📚 AQA A-Level Physics June 2018 Insert 5: Stopping Potential vs Frequency | AQA 2018年6月物理A级 Insert 5：遏止电压与频率
The photoelectric effect is a cornerstone of quantum physics, and the June 2018 AQA A-level Physics paper included an insert (Figure 5) that displayed a graph of stopping potential against frequency for electrons emitted from a metal surface. This graph is not just a piece of data; it is a key tool for determining fundamental constants such as Planck’s constant and the work function of the metal. This article explains the physics behind the graph, how to analyse it accurately, and how to avoid common exam mistakes.
光电效应是量子物理的基石，而在2018年6月AQA物理A级试卷中，插页5（Figure 5）展示了一张金属表面发射电子时遏止电压随频率变化的图表。这张图不仅仅是数据，它更是确定普朗克常数和金属逸出功等基本常数的关键工具。本文将解释该图背后的物理原理、如何准确分析它，以及如何避免常见的考试错误。
1. The Photoelectric Effect and Einstein’s Equation | 光电效应与爱因斯坦方程
The photoelectric effect occurs when electromagnetic radiation of high enough frequency falls on a metal surface, causing electrons to be emitted. Einstein’s photoelectric equation relates the energy of a single photon, \(hf\), to the work function, \(\phi\), and the maximum kinetic energy of the emitted electron, \(K_{\text{max}}\). The equation is \(hf = \phi + K_{\text{max}}\).
光电效应发生在频率足够高的电磁辐射照射金属表面时，导致电子被发射出来。爱因斯坦光电效应方程将一个光子的能量 \(hf\) 与逸出功 \(\phi\) 以及发射电子的最大动能 \(K_{\text{max}}\) 联系起来，即 \(hf = \phi + K_{\text{max}}\)。
hf = φ + Kmax
Here, \(h\) is Planck’s constant, \(f\) is the frequency of the incident light, and \(\phi\) is the work function—the minimum energy required to free an electron from the metal surface. \(K_{\text{max}}\) is the maximum kinetic energy of the photoelectrons, which can be measured using a stopping potential.
其中，\(h\) 是普朗克常数，\(f\) 是入射光的频率，\(\phi\) 是逸出功—从金属表面释放一个电子所需的最小能量。\(K_{\text{max}}\) 是光电子的最大动能，可以通过遏止电压来测量。
2. The Stopping Potential and the Electron-Volt | 遏止电压与电子伏特
To measure the maximum kinetic energy of photoelectrons, a potential difference is applied that opposes the flow of electrons. The stopping potential, \(V_s\), is the reverse voltage that just stops even the most energetic photoelectrons from reaching the anode. At this point, the electric potential energy lost by the electron equals its initial kinetic energy: \(eV_s = K_{\text{max}}\).
为了测量光电子的最大动能，我们施加一个反向电压来阻止电子流动。遏止电压 \(V_s\) 是恰好能使最动能的光电子也无法到达阳极的反向电压。在此情况下，电子失去的电势能等于其初始动能：\(eV_s = K_{\text{max}}\)。
eVs = Kmax
The unit volt is closely linked to the electron-volt (eV). An electron accelerated through a potential difference of 1 V gains an energy of 1 eV, which is equal to \(1.6 \times 10^{-19}\) J. Using this, the stopping potential in volts directly gives the maximum kinetic energy in electron-volts—a convenient correspondence in atomic physics.
电压单位与电子伏特（eV）密切相关。一个电子通过1 V电势差加速后获得的能量为1 eV，等于 \(1.6 \times 10^{-19}\) J。利用这一点，以伏特为单位的遏止电压直接给出了以电子伏特为单位的最大动能—这在原子物理学中是一组非常方便的对应关系。
3. Deriving the Linear Relationship | 推导线性关系
Combining Einstein’s photoelectric equation with the stopping potential equation gives \(hf = \phi + eV_s\). Rearranging for \(V_s\) produces a linear equation in terms of frequency:
将爱因斯坦光电效应方程与遏止电压方程结合，得到 \(hf = \phi + eV_s\)。将其改写为关于频率 \(V_s\) 的线性方程：
Vs = (h/e)f − φ/e
This is of the form \(y = mx + c\), where \(y = V_s\), \(x = f\), the gradient \(m = h/e\), and the intercept \(c = -\phi/e\). Therefore, plotting \(V_s\) against \(f\) should give a straight line whose gradient is a direct measure of Planck’s constant divided by the elementary charge.
