstein’s photon model. In the WJEC A-Level Physics specification, you must understand the experimental observations, Einstein’s photoelectric equation, concepts such as work function and threshold frequency, and how to analyse graphs of kinetic energy against frequency. This article will guide you through all the key points, ensuring you are fully prepared for your exams.

光电效应是现代物理学中最重要的现象之一，为光的粒子性提供了强有力的证据。当频率足够高的电磁辐射照射到金属表面时，电子会被发射出来。这种现象无法用经典波动理论解释，但爱因斯坦的光子模型却完美地进行了说明。在 WJEC A-Level 物理大纲中，你必须理解实验观察结果、爱因斯坦光电方程、功函数和阈频等概念，以及如何分析动能随频率变化的图像。本文将带你梳理所有关键知识点，确保你为考试做好充分准备。

1. Introduction to the Photoelectric Effect | 光电效应简介

The photoelectric effect refers to the emission of electrons from a metal surface when it is exposed to electromagnetic radiation of a sufficiently high frequency. First observed by Heinrich Hertz in 1887, the effect puzzled physicists because the kinetic energy of the emitted electrons did not depend on the intensity of the light, as wave theory would predict. Instead, it depended on the frequency of the radiation. Below a certain frequency, no electrons were emitted at all, regardless of how intense the light was. This marked the beginning of quantum mechanics.

光电效应是指当频率足够高的电磁辐射照射到金属表面时，金属会发射电子的现象。赫兹于 1887 年首次观察到这一现象，它令物理学家们困惑不已，因为发射电子的动能并不像波动理论所预测的那样取决于光的强度，而是取决于辐射的频率。在某个频率以下，无论光有多强，都不会有电子发射出来。这标志着量子力学的开端。

2. Experimental Setup and Key Observations | 实验装置与关键观察

A typical photoelectric experiment uses a vacuum tube containing an emitter metal and a collector plate. Monochromatic light illuminates the emitter, and the emitted photoelectrons are collected, producing a current. By applying a reverse potential (stopping voltage Vₛ), the current can be reduced to zero. The key observations are: (1) emission is instantaneous, even at low intensities; (2) there is a threshold frequency f₀ below which no emission occurs; (3) the maximum kinetic energy Eₖ(max) of the electrons increases linearly with frequency and is independent of intensity; (4) the photocurrent is directly proportional to the intensity of light above the threshold frequency.

典型的光电效应实验使用一个包含发射金属和收集极的真空管。单色光照射在发射极上，所产生的光电子被收集起来，从而形成电流。通过施加反向电压（截止电压 Vₛ），可以将电流降至零。关键的观察结果是：（1）发射是瞬时的，即使在低光强下也是如此；（2）存在一个阈频 f₀，低于该频率时没有电子发射；（3）电子的最大动能 Eₖ(max) 随频率线性增加，且与光强无关；（4）在阈频以上，光电流与光强度成正比。

3. The Photon Model of Light | 光的光子模型

To explain the photoelectric effect, Einstein proposed that light consists of discrete packets of energy called photons. Each photon has an energy E = h f, where h is Planck’s constant (6.63 × 10⁻³⁴ J s) and f is the frequency of the radiation. A single photon interacts with a single electron, transferring its entire energy instantaneously. This photon picture immediately explains the existence of a threshold frequency: an electron needs a minimum amount of energy to escape the metal surface.

为了解释光电效应，爱因斯坦提出光是由称为光子的分立能量包组成的。每个光子具有能量 E = h f，其中 h 为普朗克常量（6.63 × 10⁻³⁴ J s），f 为辐射频率。单个光子与单个电子相互作用，瞬间传递其全部能量。这种光子图景立刻解释了阈频的存在：电子需要获得最低限度的能量才能脱离金属表面。

4. Work Function and Threshold Frequency | 功函数与阈频

The minimum energy required to free an electron from the surface of a metal is called the work function, symbol Φ (phi). It is a characteristic property of the metal, typically measured in electronvolts (eV). The threshold frequency f₀ is directly related to the work function by Φ = h f₀. If the photon energy h f is less than Φ, no electron can be emitted. Different metals have different work functions, which is why some materials are more suitable for photocathodes than others.

将电子从金属表面释放所需的最低能量称为功函数，符号为 Φ。它是金属的一种特征属性，通常以电子伏特（eV）为单位。阈频 f₀ 与功函数直接相关：Φ = h f₀。如果光子能量 h f 小于 Φ，则没有电子可以发射。不同金属有不同的功函数，这就是为什么有些材料比其它材料更适合用作光电阴极。

5. Einstein’s Photoelectric Equation | 爱因斯坦光电方程

The heart of the photoelectric effect is Einstein’s equation:

Eₖ(max) = h f − Φ

This states that the maximum kinetic energy of a photoelectron is the photon energy minus the work function. Any leftover photon energy after escaping the metal becomes kinetic energy. If h f = Φ, Eₖ(max) = 0. For h f < Φ, no electrons are emitted. The equation explains the linear relationship between f and Eₖ(max), with slope equal to h.

