The Photoelectric Effect: A Complete Guide for A-Level Physics

The Photoelectric Effect: When Light Behaves Like a Particle | 光电效应：当光表现得像粒子

Imagine shining a beam of light onto a metal surface and watching electrons fly off. Simple enough, right? Yet in 1887, Heinrich Hertz noticed something deeply puzzling: ultraviolet light could knock electrons loose from a metal surface, while visible light — no matter how intense — could not. This observation would eventually overturn centuries of classical physics and usher in the age of quantum mechanics.

The photoelectric effect is the emission of electrons from a metal surface when electromagnetic radiation of sufficiently high frequency falls upon it. What makes this phenomenon so remarkable is not that it happens, but how it happens — in ways that classical wave theory simply cannot explain. In this guide, we will explore the experimental observations, the theoretical framework Einstein developed to explain them, and the mathematical principles that tie everything together.

Experimental Observations

When physicists systematically studied the photoelectric effect, they discovered four key characteristics that demanded an explanation:

1. Threshold Frequency — For any given metal, there exists a minimum frequency of incident light below which no electrons are emitted, no matter how intense the light source. For zinc, this threshold lies in the ultraviolet region. For sodium and potassium, the threshold falls within visible light. If you shine red light on zinc — no intensity, no duration, no combination of lenses — will ever cause a single electron to leave. But ultraviolet light, even at the faintest intensity, produces electrons immediately.

2. Instantaneous Emission — There is no measurable time delay between the light striking the metal and the emission of electrons. Classical wave theory predicts that an electron would need to absorb energy gradually from the wave front until it accumulated enough to escape — a process that could take seconds or minutes for low-intensity light. Experiment shows emission begins within 10⁻⁹ seconds of illumination, even at the lowest intensities.

3. Kinetic Energy Depends on Frequency, Not Intensity — The maximum kinetic energy of emitted photoelectrons increases linearly with the frequency of the incident light. Doubling the intensity of the light doubles the number of electrons emitted (the photocurrent), but does nothing to their individual kinetic energies. A brighter light produces more electrons, but not faster ones. Only increasing the frequency can produce electrons with higher kinetic energies.

4. Intensity Affects Only the Number of Electrons — Provided the incident light is above the threshold frequency, the rate at which electrons are emitted (the photocurrent) is directly proportional to the intensity of the light. This makes intuitive sense: more photons per second means more electron-photon interactions per second.

Why Classical Wave Theory Failed

Classical physics describes light as a continuous electromagnetic wave. According to this view, the energy carried by the wave spreads out uniformly across its wave front. An electron in the metal absorbs this energy gradually, and once it accumulates enough to overcome the work function of the metal, it escapes.

This model makes three predictions, all of which are contradicted by experiment:

Any frequency should work given enough intensity. In the wave picture, if you wait long enough, even low-frequency light should deposit enough energy onto an electron for escape. Experiment says otherwise: below the threshold frequency, no electrons are ever emitted.
There should be a measurable time delay at low intensities. If intensity is reduced, the energy arriving per unit area per second decreases, and the “accumulation time” should increase. Experiment shows emission is always instantaneous.
Intensity should affect kinetic energy. A more intense wave carries more energy per unit area, so electrons should emerge with higher kinetic energies. Experiment shows kinetic energy depends solely on frequency, not intensity.

The wave model was broken. Physics needed a new idea.

Einstein’s Photon Model (1905)

In his annus mirabilis paper on the photoelectric effect, Albert Einstein proposed a radical departure from classical thinking: light is not a continuous wave but consists of discrete packets of energy called photons. Each photon carries an energy given by:

E = hf

where h is Planck’s constant (6.63 × 10⁻³⁴ J·s) and f is the frequency of the radiation. This single equation — deceptively simple — resolved every paradox of the photoelectric effect.

