📚 The Millikan Oil-Drop Experiment and the Elementary Charge | 密立根油滴实验与元电荷测定
Between 1909 and 1913, the American physicist Robert Andrews Millikan performed a series of experiments that would decisively measure the charge of the electron. His oil-drop experiment is regarded as one of the most elegant and important measurements in the history of physics, for it demonstrated that electric charge exists only in discrete multiples of a fundamental unit.
1909 年至 1913 年间，美国物理学家罗伯特·安德鲁斯·密立根进行了一系列实验，最终精确测定了电子的电荷量。他的油滴实验被视为物理学史上最精巧、最重要的测量之一，因为它证明了电荷只能以某一基本单位的整数倍存在。
1. Historical Context | 历史背景
By the late nineteenth century, evidence for the existence of the electron was already strong. J. J. Thomson had measured the charge-to-mass ratio e/m of cathode rays in 1897, but neither e nor m was known separately. A direct measurement of the elementary charge e was therefore of fundamental importance.
十九世纪末，电子存在的证据已经相当充分。1897 年 J. J. 汤姆孙测量了阴极射线的荷质比 e/m，但人们仍然不知道单个电荷 e 和质量 m 的具体数值。因此，直接测定基本电荷 e 具有根本性的重要意义。
Millikan's achievement was to observe the motion of a single microscopic oil droplet under the combined influence of gravity, buoyancy, air drag and an electric field. By balancing these forces, he could determine the charge carried by the droplet with remarkable precision.
密立根的成就，在于观察单个微小油滴在重力、浮力、空气阻力和电场共同作用下的运动。通过平衡这些力，他能够以惊人的精度确定油滴所携带的电荷量。
2. Key Concepts and Quantities | 关键概念与物理量
Before analysing the experiment, we must define the physical quantities involved.
在分析实验之前，必须先明确所涉及的物理量。
- e — the elementary charge, approximately 1.602 × 10⁻¹⁹ C.
- e — 元电荷，约为 1.602 × 10⁻¹⁹ C（库仑）。
- q — the net charge on an individual oil droplet; always q = n e, where n is an integer.
- q — 单个油滴所带的净电荷；始终满足 q = n e，其中 n 为整数。
- E — the uniform electric field between the parallel plates, E = V / d.
- E — 平行板之间的匀强电场，E = V / d（d 为板间距）。
- m — the mass of the oil droplet.
- m — 油滴的质量。
- ρ_oil — the density of the oil; ρ_air — the density of air.
- ρ_oil — 油的密度；ρ_air — 空气的密度。
- r — the radius of the droplet, assumed spherical.
- r — 油滴的半径，假设为球形。
- v_terminal — the terminal speed of the droplet during free fall or rise under the electric field.
- v_terminal — 油滴在自由下落或电场中上升时的终端速度。
E = V / d
The electric force on the droplet is Fe = q E. The gravitational force is Fg = m g. The buoyant force is Fb = ρ_air Vdrop g, which is tiny compared with m g because ρ_oil >> ρ_air, but is not ignored in the most accurate treatment.
油滴所受电场力为 Fe = q E。重力为 Fg = m g。浮力为 Fb = ρ_air Vdrop g。由于油的密度远大于空气密度，浮力相比重力很小，但在最精确的计算中不能忽略。
3. Experimental Setup | 实验装置
The apparatus consists of two horizontal metal plates separated by a small distance d, typically about 6 mm. A voltage V applied across the plates creates a uniform electric field E = V/d. The region between the plates is illuminated by a strong light source and observed through a low-power microscope fitted with a graticule.
实验装置由两块水平金属平行板组成，板间距 d 很小，通常约为 6 mm。在两板间施加电压 V，产生匀强电场 E = V/d。两板之间的区域由强光源照亮，并通过带有刻度尺的低倍显微镜观察。
Oil is sprayed from an atomiser above the upper plate. Some droplets pass through a small hole in the upper plate and enter the space between the plates. The microscope allows the experimenter to select a single droplet and follow its motion.
油液通过上方的喷雾器喷出。部分油滴穿过上板的小孔进入两板之间的空间。显微镜使实验者能够选择单个油滴并追踪其运动。
- The upper plate can be charged positively or negatively as required.
- 上极板可按需要带正电或负电。
- The whole chamber is enclosed to reduce air currents.
- 整个腔室是封闭的，以减少气流干扰。
- X-rays or a radioactive source are used to ionise the air, altering the charge on a droplet between runs.