光电效应的核心是爱因斯坦方程：

Eₖ(max) = h f − Φ

该方程表明，光电子的最大动能等于光子能量减去功函数。脱离金属后剩余的任何光子能量都转化为动能。如果 h f = Φ，则 Eₖ(max) = 0；如果 h f < Φ，则没有电子发射。该方程解释了 f 与 Eₖ(max) 之间的线性关系，其斜率等于 h。

6. Maximum Kinetic Energy and Stopping Potential | 最大动能与截止电压

The maximum kinetic energy can be measured by applying a retarding potential, known as the stopping potential Vₛ. The work done by the electric field in stopping the fastest electrons is e Vₛ, where e is the elementary charge (1.60 × 10⁻¹⁹ C). Therefore:

e Vₛ = Eₖ(max) = h f − Φ

Thus, the stopping potential is related to frequency by Vₛ = (h/e) f − Φ/e. A graph of Vₛ against f gives a straight line with gradient h/e and y-intercept −Φ/e, which allows an experimental determination of Planck’s constant.

最大动能可以通过施加一个反向电压（即截止电压 Vₛ）来测量。电场在阻止最快电子时所做的功为 e Vₛ，其中 e 为元电荷（1.60 × 10⁻¹⁹ C）。因此：

e Vₛ = Eₖ(max) = h f − Φ

所以，截止电压与频率的关系为 Vₛ = (h/e) f − Φ/e。绘制 Vₛ 对 f 的图像会得到一条直线，其斜率为 h/e，y 轴截距为 −Φ/e，这使得我们可以通过实验测定普朗克常量。

7. Intensity, Frequency and Photocurrent | 光强、频率与光电流

A common misconception is that increasing the intensity of light increases the kinetic energy of the photoelectrons. In the photon model, intensity is related to the number of photons per second, not the energy of individual photons. Therefore, raising the intensity increases the photocurrent (more electrons emitted per second) but does not affect the maximum kinetic energy. Only a higher frequency can boost Eₖ(max). At frequencies below f₀, even the brightest light produces no photoelectrons at all.

一个常见的误解是，增加光强会增加光电子的动能。在光子模型中，光强与每秒光子数有关，而与单个光子的能量无关。因此，提高光强会增加光电流（每秒发射更多电子），但不会影响最大动能。只有更高的频率才能提升 Eₖ(max)。在低于 f₀ 的频率下，即使是最亮的光也完全不会产生光电子。

8. Instantaneous Emission — Evidence for Photons | 瞬时发射——光子的证据

Experiments show that photoelectrons are emitted as soon as the light is switched on, with no measurable time delay, even at extremely low intensities. Classical wave theory would require an electron to accumulate energy gradually over several wave cycles. With the photon model, a single photon delivers the required energy in one interaction, so emission is effectively instant if the frequency meets the threshold condition. This instantaneous nature was a crucial piece of evidence confirming the particle behaviour of light.

实验表明，一旦打开光源，光电子就立即发射，没有任何可测量的时间延迟，即使在极低的光强下也是如此。按照经典波动理论，电子需要经历多个波周期才能逐渐累积能量。而在光子模型中，单个光子在一次相互作用中就传递了所需的能量，因此只要频率满足阈频条件，发射实际上就是瞬时的。这种瞬时特性是证实光具有粒子行为的关键证据之一。

9. Failure of the Classical Wave Theory | 经典波动理论的失败

The following table summarises why the classical wave model cannot explain the photoelectric observations:

Observation	Wave Theory Prediction	Photon Model Explanation
Existence of threshold frequency	Any frequency should work if intense enough	Photon energy must be ≥ work function
K_max depends on frequency, not intensity	K_max increases with intensity	K_max = hf − Φ, independent of photon number
No time lag	Time delay while energy accumulates	One-photon one-electron interaction is instant

下表总结了为什么经典波动模型无法解释光电效应的观察结果：

观察现象	波动理论预测	光子模型解释
阈频的存在	只要强度足够，任何频率都应产生发射	光子能量必须 ≥ 功函数
K_max 取决于频率而非强度	K_max 随强度增加	K_max = hf − Φ，与光子数量无关
无时间延迟	在能量累积期间存在时间延迟	单光子-单电子相互作用是瞬时的

10. Working with Graphs: Ek vs f | 图表分析：Ek 随 f 变化

A common exam task is to analyse a graph of maximum kinetic energy versus frequency. The straight-line equation is Eₖ(max) = h f − Φ. The gradient gives Planck’s constant h. The x-intercept is the threshold frequency f₀, where Eₖ(max) = 0. The y-intercept is −Φ, allowing calculation of the work function. Changing the metal shifts the line parallel to itself because h is constant but Φ changes. Changing the intensity does not affect this graph at all; it only changes the photocurrent magnitude.

