The Photoelectric Effect and Its Experimental Laws | IB物理：光电效应及其实验规律

📚 The Photoelectric Effect and Its Experimental Laws | IB物理：光电效应及其实验规律

The photoelectric effect is one of the most pivotal phenomena in modern physics, providing direct evidence for the quantum nature of light. This article systematically reviews the experimental observations, the failure of classical wave theory, and Einstein’s photon explanation, all of which are essential for IB Physics HL/SL examinations.

光电效应是现代物理学中最具关键意义的现象之一，它直接证明了光的量子本质。本文系统梳理光电效应的实验规律、经典波动理论的困境以及爱因斯坦的光子解释，这些内容皆为IB物理HL/SL考试的核心考点。

1. What Is the Photoelectric Effect? | 什么是光电效应？

When electromagnetic radiation with sufficient frequency shines on a metal surface, electrons can be emitted from that surface. These emitted electrons are called photoelectrons, and the phenomenon itself is termed the photoelectric effect.

当足够频率的电磁波照射金属表面时，金属表面会释放出电子。这些被释放的电子被称为光电子，这一现象本身则被称为光电效应。

It is important to distinguish photoelectrons from other types of emitted electrons: photoelectrons are specifically those liberated from a material due to the absorption of electromagnetic radiation, typically visible or ultraviolet light.

需要将光电子与其他类型的逸出电子区分开来：光电子特指因吸收电磁辐射（通常是可见光或紫外线）而从材料中释放出来的电子。

2. Experimental Apparatus | 实验装置

The classic photoelectric effect experiment uses an evacuated glass tube containing two electrodes: a photocathode (often caesium or zinc) and an anode. A variable voltage source is connected across the tube, along with a sensitive ammeter and a voltmeter.

经典的光电效应实验使用一个真空玻璃管，内部包含两个电极：一个光阴极（常用铯或锌制成）和一个阳极。可调电压源连接在管的两端，并配有灵敏的电流表和电压表。

When monochromatic light of a chosen frequency is incident on the cathode, photoelectrons are ejected. Some of these electrons travel to the anode, producing a measurable photocurrent. By adjusting the magnitude and direction of the applied voltage, the kinetic energy of the emitted electrons can be investigated.

当选定频率的单色光照射到阴极时，光电子被发射出来。其中部分电子飞向阳极，形成可测量的光电流。通过调节外加电压的大小和方向，可以研究逸出电子的动能特性。

Photocurrent I = n × e × v_d | 光电流 I = n × e × v_d

where n is the number of photoelectrons emitted per unit time reaching the anode, e is the elementary charge (1.60 × 10⁻¹⁹ C), and v_d is the effective drift speed of the electron population.

其中 n 是单位时间内到达阳极的光电子数，e 是元电荷（1.60 × 10⁻¹⁹ C），v_d 是电子群的有效漂移速度。

3. Experimental Law 1: The Existence of a Threshold Frequency | 实验规律一：截止频率的存在

For any given metal surface, there exists a minimum frequency of incident light, \(f₀\), below which no photoelectrons are emitted at all, regardless of the light intensity or the duration of illumination. This minimum frequency is called the threshold frequency.

对于任何给定的金属表面，存在一个最小入射光频率 \(f₀\)，低于此频率时无论光强多大、照射时间多长，都不会发射任何光电子。这个最小频率被称为截止频率（也称极限频率）。

f₀ = φ / h

where φ is the work function of the metal (minimum energy required to liberate an electron from the surface) and h is Planck’s constant (6.63 × 10⁻³⁴ J·s).

其中 φ 是金属的逸出功（从表面释放一个电子所需的最小能量），h 是普朗克常数（6.63 × 10⁻³⁴ J·s）。

For example, zinc has a threshold frequency of approximately 1.04 × 10¹⁵ Hz, corresponding to ultraviolet light. Visible light does not cause emission from clean zinc under normal conditions.

例如，锌的截止频率约为 1.04 × 10¹⁵ Hz，对应紫外线波段。在普通条件下，可见光无法使洁净的锌产生光电发射。

4. Experimental Law 2: Instantaneous Emission | 实验规律二：发射的瞬时性

When light of a frequency above the threshold frequency strikes the metal, photoelectrons are emitted instantaneously, within less than 10⁻⁹ seconds of illumination. There is no measurable time delay for the accumulation of energy.