- 使用 X 射线或放射源电离空气，以便在不同轮次之间改变油滴的电荷量。
4. Stokes' Law and the Droplet Radius | 斯托克斯定律与油滴半径
A tiny sphere moving through a viscous fluid experiences a drag force given by Stokes' law:
微小球体在粘性流体中运动时会受到粘滞阻力，其表达式由斯托克斯定律给出：
Fdrag = 6 π η r v
where η is the viscosity of air. When the droplet falls under gravity, it rapidly reaches a terminal velocity v₁ where the drag force balances the effective weight.
其中 η 是空气的粘滞系数（粘度）。当油滴在重力作用下下落时，它会很快达到终端速度 v₁，此时粘滞阻力与有效重力平衡。
Let P = m g be the weight and B = ρ_air (4/3)π r³ g be the buoyancy. The effective downward force is:
设 P = m g 为重力，B = ρ_air (4/3)π r³ g 为浮力，则向下的合力为：
(m - Mair) g = 6 π η r v₁
where Mair is the mass of air displaced by the droplet. Substituting m = ρ_oil (4/3)π r³ and Mair = ρ_air (4/3)π r³:
其中 Mair 为被油滴排开的空气质量。代入 m = ρ_oil (4/3)π r³ 和 Mair = ρ_air (4/3)π r³，可得：
(4/3) π r³ (ρ_oil − ρ_air) g = 6 π η r v₁
Rearranging, the radius of the droplet is:
整理后，可得油滴的半径为：
r = √( 9 η v₁ / (2 (ρ_oil − ρ_air) g ) )
Thus, by measuring the terminal fall velocity v₁, we know the radius r and therefore the mass m of the droplet directly.
因此，通过测量自由下落的终端速度 v₁，就可以确定油滴半径 r，进而直接得到油滴质量 m。
5. Force Balance with the Electric Field | 电场中的力平衡
Once the droplet is falling under gravity, a voltage is applied across the plates. The electric field acts upward if the droplet is negatively charged and the lower plate is positive. If the field is strong enough, the droplet rises with terminal velocity v₂.
当油滴在重力作用下下落时，在两板间施加电压。如果油滴带负电且下板为正，则电场力向上。当场强足够大时，油滴将以终端速度 v₂ 上升。
When the droplet is rising at terminal speed v₂, the upward forces are:
当油滴以终端速度 v₂ 上升时，向上的力为：
q E = (4/3) π r³ (ρ_oil − ρ_air) g + 6 π η r v₂
Notice that the drag force now acts downward because the droplet is moving upward. Combining this result with the falling case v₁ above gives:
注意：由于油滴向上运动，此时粘滞阻力方向向下。将这一结果与前面自由下落的情形 v₁ 联立，可以得到：
q E = 6 π η r (v₁ + v₂)
Since E = V/d, the charge q on the droplet is:
又因 E = V/d，所以油滴上的电荷 q 为：
q = 6 π η r d (v₁ + v₂) / V
This is the principal working equation of the experiment.
这就是油滴实验的主要工作方程。
6. The Milikan Falling-and-Rising Method | 密立根升降法
In practice, Millikan adopted the following procedure:
在实际操作中，密立根采用以下步骤：
- Select a droplet in the field of view, and measure the time t₁ for it to fall a known distance s. Then v₁ = s / t₁.
- 在视野中选定一个油滴，测量它下落已知距离 s 所需的时间 t₁，则 v₁ = s / t₁。
- Apply the electric field and measure the time t₂ for the same droplet to rise the same distance s. Then v₂ = s / t₂.
- 施加电场，测量同一油滴上升同样距离 s 所需的时间 t₂，则 v₂ = s / t₂。
- Repeat the measurements for the same droplet under different voltages and ionisation conditions.
- 在同一油滴上，于不同电压和不同电离条件下重复测量。
Because the droplet remains the same, its radius r and mass m never change during these successive runs. What changes is the charge q on the droplet whenever ions are captured from the ionised air.
由于油滴本身没有改变，其半径 r 和质量 m 在连续多轮观测中保持不变。会变化的只是油滴所带的电荷量 q——每当油滴从电离空气中捕获离子时，q 就会改变。
7. Observing the Quantisation of Charge | 观察电荷的量子化
Suppose that for one particular droplet Millikan obtains a series of charge values: q₁, q₂, q₃, … . If he subtracts the smallest value from each of the others, he often finds differences that are integer multiples of a common value. This common value is precisely the elementary charge e.
