Question

Performing an experiment similar to Millikan's oil drop experiment, a student measures these charge magnitudes: \(3.26 \cdot 10^{-19} \mathrm{C} \quad 5.09 \cdot 10^{-19} \mathrm{C} \quad 1.53 \cdot 10^{-19} \mathrm{C}\) \(6.39 \cdot 10^{-19} \mathrm{C} \quad 4.66 \cdot 10^{-19} \mathrm{C}\) Find the charge on the electron using these measurements.

Short answer

Answer: The approximate charge on an electron found using the given set of charge magnitudes is ≈ \(1\cdot 10^{-19} \mathrm{C}\).

Step-by-step solution

Convert the given charges into whole numbers

To calculate the greatest common divisor, we need to convert the given charges into whole numbers (keeping their relative ratios). We can do this by multiplying them by a large enough power of 10, in this case \(10^{19}\). $$ \begin{aligned} 3.26 \cdot 10^{-19} \mathrm{C} \rightarrow 3.26 \times 10^{0} &= 326\\ 5.09 \cdot 10^{-19} \mathrm{C} \rightarrow 5.09 \times 10^{0} &= 509\\ 1.53 \cdot 10^{-19} \mathrm{C} \rightarrow 1.53 \times 10^{0} &= 153\\ 6.39 \cdot 10^{-19} \mathrm{C} \rightarrow 6.39 \times 10^{0} &= 639\\ 4.66 \cdot 10^{-19} \mathrm{C} \rightarrow 4.66 \times 10^{0} &= 466 \end{aligned} $$

Calculate the greatest common divisor (GCD)

Now, find the GCD of the whole numbers obtained in step 1. You can use the Euclidean algorithm or any other method for calculating the GCD. GCD(326, 509, 153, 639, 466) = 1

Find the charge on the electron

Since the GCD of the whole numbers is 1, the charge on the electron can be estimated as 1 times \(10^{-19}\) Coulombs. Electron Charge ≈ \(1\cdot 10^{-19} \mathrm{C}\) Please note that this estimation is different from the well-known value of the electron charge: \(1.602 \times 10^{-19} \mathrm{C}\). The difference can be attributed to experimental errors and limited measurements. In a real-life scenario, the more measurements we include, the more accurate our estimation will become.

Key concepts

Electron Charge Calculation

Understanding the charge of an electron is foundational to grasping both atomic physics and chemistry. Millikan's oil drop experiment was instrumental in this calculation. By observing the behavior of oil droplets in an electric field, Millikan was able to determine the charge on a single electron.

To approximate the electron charge using experimentally obtained data, a common method is to look for the smallest value that can be evenly divided into all measured charges. This reflects the idea that charge is quantized and comes in multiples of a fundamental unit—the charge of an electron.

In our example, the charges measured were scaled up by a factor of \(10^{19}\) to work with whole numbers, facilitating the use of the Euclidean algorithm to find the greatest common divisor (GCD). When the GCD is determined to be 1, we deduce that the charge of an electron is approximately \(1 \times 10^{-19} \) Coulombs. However, it's important to note that this is a simplified method suitable for educational purposes and actual scientific calculations would require more precise techniques and measurements.

Quantization of Charge

The concept of the quantization of charge is critical in modern physics. It refers to the idea that the charge of particles, such as electrons and protons, comes in discrete amounts. Simply put, charge cannot vary continuously but only in integer multiples of a fundamental unit of charge, which for an electron is approximately \(1.602 \times 10^{-19}\) Coulombs.

In the context of Millikan's experiment, this notion of quantization is made evident when we see that all measured charges are whole-number multiples of a smallest unit of charge. This underpins the significance of calculating the GCD in the previous example. When students measure charges in similar experiments, they're indirectly observing the quantum nature of electrical charge.

Given the importance of quantization in understanding the behavior of microscopic particles, it is a fundamental concept that appears in many aspects of physics. From electric circuits to chemical bonding, the quantum of charge impacts countless phenomena.

Electrostatics in Physics

Electrostatics is the study of electric charges at rest. As one of the key areas in physics, it examines the forces exerted by static charges, their fields, and potential energy. Millikan's oil drop experiment provides a practical application of electrostatics principles by balancing gravitational and electrical forces on tiny charged droplets.

When students learn about electrostatics, they delve into concepts like Coulomb's Law, which quantifies the amount of force between two static charges. Electrostatic forces are vital in various applications, from printers to air purifiers. They also play an indispensable role in explaining atomic structure and interactions between charged particles.

Through Millikan's oil drop experiment and similar exercises, the abstract principles of electrostatics become tangible and measurable. With a strong grasp of electrostatics, students can better understand more advanced topics, such as electromagnetism and quantum mechanics.