In Einstein’s picture, each photoelectron is liberated by a single photon in a one-to-one interaction. The photon transfers all its energy to the electron in a single, instantaneous event. The electron uses some of this energy to overcome the attractive forces binding it to the metal — this minimum required energy is called the work function (φ) of the metal. Any surplus becomes the electron’s kinetic energy.

This leads to Einstein’s photoelectric equation:

hf = φ + KE_max

Or equivalently:

KE_max = hf − φ

The equation elegantly explains every experimental observation:

Threshold frequency exists because: when hf < φ, the photon's energy is insufficient to overcome the work function. No electron can be liberated, regardless of how many photons arrive. The threshold frequency f₀ is simply f₀ = φ / h.
Emission is instantaneous because: the energy transfer is an all-or-nothing event. There is no accumulation — either the photon has enough energy or it doesn’t.
Kinetic energy depends on frequency because: KE_max = hf − φ. Higher frequency photons carry more energy, so the surplus kinetic energy after overcoming φ is larger.
Intensity affects only photocurrent because: intensity is a measure of photon count per unit area per second. More photons = more one-to-one interactions = more electrons, but each electron still receives exactly hf per photon.

The Stopping Potential Experiment

The relationship between electron kinetic energy and light frequency is measured experimentally using a photocell and a variable reverse voltage, known as the stopping potential (V_s).

In this experiment, a metal cathode is illuminated with monochromatic light of known frequency. The emitted photoelectrons travel to an anode, creating a photocurrent. A variable power supply applies a reverse potential difference that opposes the electron flow. As the reverse voltage increases, fewer electrons reach the anode, and the photocurrent decreases. The voltage at which the photocurrent falls to zero is the stopping potential — it is a direct measure of the maximum kinetic energy of the photoelectrons:

eV_s = KE_max = hf − φ

Rearranging gives:

V_s = (h/e)f − φ/e

This is a linear relationship between V_s and f. By measuring V_s for several different frequencies of incident light and plotting V_s against f, we obtain a straight line whose gradient is h/e and whose x-intercept is the threshold frequency f₀. This experiment, first performed by Robert Millikan in 1916, provided the most precise determination of Planck’s constant at the time.

Key features of the V_s-f graph:

Gradient: h/e — the same for all metals (a universal constant).
x-intercept: The threshold frequency f₀ — different for each metal.
y-intercept: −φ/e — the negative of the work function divided by electron charge.
Linearity: The straight line confirms KE_max ∝ f, as Einstein predicted.

Work Functions of Common Metals

The work function φ is a material-specific property — the minimum energy needed to extract an electron from the metal’s surface. Here are typical values:

Metal	Work Function φ (eV)	Work Function φ (J)	Threshold Wavelength λ₀ (nm)
Sodium (Na)	2.3	3.68 × 10⁻¹⁹	539
Potassium (K)	2.3	3.68 × 10⁻¹⁹	539
Calcium (Ca)	2.9	4.64 × 10⁻¹⁹	428
Zinc (Zn)	4.3	6.88 × 10⁻¹⁹	288
Platinum (Pt)	6.4	1.02 × 10⁻¹⁸	194

Notice that alkali metals (sodium, potassium) have low work functions, so their threshold frequencies lie in visible light. This is also why these metals are used in photomultiplier tubes and night-vision devices. Metals like zinc and platinum require ultraviolet light to trigger photoemission.

Worked Example

Question: Ultraviolet light of wavelength 200 nm is incident on a zinc surface (φ = 4.3 eV). Calculate (a) the energy of a single photon, (b) the maximum kinetic energy of emitted photoelectrons in both joules and electronvolts, and (c) the stopping potential.