当频率高于截止频率的光照射金属时，光电子瞬时被发射，延迟时间小于 10⁻⁹ 秒。不存在可测量的能量积累时间延迟。

This is a remarkable result because classical wave theory predicts that sufficient energy from a low-intensity wave would need to be accumulated over time before emission could occur. For a typical metal illuminated by faint light, this predicted delay would be several seconds or even minutes — yet no such delay is observed.

这是一个非常显著的结果，因为经典波动理论预测：对于低强度波，需要经过一段时间的能量积累后才可能发射电子。对于微弱光照下的典型金属，这一理论预测的延迟时间可能长达数秒甚至数分钟——然而实际观测中不存在任何这样的延迟。

5. Experimental Law 3: Maximum Kinetic Energy Depends on Frequency, Not Intensity | 实验规律三：最大动能取决于频率而非光强

The maximum kinetic energy of emitted photoelectrons, \(K_{\text{max}}\), increases linearly with the frequency of the incident light, but is independent of the light intensity.

发射光电子的最大动能 \(K_{\text{max}}\) 随入射光频率的增加而线性增大，但与光强无关。

Experimentally, this is determined using a reverse (stopping) voltage. By applying a potential difference that opposes the motion of photoelectrons, the current can be reduced to zero. The reverse voltage at which the photocurrent becomes zero is called the stopping potential, \(V_s\).

在实验上，这是通过反向截止电压来测量的。施加一个阻碍光电子运动的电压，可使光电流降为零。使光电流恰好为零的反向电压被称为遏止电压 \(V_s\)。

\(K_{\text{max}}\) = e × \(V_s\)

where e is the elementary charge. Since the stopping potential can be measured precisely, the maximum kinetic energy is directly determined.

其中 e 是元电荷。由于遏止电压可以被精确测量，最大动能也因此被直接确定。

6. Experimental Law 4: Photocurrent Is Proportional to Intensity | 实验规律四：光电流与光强成正比

When the frequency of the incident light is held constant above the threshold frequency, the saturation photocurrent (the maximum photocurrent when all emitted electrons are collected) is directly proportional to the intensity of the incident light.

当入射光的频率保持恒定且高于截止频率时，饱和光电流（所有发射电子均被收集时的最大光电流）与入射光的强度成正比。

Doubling the light intensity at a fixed frequency doubles the number of photoelectrons emitted per second, and hence doubles the saturation current. However, the stopping potential \(V_s\) remains unchanged.

在固定频率下将光强加倍，每秒发射的光电子数量加倍，因而饱和光电流也加倍。但遏止电压 \(V_s\) 保持不变。

This indicates that a greater light intensity means more photons per second, each carrying the same quantum of energy hf — not photons of greater energy.

这表明更大的光强意味着每秒钟有更多光子，而每个光子携带相同的能量量子 hf——并不是单个光子的能量更大。

7. The Failure of Classical Wave Theory | 经典波动理论的失败

Classical wave theory treats light as a continuous electromagnetic wave whose energy depends on its amplitude (intensity). This theory fails to explain any of the four experimental laws above.

经典波动理论将光视为连续的电磁波，其能量取决于振幅（强度）。该理论无法解释上述任何一条实验规律。

Threshold frequency unexplained: In wave theory, any frequency could transfer energy to electrons; given enough time, even low-frequency light should cause emission. However, experiments show that below \(f₀\), emission never occurs.
截止频率无法解释：根据波动理论，任何频率的光都能向电子传递能量；只要有足够时间，即使是低频光也应引发发射。然而实验显示，低于 \(f₀\) 时永远不会发生发射。

Time delay contradiction: Classical theory predicts a measurable delay for low-intensity light, but photoelectric emission is instantaneous.
延迟时间矛盾：经典理论预测低强度光需要可测量的延迟，但光电发射是瞬时的。

Kinetic energy problem: Classical theory predicts that greater intensity should yield electrons with greater kinetic energy, but experiments show \(K_{\text{max}}\) is independent of intensity.
动能问题：经典理论预测更强的光应使电子获得更大的动能，但实验表明 \(K_{\text{max}}\) 与光强无关。

Therefore, the wave model was fundamentally incompatible with the photoelectric effect — a crisis that demanded a revolutionary new viewpoint.