假设对于某一个油滴，密立根得到了一系列电荷值：q₁、q₂、q₃……如果他用其他各值减去最小值，通常会得到某个共同值的整数倍之差。这个共同值正是基本电荷 e。
More systematically, the charge on a droplet is always found to be:
更系统地讲，油滴上的电荷总是被表示为：
q = n e, n = ±1, ±2, ±3, …
where e = 1.60 × 10⁻¹⁹ C. In his 1913 paper, Millikan wrote: "The charge on the electron is e = 4.774 × 10⁻¹⁰ statcoulombs", which in modern SI units is approximately 1.592 × 10⁻¹⁹ C, very close to the accepted value.
其中 e = 1.60 × 10⁻¹⁹ C。在 1913 年的论文中，密立根写道："电子的电荷为 e = 4.774 × 10⁻¹⁰ 静库仑"，换算为现代国际单位制约为 1.592 × 10⁻¹⁹ C，已经非常接近今天公认的数值。
The statistical nature of this result is important: Millikan did not rely on one droplet. He examined dozens of droplets over many months, and for each one, the values of q were integer multiples of the same fundamental charge e. This reproducibility across different droplets of entirely different sizes proved that charge is quantised in nature, not merely an artefact of one droplet.
这一结果的统计性很重要：密立根并不依赖单个油滴。他在数月间观测了数十个油滴，而每一个油滴上的电荷值都是同一个基本电荷 e 的整数倍。不同大小、完全不同的油滴之间具有这种可重复性，证明了电荷的量子化是自然界的本质特性，而非某个油滴的偶然结果。
8. Sources of Error and Corrections | 误差来源与修正
Several sources of systematic error must be considered carefully.
以下几类系统误差需要认真加以考虑。
- Viscosity of air varies with temperature — η must be measured at the actual temperature inside the chamber.
- 空气粘度随温度变化 — 必须在腔室实际温度下测定 η。
- Stokes' law is only valid for continuous media — when the droplet radius r is comparable to the mean free path of air molecules, a correction (Cunningham correction) must be applied.
- 斯托克斯定律仅适用于连续介质 — 当油滴半径 r 与空气分子平均自由程相当时，需要引入坎宁安修正。
- Evaporation of the oil droplet — the radius r may change slowly, so the experimentalist must re-measure v₁ between runs.
- 油滴蒸发 — 油滴半径 r 可能缓慢改变，因此实验者必须在多轮测量之间重新测定 v₁。
- Brownian motion — random molecular bombardment causes the droplet to jitter, affecting the timing precision. Millikan minimized this by choosing droplets with a suitable radius, about 1 μm.
- 布朗运动 — 分子的随机撞击使油滴抖动，影响计时精度。密立根通过选择半径约为 1 μm 的合适油滴来减小布朗运动的影响。
- Convection currents in the air — these are eliminated by performing the experiment in a well-sealed chamber of uniform temperature.
- 空气对流 — 通过在密封良好、温度均匀的腔室中进行实验来消除对流。
9. Results and Their Significance | 实验结果及其意义
Millikan's final value for e was remarkably precise for its time. Current accepted value:
密立根最终测得的 e 值在当时达到了惊人的精度。现代公认值为：
e = 1.602 176 634 × 10⁻¹⁹ C
The result has three deep implications:
这一结果具有三层深远意义：
| Implication | 意义 | Explanation | 解释 |
| Atomic structure | The electron carries one unit of elementary charge; ions carry integer multiples. |
| 原子结构 | 电子携带一个单位的基本电荷；离子携带其整数倍。 |
| Quantisation of charge | No charge smaller than e has ever been observed in free particles. |
| 电荷量子化 | 在自由粒子中，从未观察到比 e 更小的电荷。 |
| Determination of other constants | Combined with Thomson's e/m, Millikan's e gives the mass m of the electron directly: me = e / (e/m). |
| 确定其他物理常数 | 结合汤姆孙的 e/m 与密立根的 e，可直接得到电子质量 me = e / (e/m)。 |
10. Worked Example | 计算示例
A student repeats the experiment with a single droplet. The following observations are made:
某学生用一个油滴重复该实验，得到如下观测数据：
Plate separation d = 6.00 × 10⁻³ m. Density of oil ρ = 875 kg m⁻³. Viscosity of air η = 1.83 × 10⁻⁵ Pa s. The droplet falls s = 1.20 × 10⁻³ m in t₁ = 12.0 s with no electric field. Under a voltage V = 300 V, the same droplet rises the same distance in t₂ = 8.0 s.