这符合 \(y = mx + c\) 的形式，其中 \(y = V_s\)，\(x = f\)，梯度 \(m = h/e\)，截距 \(c = -\phi/e\)。因此，以 \(f\) 为横轴、\(V_s\) 为纵轴作图应得到一条直线，其梯度直接反映普朗克常数与基本电荷之比。
4. The Graph in Insert 5 | 插页5中的图像
The insert in the June 2018 AQA paper provided a graph of \(V_s\) against \(f\). The data points fell on a straight line, as predicted by the equation above. The graph was labelled with frequency on the horizontal axis and stopping potential on the vertical axis. Two key features were visible: the intercept on the vertical axis gave \(-\phi/e\), and the intercept on the horizontal axis (where \(V_s = 0\)) gave the threshold frequency, \(f_0\).
2018年6月AQA试卷的插页提供了一张 \(V_s\) 对 \(f\) 的图像。数据点落在一条直线上，正如上述方程所预测。图像横轴标注为频率，纵轴标注为遏止电压。图中能看到两个关键特征：纵轴截距给出 \(-\phi/e\)，而横轴截距（\(V_s = 0\) 处）给出了极限频率 \(f_0\)。
At the threshold frequency, the maximum kinetic energy of the photoelectrons is zero, so \(hf_0 = \phi\). This means that the frequency at which the graph crosses the horizontal axis provides a direct way to determine the work function of the metal.
在极限频率下，光电子的最大动能为零，因此 \(hf_0 = \phi\)。也就是说，图像与横轴相交处的频率为确定金属逸出功提供了一种直接方法。
5. Calculating Planck’s Constant from the Gradient | 从梯度计算普朗克常数
The gradient of the \(V_s\) vs \(f\) graph is equal to \(h/e\). Therefore, by measuring the gradient using two well-separated points on the best-fit line, Planck’s constant can be found using \(h = e \times \text{gradient}\).
\(V_s\) 对 \(f\) 图像的梯度等于 \(h/e\)。因此，通过在最佳拟合直线上选取两个间隔较大的点来测量梯度，即可使用 \(h = e \times \text{梯度}\) 求出普朗克常数。
h = e × (ΔVs / Δf)
In the 2018 insert, the data points were typically such that a line through the points had a gradient of approximately \(4.14 \times 10^{-15}\) V/Hz. Multiplying by the elementary charge \(e = 1.60 \times 10^{-19}\) C gives \(h = 6.6 \times 10^{-34}\) J·s, which is close to the accepted value. It is essential to use the line of best fit, not the raw data points, and to choose points that are far apart to reduce percentage uncertainty.
在2018年插页中，数据点通常使通过各点的直线梯度接近 \(4.14 \times 10^{-15}\) V/Hz。乘以基本电荷 \(e = 1.60 \times 10^{-19}\) C 后，得到 \(h = 6.6 \times 10^{-34}\) J·s，这与公认值非常接近。重要的是使用最佳拟合直线而不是原始数据点，并选择相隔较远的点来减少百分比不确定度。
6. Calculating the Work Function from the Intercept | 从截距计算逸出功
The intercept of the graph on the \(V_s\)-axis (where \(f = 0\)) is \(-\phi/e\). In the June 2018 insert, extrapolating the line back to the y-axis gave a negative intercept. The magnitude of this intercept, multiplied by \(e\), gives the work function \(\phi\) in joules. Often, the work function is quoted in electron-volts instead, which is numerically equal to the magnitude of the intercept in volts.
图像在 \(V_s\) 轴上的截距（\(f = 0\) 处）为 \(-\phi/e\)。在2018年6月插页中，将直线反向延长到y轴会得到负截距。该截距的绝对值乘以 \(e\) 便得到以焦耳为单位的逸出功 \(\phi\)。通常，逸出功也会以电子伏特为单位给出，其数值等于以伏特为单位的截距绝对值。
φ = e × |intercept
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