一个常见的考题是分析最大动能随频率变化的图像。直线方程为 Eₖ(max) = h f − Φ。斜率给出普朗克常量 h。x 轴截距为阈频 f₀，此时 Eₖ(max) = 0。y 轴截距为 −Φ，从而可以计算出功函数。更换金属会使直线平行移动，因为 h 是常数而 Φ 不同。改变光强完全不影响这张图；它只改变光电流的大小。

11. Solved Example: Calculating Stopping Potential | 解题示例：计算截止电压

Question: Ultraviolet light of frequency 1.20 × 10¹⁵ Hz falls on a sodium surface with a work function of 2.46 eV. Calculate (a) the maximum kinetic energy of the photoelectrons in eV and joules, and (b) the stopping potential. (h = 6.63 × 10⁻³⁴ J s, 1 eV = 1.60 × 10⁻¹⁹ J)

Solution: Photon energy E = h f = (6.63 × 10⁻³⁴) × (1.20 × 10¹⁵) = 7.956 × 10⁻¹⁹ J. Convert to eV: 7.956 × 10⁻¹⁹ / 1.60 × 10⁻¹⁹ ≈ 4.97 eV.

Then Eₖ(max) = h f − Φ = 4.97 eV − 2.46 eV = 2.51 eV. In joules: 2.51 × 1.60 × 10⁻¹⁹ = 4.02 × 10⁻¹⁹ J.

Stopping potential Vₛ = Eₖ(max) / e = 2.51 V. (Because e Vₛ = Eₖ(max) in eV gives Vₛ directly in volts.)

Therefore, the stopping potential is about 2.51 V.

题目：频率为 1.20 × 10¹⁵ Hz 的紫外线照射到功函数为 2.46 eV 的钠表面。计算（a）光电子的最大动能，分别以 eV 和焦耳表示；（b）截止电压。（h = 6.63 × 10⁻³⁴ J s，1 eV = 1.60 × 10⁻¹⁹ J）

解答：光子能量 E = h f = (6.63 × 10⁻³⁴) × (1.20 × 10¹⁵) = 7.956 × 10⁻¹⁹ J。转换为 eV：7.956 × 10⁻¹⁹ / 1.60 × 10⁻¹⁹ ≈ 4.97 eV。

那么 Eₖ(max) = h f − Φ = 4.97 eV − 2.46 eV = 2.51 eV。以焦耳计：2.51 × 1.60 × 10⁻¹⁹ = 4.02 × 10⁻¹⁹ J。

截止电压 Vₛ = Eₖ(max) / e = 2.51 V。（因为 e Vₛ = Eₖ(max)，以 eV 为单位直接得出 Vₛ 以伏特表示。）

因此，截止电压约为 2.51 V。

12. Common Mistakes and Exam Tips | 常见错误与考试提示

Confusing intensity and frequency: Many students incorrectly think brighter light gives higher-energy electrons. Remember: intensity affects the number of electrons, frequency affects the energy of electrons.

混淆光强与频率：许多学生错误地认为更亮的光会提供更高能量的电子。请记住：光强影响电子的数量，频率影响电子的能量。
Unit conversion errors: Work function and photon energy are often given in eV, while Planck’s constant uses joules. Always convert to consistent units before applying E = h f.

单位转换错误：功函数和光子能量通常以 eV 给出，而普朗克常量使用焦耳。在应用 E = h f 之前，务必转换为一致的单位。
Forgetting the meaning of stopping potential: e Vₛ equals the maximum kinetic energy. If your calculated Vₛ seems too large, check whether you divided by e correctly.

忘记截止电压的含义：e Vₛ 等于最大动能。如果你计算出的 Vₛ 显得过大，请检查你是否正确地除以了 e。
Misreading graphs: On an Eₖ vs f graph, the gradient is h, not h/e. Only on a Vₛ vs f graph is the gradient h/e. Read axes carefully.

读图错误：在 Eₖ 对 f 的图中，斜率是 h，而不是 h/e。只有在 Vₛ 对 f 的图中，斜率才为 h/e。请仔细阅读坐标轴。
Assuming emission below f₀: No matter how high the intensity, if f < f₀ there will be no photoelectrons. This is a common trick in multiple-choice questions.

误认为在 f₀ 以下也能发射：无论光强多高，只要 f < f₀，就不会有光电子。这是选择题中常见的陷阱。

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Photoelectric Effect Explained | 光电效应 考点精讲