板间距 d = 6.00 × 10⁻³ m。油密度 ρ = 875 kg m⁻³。空气粘度 η = 1.83 × 10⁻⁵ Pa·s。无电场时油滴下落 s = 1.20 × 10⁻³ m 用时 t₁ = 12.0 s。在电压 V = 300 V 下，同一油滴上升同样的距离用时 t₂ = 8.0 s。
Step 1 — find the terminal velocities:
第 1 步——求终端速度：
v₁ = s / t₁ = 1.00 × 10⁻⁴ m s⁻¹, v₂ = s / t₂ = 1.50 × 10⁻⁴ m s⁻¹
Step 2 — find the radius of the droplet:
第 2 步——求油滴半径：
r² = 9 η v₁ / (2 ρ g) = (9 × 1.83 × 10⁻⁵ × 1.00 × 10⁻⁴) / (2 × 875 × 9.81)
r = 9.80 × 10⁻⁷ m = 0.98 μm
(Here ρ_air is neglected since ρ_oil >> ρ_air.)
（此处忽略空气密度，因为 ρ_oil >> ρ_air。）
Step 3 — find the charge q:
第 3 步——求电荷 q：
q = 6 π η r d (v₁ + v₂) / V
q = 6π × 1.83 × 10⁻⁵ × 9.80 × 10⁻⁷ × 6.00 × 10⁻³ × 2.50 × 10⁻⁴ / 300
q = 1.69 × 10⁻¹⁸ C ≈ 10.6 e
This droplet carries approximately 11 elementary charges, which is entirely reasonable when using X-rays to ionise the air.
该油滴携带约 11 个基本电荷。在使用 X 射线电离空气时，这是完全合理的结果。
11. Common Student Errors in Exam Questions | 考试中常见错误
In written examinations on this experiment, students frequently make the following mistakes.
在关于该实验的笔试中，考生经常犯以下错误。
- Forgetting to convert time and distance into SI units before substitution.
- 代入公式前忘记把时间和距离换算成国际单位。
- Confusing which velocity is v₁ and which is v₂ in the equation q = 6π η r d (v₁+v₂)/V.
- 在 q = 6π η r d (v₁+v₂)/V 公式中混淆 v₁ 和 v₂ 各自的意义。
- Forgetting that the drag force direction reverses when the droplet rises.
- 忘记油滴上升时粘滞阻力的方向反向。
- Neglecting to mention the quantisation condition q = n e when asked to describe the conclusion.
- 在要求描述结论时，忘记说明电荷量子化条件 q = n e。
- Not explaining why oil is used instead of water — oil has a very low vapour pressure and evaporates negligibly during the experiment.
- 没有解释为什么使用油而不是水——油具有极低的饱和蒸气压，在实验过程中几乎不蒸发。
12. Why Millikan's Experiment Still Matters | 密立根油滴实验的当代意义
The oil-drop experiment is a landmark in the teaching of physics because it links macroscopic measurements — fall times, voltages, separations — to a quantity of atomic scale, the charge of the electron. It also represents a masterpiece of experimental design: Millikan isolated a single microscopic object and controlled every force acting upon it.
油滴实验之所以是物理教学中的里程碑，是因为它将宏观测量——下落时间、电压、板间距——与原子尺度的量（电子电荷）联系起来。它同时代表了实验设计的杰作：密立根隔离了一个微观物体，并控制了作用于其上的每一个力。
Moreover, the experiment embodies the scientific method in a beautiful way: the hypothesis of charge quantisation was not merely asserted — it was tested against data taken from many different droplets, and every test confirmed the same fundamental constant. In modern physics, e is now a defining constant of the International System of Units, fixed exactly at 1.602 176 634 × 10⁻¹⁹ C since 2019, but the conceptual insight that charge comes in discrete packets remains Millikan's enduring legacy.
此外，该实验以优美的方式体现了科学方法：电荷量子化的假设不仅是断言，而是通过取自许多不同油滴的数据检验，每一次检验都证实了同一个基本常数。在现代物理学中，e 已成为国际单位制的定义常数，自 2019 年起被固定为 1.602 176 634 × 10⁻¹⁹ C；但电荷以离散小包形式存在的洞见仍然是密立根的不朽遗产。
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