Solution:

(a) Photon energy:

E = hf = hc/λ = (6.63 × 10⁻³⁴)(3.00 × 10⁸) / (200 × 10⁻⁹)

E = 9.95 × 10⁻¹⁹ J = 6.22 eV

(b) Maximum kinetic energy:

KE_max = hf − φ = 6.22 − 4.3 = 1.92 eV

In joules: KE_max = 1.92 × 1.60 × 10⁻¹⁹ = 3.07 × 10⁻¹⁹ J

(c) Stopping potential:

V_s = KE_max / e = 1.92 V

Key Equations at a Glance

For your A-Level exam, these are the essential relationships:

Photon energy: E = hf = hc/λ
Einstein’s photoelectric equation: hf = φ + KE_max
Stopping potential: eV_s = KE_max
Threshold frequency: f₀ = φ / h
Photocurrent: I ∝ photon intensity (above threshold)
Electronvolt conversion: 1 eV = 1.60 × 10⁻¹⁹ J

Common Misconceptions

“Increasing intensity increases the kinetic energy of photoelectrons.” No — intensity only increases the number of electrons emitted per second (the photocurrent). Each individual electron’s kinetic energy is determined solely by the frequency of the incident photon and the work function of the metal.

“Below the threshold frequency, increasing intensity might eventually release electrons.” No — if the photon energy hf is below the work function φ, no individual photon carries enough energy. A million low-energy photons cannot combine to release a single electron; the interaction is one photon per one electron.

“The photoelectric effect proves light is a particle, not a wave.” Not quite — it demonstrates that light exhibits particle-like behaviour in certain interactions. Modern physics accepts wave-particle duality: light shows wave behaviour (interference, diffraction) and particle behaviour (photoelectric effect, Compton scattering) depending on the experiment.

Exam Tips for A-Level Physics

When describing the photoelectric effect experiment, always mention the four key observations and explain why each contradicts classical wave theory.
In calculations, convert everything to SI units (joules, metres, seconds) unless working exclusively in electronvolts. The conversion factor 1 eV = 1.60 × 10⁻¹⁹ J is in your data sheet — use it.
The gradient of the V_s-f graph is h/e ≈ 4.14 × 10⁻¹⁵ V·s, and this is the same for all metals. If an exam question asks what would change if a different metal were used, the answer is: only the x-intercept (f₀) shifts; the gradient remains unchanged.
Be precise with terminology: “photoelectrons” refers to electrons emitted via the photoelectric effect (one photon, one electron). “Electron” is the general term.
Always define your symbols — especially distinguishing f (frequency, Hz) from φ (work function, J or eV).

光电效应：A-Level 物理完整指南

想象一束光照射到金属表面，电子随即飞离。听起来很简单对吗？然而在1887年，海因里希·赫兹发现了一个令人深省的现象：紫外光可以从金属表面打出电子，而可见光——无论强度多大——都无法做到。这一观察最终颠覆了几个世纪的经典物理学，并开启了量子力学的时代。

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，电子从金属表面逸出的现象。这个现象之所以如此非凡，不在于它会发生这个事实，而在于它发生的方式——经典波动理论根本无法解释的方式。在本指南中，我们将探讨实验观察、爱因斯坦为解释这些现象而建立的理论框架，以及将这些联系在一起的数学原理。

实验观察

当物理学家系统地研究光电效应时，他们发现了四个关键特征需要解释：

1. 阈值频率 — 对于任何给定的金属，存在一个入射光的最小频率，低于此频率时，无论光源强度多大，都不会有电子逸出。对于锌来说，这个阈值在紫外区域。对于钠和钾，阈值落在可见光范围内。如果你用红光照射锌——无论什么强度、多长时间、什么透镜组合——永远无法使任何一个电子离开。但紫外光，即使在最弱的强度下，也能立即产生电子。

2. 瞬时发射 — 光照射到金属与电子逸出之间没有可测量的时间延迟。经典波动理论预测，电子需要从波前逐渐吸收能量，直到积累足够的能量才能逃逸——对于低强度光来说，这个过程可能需要几秒甚至几分钟。实验表明，即使在最低强度下，电子发射也在光照后的10⁻⁹秒内开始。

3. 动能取决于频率而非强度 — 逸出光电子的最大动能随入射光频率的增加而线性增加。将光强度加倍会使逸出电子的数量（光电流）加倍，但不会改变每个电子的动能。更亮的光产生更多的电子，而不是更快的电子。只有增加频率才能产生具有更高动能的电子。

