Question

Millikan determined the charge on the electron by studying the static charges on oil drops falling in an electric field (Figure 2.5$)$. A student carried out this experiment using several oil drops for her measurements and calculated the charges on the drops. She obtained the following data: $$\overline{)\begin{array}{lc}\text{ Droplet }& \text{ Calculated Charge (C) }\\ \text{ A }& 1.60\times {10}^{-19}\\ \text{B }& 3.15\times {10}^{-19}\\ \text{C }& 4.81\times {10}^{-19}\\ \text{D }& 6.31\times {10}^{-19}\end{array}}$$ (a) What is the significance of the fact that the droplets carried different charges? (b) What conclusion can the student draw from these data regarding the charge of the electron? (c) What value (and to how many significant figures) should she report for the electronic charge?

Short answer

The different charges on the oil droplets indicate that they contain differing numbers of electrons, with each droplet having a different multiple of the smallest charge. The student can conclude that the charge of a single electron is the smallest charge in the dataset, which is $1.60\times {10}^{-19}$ C. This value is supported by the nearly whole number ratios of the other charges with respect to Droplet A's charge. Therefore, the student should report the electronic charge as $1.60\times {10}^{-19}$ C.

Step-by-step solution

Identify the smallest charge in the dataset

In order to understand the significance of the different charges, we first need to identify the smallest charge in the dataset. This is because the smallest charge usually represents the charge of a single electron. In this case, the smallest charge from the dataset is 1.60 x 10^{-19} C for Droplet A.

Determine the significance of the different charges

The fact that the droplets carried different charges indicates that they contain differing numbers of electrons, with each droplet having a different multiple of the smallest charge. The presence of multiple charges on these droplets is a result of different numbers of electrons being attached to them.

Analyze the relationship between the charges

Now, we need to analyze the relationship between the charges on the droplets to determine the conclusion the student can draw regarding the charge of the electron. The charges on the 4 droplets are as follows: Droplet A: $1.60\times {10}^{-19}$ C Droplet B: $3.15\times {10}^{-19}$ C Droplet C: $4.81\times {10}^{-19}$ C Droplet D: $6.31\times {10}^{-19}$ C We can see that as the charge of the droplets increase, they approach a common multiple of the smallest charge (Droplet A), which is likely representative of the charge of a single electron.

Determine the charge of the electron

In this step, we will calculate the charge of the electron by dividing the charges of the other droplets by the smallest charge. Droplet B charge / Droplet A charge = $\frac{3.15\times {10}^{-19}}{1.60\times {10}^{-19}}\approx 1.97$ Droplet C charge / Droplet A charge = $\frac{4.81\times {10}^{-19}}{1.60\times {10}^{-19}}\approx 3.01$ Droplet D charge / Droplet A charge = $\frac{6.31\times {10}^{-19}}{1.60\times {10}^{-19}}\approx 3.94$ The values are close to whole numbers (2, 3, and 4), which indicates that the smallest charge (Droplet A: $1.60\times {10}^{-19}$ C) is likely the charge of a single electron.

Report the value of the electronic charge

The student should report the value of the electronic charge as the smallest charge in the dataset, to the given significant figures: $1.60\times {10}^{-19}$ C. This value is supported by the nearly whole number ratios of the other charges with respect to Droplet A's charge, which suggests that the droplets contain different numbers of electrons but the basic unit of charge (the charge of one electron) remains consistent.

Key concepts

Charge of an Electron

The charge of an electron is a fundamental physical constant that plays a crucial role in the world of atomic and subatomic particles. Measuring this tiny charge was first accomplished by Robert Millikan through his famous oil drop experiment, where he isolated and measured the charge on tiny oil droplets. The smallest charge he found was consistently around $1.60\times {10}^{-19}$ coulombs (C), which is considered the charge of a single electron.

In the experiment, various droplets were found to have different charges. These charges were multiples of this smallest charge, suggesting that each droplet carried a different number of electrons. Therefore, by analyzing these charges, it becomes apparent that the charge of any object is quantized, meaning it exists in discrete "packets" equal to the charge of an electron.

• The smallest charge observed in a system of particles is typically the charge of one electron.
• The charge of an electron is a fixed constant: $1.60\times {10}^{-19}\text{C}$.

Significant Figures

Significant figures are used to express the precision of a calculated or measured value. They indicate which digits in a number are important in reflecting the true measurement. When reporting the charge of an electron, as determined through an experiment, it is crucial to use the appropriate number of significant figures to show the accuracy of the measurement.

In the Millikan oil drop experiment example, a droplet charge is reported as $1.60\times {10}^{-19}\text{C}$, indicating three significant figures. This means the measurement is precise to three digits, including all known digits plus one estimated digit.

• Use significant figures to convey the accuracy of a measurement.
• Ensure consistency in the number of significant figures reported across results.
• Significant figures help in maintaining consistency and reliability in scientific reporting.

Experimental Physics

Experimental physics refers to the use of empirical data and observations to understand physical phenomena. The aim is to test theories, understand matter and energy, as well as make empirical deductions about physical laws.

The Millikan oil drop experiment is a classic example of experimental physics. Millikan's meticulous manipulation of electric fields allowed him to observe and measure the tiny charges on oil droplets. His approach provided clear empirical evidence of the quantization of electric charge.

• Involves hands-on data collection through experiments.
• Used to verify the validity of physics theories.
• Highlights the importance of precision and control in scientific investigations.

Quantization of Charge

The quantization of charge is a principle that states charge comes in discrete units called "quanta." An electron, with its charge of $1.60\times {10}^{-19}\text{C}$, is the fundamental unit of charge, and any excess or deficit of charge on an object corresponds to a whole number of electrons.

In the context of the Millikan experiment, different oil drops carried charges that were simple whole-number multiples of this basic unit of charge. This shows that charge is not continuous but instead quantized, a finding that has profound implications in both classical and quantum physics.

• Charges exist in discrete units rather than any small increment.
• Electrons are the smallest carriers of non-zero charge.
• Quantization supports various modern physics theories like quantum mechanics.