因此，波动模型与光电效应在根本上不兼容——这一困境呼唤着一个革命性的新视角。

8. Einstein’s Photon Hypothesis | 爱因斯坦的光子假说

In 1905, Albert Einstein proposed that electromagnetic radiation is quantised: light consists of discrete packets of energy called photons. Each photon carries energy.

1905年，阿尔伯特·爱因斯坦提出电磁辐射是量子化的：光由称为光子的离散能量包组成。每个光子携带的能量为

\(E\) = hf = hc / λ

where h is Planck’s constant, f is the frequency, c is the speed of light (3.00 × 10⁸ m/s), and λ is the wavelength of the radiation.

其中 h 是普朗克常数，f 是频率，c 是光速（3.00 × 10⁸ m/s），λ 是辐射波长。

The key insight is that an electron interacts with light by absorbing one whole photon at a time, not by accumulating energy from many waves. If a single photon has energy greater than or equal to the work function, the electron is liberated; otherwise, absorption simply does not occur, regardless of how many photons strike the surface.

关键的洞见在于：电子与光的交互是一次性吸收一个完整光子，而非从许多波中积累能量。如果单个光子的能量大于或等于逸出功，电子就被释放；否则吸收根本不会发生，无论有多少光子照射表面。

9. The Photoelectric Equation | 爱因斯坦光电方程

Applying the principle of conservation of energy to the single-photon absorption process yields the famous Einstein photoelectric equation:

将能量守恒原理应用于单光子吸收过程，我们得到著名的爱因斯坦光电方程：

hf = φ + \(K_{\text{max}}\)

or equivalently:

等价地：

\(K_{\text{max}}\) = hf − φ

Here, hf is the energy of the incident photon, φ is the work function of the metal, and \(K_{\text{max}}\) is the maximum kinetic energy of the emitted electron.

其中 hf 是入射光子的能量，φ 是金属的逸出功，\(K_{\text{max}}\) 是发射电子的最大动能。

The work function itself is related to the threshold frequency by φ = hf₀. Substituting into the photoelectric equation gives another useful form:

逸出功本身与截止频率的关系为 φ = hf₀。代入光电方程可得到另一个有用的形式：

\(K_{\text{max}}\) = h(f − f₀)

This equation directly predicts that \(K_{\text{max}}\) is a linear function of frequency with slope h, consistent with experimental observations.

该方程直接预测 \(K_{\text{max}}\) 是频率的线性函数，斜率为 h，与实验观测完全一致。

10. Graphical Analysis and the Determination of Planck’s Constant | 图像分析与普朗克常数的测定

The relationship between the stopping potential \(V_s\) and the frequency of incident light is a straight line. Using \(e × V_s\) = hf − φ, we obtain:

遏止电压 \(V_s\) 与入射光频率之间的关系是一条直线。利用 \(e × V_s\) = hf − φ，我们得到：

\(V_s\) = (h/e) × f − (φ/e)

Plotting \(V_s\) on the vertical axis and f on the horizontal axis gives a straight line with slope h/e and y-intercept −φ/e. The x-intercept equals the threshold frequency f₀.

以 \(V_s\) 为纵轴、f 为横轴作图，得到一条直线，其斜率为 h/e，纵轴截距为 −φ/e，横轴截距等于截止频率 f₀。

Graph Feature \| 图像特征	Meaning \| 物理意义
Slope = h/e \| 斜率 = h/e	Enables determination of Planck’s constant \| 可测定普朗克常数
x-intercept = f₀ \| 横轴截距 = f₀	Threshold frequency of the metal \| 金属的截止频率
y-intercept = −φ/e \| 纵轴截距 = −φ/e	Work function of the metal \| 金属的逸出功

Since e is known to be 1.60 × 10⁻¹⁹ C, the slope of the \(V_s\)-f graph allows physicists to measure Planck’s constant with high precision — a powerful experimental confirmation of quantum theory.