4. 强度只影响电子数量 — 只要入射光高于阈值频率，电子逸出的速率（光电流）就与光强度成正比。这在直觉上是合理的：每秒更多的光子意味着每秒更多的电子-光子相互作用。

为什么经典波动理论失败了

经典物理学将光描述为连续的电磁波。根据这一观点，波携带的能量均匀分布在其波前上。金属中的电子逐渐吸收这种能量，一旦积累足够的能量克服金属的功函数，电子就会逃逸。

这个模型做出了三个预测，但全部被实验推翻：

任何频率在足够强度下都应该有效。 在波动图像中，如果等待足够长的时间，即使是低频光也应该将足够的能量沉积到电子上使其逃逸。实验表明：低于阈值频率时，永远不会有电子逸出。
低强度下应该有可测量的时间延迟。 如果降低强度，每秒每单位面积到达的能量减少，”积累时间”应该增加。实验表明电子发射总是瞬时的。
强度应该影响动能。 更强的波携带更多能量，电子应该以更高的动能逸出。实验表明动能仅取决于频率而非强度。

波动模型被打破了。物理学需要新的思想。

爱因斯坦的光子模型（1905年）

在他关于光电效应的奇迹年论文中，阿尔伯特·爱因斯坦提出了一个与经典思维彻底偏离的观点：光不是连续的波，而是由称为光子的分立能量包组成。每个光子携带的能量由下式给出：

E = hf

其中 h 是普朗克常数（6.63 × 10⁻³⁴ J·s），f 是辐射频率。这个单一的方程——看似简单——解决了光电效应的每一个悖论。

在爱因斯坦的图像中，每个光电子由一个光子在一次一对一相互作用中释放。光子在一个瞬时事件中将所有能量转移给电子。电子用其中一部分能量来克服将其束缚在金属中的吸引力——这个最小所需能量称为金属的功函数（φ）。剩余的能量成为电子的动能。

这导出了爱因斯坦光电方程：

hf = φ + KE_max

或等价地：

KE_max = hf − φ

该方程优雅地解释了每一个实验观察：

阈值频率存在的原因：当 hf < φ 时，光子的能量不足以克服功函数。无论多少光子到达，都无法释放电子。阈值频率 f₀ 就是 f₀ = φ / h。
发射是瞬时的原因：能量转移是一个全有或全无的事件。没有积累——光子要么有足够的能量，要么没有。
动能取决于频率的原因：KE_max = hf − φ。更高频率的光子携带更多能量，因此克服 φ 后的剩余动能更大。
强度只影响光电流的原因：强度是每秒每单位面积光子数的量度。更多光子 = 更多一对一相互作用 = 更多电子，但每个电子仍然每次只接收 hf 的能量。

遏止电压实验

电子动能与光频率之间的关系是通过使用光电管和可变反向电压（称为遏止电压 V_s）实验测量的。

在这个实验中，金属阴极被已知频率的单色光照射。逸出的光电子向阳极移动，产生光电流。可变电源施加反向电势差来阻止电子流动。随着反向电压的增加，到达阳极的电子减少，光电流减小。光电流降为零时的电压就是遏止电压——它直接量度光电子的最大动能：

eV_s = KE_max = hf − φ

整理得：

V_s = (h/e)f − φ/e

这是 V_s 与 f 之间的线性关系。通过测量几种不同频率入射光的 V_s，并将 V_s 对 f 作图，我们得到一条直线，其梯度为 h/e，x轴截距为阈值频率 f₀。这个实验由罗伯特·密立根于1916年首次完成，为当时精确测定普朗克常数提供了方法。

V_s-f 图的关键特征：

梯度： h/e — 对所有金属相同（普适常数）。
x轴截距： 阈值频率 f₀ — 不同金属不同。
y轴截距： −φ/e — 功函数除以电子电荷的负值。
线性： 直线证实 KE_max ∝ f，正如爱因斯坦所预测的。

