Question

The electron microscope has been widely used to obtain highly magnified images of biological and other types of materials. When an electron is accelerated through a particular potential field, it attains a speed of \(9.47 \times 10^{6} \mathrm{~m} / \mathrm{s}\) What is the characteristic wavelength of this electron? Is the wavelength comparable to the size of atoms?

Short answer

The characteristic wavelength of the electron with a speed of \(9.47 \times 10^{6} \mathrm{~m} / \mathrm{s}\) is approximately \(7.68 \times 10^{-11}\mathrm{m}\). Since the size of atoms typically ranges from \(1 \times 10^{-10}\mathrm{m}\) to \(5 \times 10^{-10}\mathrm{m}\), the electron's wavelength is indeed comparable to the size of atoms. Consequently, electron microscopes can generate highly magnified images of biological and other types of materials.

Step-by-step solution

Recall the de Broglie wavelength formula

In order to find the characteristic wavelength of the electron, we need to utilize the de Broglie wavelength formula, which is given by: \[\lambda = \frac{h}{p}\] Where \(\lambda\) is the characteristic wavelength, \(h\) is the Planck's constant, and \(p\) is the momentum of the electron.

Calculate the momentum of the electron

The momentum of the electron can be found by multiplying its mass by its velocity. The mass of an electron is approximately \(9.11 \times 10^{-31} \mathrm{kg}\), and the velocity is given in the exercise as \(9.47 \times 10^{6} \mathrm{~m} / \mathrm{s}\). To find the electron's momentum, we multiply its mass and velocity: \[p = mv = (9.11 \times 10^{-31} \mathrm{kg})(9.47 \times 10^6 \mathrm{m/s}) = 8.62 \times 10^{-24}\mathrm{kg\ m/s}\]

Calculate the characteristic wavelength

Now that we have the momentum of the electron, we can find its characteristic wavelength using the de Broglie formula. The Planck's constant is approximately \(6.626 \times 10^{-34} \mathrm{J\cdot s}\). Therefore, the wavelength is: \[\lambda = \frac{h}{p} = \frac{6.626 \times 10^{-34}\mathrm{J\cdot s}}{8.62 \times 10^{-24}\mathrm{kg\ m/s}} = 7.68 \times 10^{-11}\mathrm{m}\]

Compare the wavelength to the size of atoms

The size of atoms typically ranges from \(1 \times 10^{-10}\mathrm{m}\) to \(5 \times 10^{-10}\mathrm{m}\). Comparing the characteristic wavelength of the electron, which is \(7.68 \times 10^{-11}\mathrm{m}\), to the size of atoms, we find that the wavelength of the electron is in fact smaller than the size of atoms. So, the electron's wavelength is comparable to the size of atoms. This is why electron microscopes can generate highly magnified images of biological and other materials, as their characteristic wavelength is similar or smaller than the size of the object they are observing.

Key concepts

Electron Microscope

An electron microscope is a powerful instrument that uses a beam of electrons to create an image of the specimen. Unlike traditional light microscopes, which use visible light, electron microscopes utilize electrons due to their much shorter wavelength. This allows for significantly higher magnification and resolution.

Here’s how it works:

• Electrons are accelerated through significant potential differences, reaching high speeds.
• When electrons interact with the specimen, they are scattered and detected to form a detailed image.
• The short de Broglie wavelengths of these electrons make it possible to visualize objects at the nanometer scale, such as individual atoms or molecules.

The capability to visualize such small details is crucial for applications in material science, biology, and nanotechnology. The electron microscope can reveal structures that are hundreds of times smaller than what the best light microscopes can see.

Electron Momentum

Momentum is a critical concept in understanding the behavior of electrons, especially in the context of the de Broglie wavelength.

Momentum is defined in classical physics as the product of mass and velocity (\( p = mv \)). For an electron with a tiny mass of \( 9.11 \times 10^{-31} \text{ kg} \), moving at high velocities, this still results in a measurable momentum.

To calculate the momentum of an electron:

• Identify the velocity, which may be determined by accelerating the electron with an electric field.
• Use the known mass of the electron to calculate its momentum using the formula \( p = mv \).

Knowing an electron's momentum is essential because it allows us to calculate its de Broglie wavelength. As the speed and momentum increase, the wavelength decreases, enabling the electron to probe smaller scales.

Planck's Constant

Planck's constant (\( h \)) is a fundamental physical constant that plays a crucial role in the field of quantum mechanics. Its value is approximately \( 6.626 \times 10^{-34} \text{ J}\cdot\text{s} \).

This constant is integral to the de Broglie wavelength formula \( \lambda = \frac{h}{p} \).

In essence, Planck's constant relates the wavelength of a particle to its momentum. It illustrates a key quantum principle: particles display both wave-like and particle-like properties.

• This duality becomes particularly significant at atomic and subatomic scales, where quantum effects cannot be ignored.
• Planck's constant is a bridge between the equations of classical physics and the principles of quantum mechanics, evidencing that this world is governed by probabilities rather than certainties.

Understanding Planck's constant is essential for explaining phenomena such as electron diffraction and the operation of electron microscopes.