由于 e 是已知量（1.60 × 10⁻¹⁹ C），\(V_s\)-f 图像的斜率使物理学家能够以高精度测量普朗克常数——这是对量子理论的强有力实验证实。

11. Worked Example | 典型例题解析

Problem: Light of wavelength 250 nm is incident on a metal surface with a work function of 2.80 eV. Determine: (a) the photon energy in eV; (b) the maximum kinetic energy of the emitted electrons in eV; (c) the stopping potential.

例题：波长为250 nm的光照射在逸出功为2.80 eV的金属表面上。求：(a) 光子能量（以eV为单位）；(b) 发射电子的最大动能（以eV为单位）；(c) 遏止电压。

Solution:

解答：

(a) First convert the wavelength to frequency: f = c/λ = (3.00 × 10⁸) ÷ (250 × 10⁻⁹) = 1.20 × 10¹⁵ Hz. Then the photon energy is \(E\) = hf = (6.63 × 10⁻³⁴) × (1.20 × 10¹⁵) = 7.96 × 10⁻¹⁹ J. Converting to eV: 7.96 × 10⁻¹⁹ ÷ (1.60 × 10⁻¹⁹) = 4.97 eV.

(a) 先换算波长到频率：f = c/λ = (3.00 × 10⁸) ÷ (250 × 10⁻⁹) = 1.20 × 10¹⁵ Hz。则光子能量为 \(E\) = hf = (6.63 × 10⁻³⁴) × (1.20 × 10¹⁵) = 7.96 × 10⁻¹⁹ J。换算为eV：7.96 × 10⁻¹⁹ ÷ (1.60 × 10⁻¹⁹) = 4.97 eV。

(b) Using the photoelectric equation: \(K_{\text{max}}\) = hf − φ = 4.97 − 2.80 = 2.17 eV.

(b) 使用光电方程：\(K_{\text{max}}\) = hf − φ = 4.97 − 2.80 = 2.17 eV。

12. Applications and Significance | 应用与深远意义

The photoelectric effect has numerous technological applications and played a central role in the development of quantum mechanics. Some common applications include:

光电效应具有众多技术应用，并在量子力学的发展中发挥了核心作用。一些常见的应用包括：

Photoelectric cells (phototubes): Used in automatic lighting controls, exposure meters, and burglar alarm systems.
光电管：用于自动照明控制、曝光计和防盗报警系统。
Image sensors: Digital cameras and night-vision devices rely on the conversion of light to electrical signals.
图像传感器：数码相机和夜视设备依赖于将光转换为电信号。
Solar panels: Although these primarily use the photovoltaic effect, the underlying photon-electron interaction principle is closely related.
太阳能电池板：虽然主要利用光伏效应，但其基本的光子-电子交互原理密切相关。

Beyond applications, the photoelectric effect provided decisive evidence for the particle nature of light, complementing interference and diffraction experiments that support the wave nature. This dual nature — wave-particle duality — is a cornerstone of modern physics and a central theme in the IB curriculum.

除了应用之外，光电效应为光的粒子性提供了决定性的证据，与支持波动性的干涉和衍射实验相辅相成。这种双重性质——波粒二象性——是现代物理学的基石，也是IB课程的核心主题。

Achieving a deep conceptual understanding of the experimental laws and Einstein’s explanation is essential for solving IB exam questions on this topic. Master the graphs, memorise the key equations, and always verify units — the photoelectric effect is one of the most rewarding topics to study.

深入理解实验规律和爱因斯坦的解释，对于解答IB考试中关于该主题的题目至关重要。掌握图像、熟记关键方程，并始终检查单位——光电效应是最值得深入学习的主题之一。

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Graph Feature \| 图像特征	Meaning \| 物理意义
Slope = h/e \| 斜率 = h/e	Enables determination of Planck’s constant \| 可测定普朗克常数
x-intercept = f₀ \| 横轴截距 = f₀	Threshold frequency of the metal \| 金属的截止频率
y-intercept = −φ/e \| 纵轴截距 = −φ/e	Work function of the metal \| 金属的逸出功

The Photoelectric Effect and Its Experimental Laws | IB物理：光电效应及其实验规律