常见金属的功函数

功函数 φ 是材料特定的属性——将电子从金属表面提取出来所需的最小能量。以下是典型值：

金属	功函数 φ (eV)	功函数 φ (J)	阈值波长 λ₀ (nm)
钠 (Na)	2.3	3.68 × 10⁻¹⁹	539
钾 (K)	2.3	3.68 × 10⁻¹⁹	539
钙 (Ca)	2.9	4.64 × 10⁻¹⁹	428
锌 (Zn)	4.3	6.88 × 10⁻¹⁹	288
铂 (Pt)	6.4	1.02 × 10⁻¹⁸	194

注意碱金属（钠、钾）具有低功函数，因此它们的阈值频率落在可见光范围内。这也是为什么这些金属被用于光电倍增管和夜视设备中。像锌和铂这样的金属需要紫外光才能触发光电发射。

例题讲解

问题： 波长为200 nm的紫外光照射在锌表面（φ = 4.3 eV）。计算 (a) 单个光子的能量，(b) 逸出光电子的最大动能（以焦耳和电子伏特表示），以及 (c) 遏止电压。

解答：

(a) 光子能量：

E = hf = hc/λ = (6.63 × 10⁻³⁴)(3.00 × 10⁸) / (200 × 10⁻⁹)

E = 9.95 × 10⁻¹⁹ J = 6.22 eV

(b) 最大动能：

KE_max = hf − φ = 6.22 − 4.3 = 1.92 eV

以焦耳表示：KE_max = 1.92 × 1.60 × 10⁻¹⁹ = 3.07 × 10⁻¹⁹ J

(c) 遏止电压：

V_s = KE_max / e = 1.92 V

关键公式一览

对于你的A-Level考试，以下是基本关系：

光子能量： E = hf = hc/λ
爱因斯坦光电方程： hf = φ + KE_max
遏止电压： eV_s = KE_max
阈值频率： f₀ = φ / h
光电流： I ∝ 光子强度（高于阈值时）
电子伏特换算： 1 eV = 1.60 × 10⁻¹⁹ J

常见误区

“增加强度会增加光电子的动能。” 错误——强度只增加每秒逸出的电子数量（光电流）。每个电子个体的动能仅由入射光子的频率和金属的功函数决定。

“低于阈值频率时，增加强度最终可能释放电子。” 错误——如果光子能量 hf 低于功函数 φ，没有单个光子携带足够的能量。一百万个低能光子不能联合释放一个电子；相互作用是一个光子对一个电子。

“光电效应证明光是一种粒子，而不是波。” 不完全正确——它证明光在某些相互作用中表现出粒子样行为。现代物理学接受波粒二象性：光根据实验类型可以展示波动行为（干涉、衍射）和粒子行为（光电效应、康普顿散射）。

A-Level 物理考试提示

描述光电效应实验时，始终提到四个关键观察，并解释为什么每个观察都与经典波动理论相矛盾。
计算时将所有量转换为国际单位制（焦耳、米、秒），除非完全在电子伏特系统中工作。换算系数 1 eV = 1.60 × 10⁻¹⁹ J 在你的数据表中——使用它。
V_s-f 图的梯度是 h/e ≈ 4.14 × 10⁻¹⁵ V·s，这对所有金属都相同。如果考试题目问使用不同金属会有什么变化，答案是：只有 x 轴截距（f₀）移动；梯度保持不变。
使用精确的术语：”光电子”（photoelectrons）指通过光电效应逸出的电子（一个光子，一个电子）。”电子”（electron）是通用术语。
始终定义你的符号——特别要区分 f（频率，Hz）和 φ（功函数，J 或 eV）。

The Photoelectric Effect: A Complete Guide for A-Level Physics | 光电效应：A-Level物理完整指南

The Photoelectric Effect: When Light Behaves Like a Particle | 光电效应：当光表现得像粒子