Question

Electron Microscopes Electron microscopes provide greater magnification than traditional light microscopes by using focused beams of electrons instead of visible light. It is the extremely short wavelengths of the electron beams that make electron microscopes so powerful. The wavelength \(\lambda\) (in meters) of any object in motion is given by \(\lambda=\frac{6.626 \times 10^{-34}}{m v}\), where \(m\) is the mass (in kilograms) of the object and \(v\) is its velocity (in meters per second). Find the wavelength of an electron with a mass of \(9.11 \times 10^{-31}\) kilogram and a velocity of \(5.9 \times 10^{6}\) meters per second. (Submitted by Brian McIntyre, Senior Laboratory Engineer for the Optics Electron Microscopy Facility at the University of Rochester.)

Short answer

The wavelength of an electron with a mass of \(9.11 \times 10^{-31}\) kilogram and a velocity of \(5.9 \times 10^{6}\) meters per second is approximately \(1.29 \times 10^{-10}\) m or 0.129 nm.

Step-by-step solution

Identifying the formula

The exercise provides the formula to calculate the wavelength of any moving object, which is \(\lambda = \frac{h}{mv}\), where h is Planck’s constant, m is the mass of the object and v is the velocity of the object.

Plugging in the values

Substitute the given values: h = \(6.626 \times 10^{-34}\) Js, m = \(9.11 \times 10^{-31}\) kg, and v = \(5.9 \times 10^{6}\) m/s, into the formula. So, \(\lambda = \frac{6.626 \times 10^{-34}}{9.11 \times 10^{-31} \times 5.9 \times 10^{6}}\).

Calculating the wavelength

After doing the necessary mathematical operations, we get the wavelength, λ, to be approximately \(1.29 \times 10^{-10}\) m which is equivalent to 0.129 nanometers.

Key concepts

Planck's constant

Planck's constant is a fundamental constant in physics. It plays a crucial role in the field of quantum mechanics, representing the relationship between the energy of photons and their frequency. This constant is denoted by the symbol \( h \), and its value is approximately \( 6.626 \times 10^{-34} \) Joule seconds (Js). This small number demonstrates just how tiny the energy packets called photons are.
In quantum mechanics, Planck's constant is vital because it establishes a foundational limit to certain measurements. For instance, it indicates that energy isn't a smooth flow but is quantized, or broken into discrete energy levels. This concept revolutionized our understanding of the micro-world, showing that on a microscopic scale, traditional physics no longer applies.

• Planck's constant connects energy (E) to the frequency (f) of a photon through the formula \( E = h \times f \).
• It symbolizes the smallest action in nature that can be calculated using quantum mechanics.

Understanding Planck's constant helps learners appreciate the transition from classical to quantum physics, expanding their view of how energy behaves at an atomic and subatomic level.

Electron microscopes

Electron microscopes are fascinating tools used to see objects at a very small scale, far beyond the capability of traditional light microscopes. They are essential in fields such as biology, materials science, and nanotechnology due to their high magnification capabilities. Unlike light microscopes, electron microscopes use beams of electrons instead of light. This allows them to achieve much greater detail.
One significant advantage of using electrons over light is the wavelength difference. Electrons can have extremely short wavelengths when accelerated to high speeds, resulting in higher resolution images. Electron microscopes can reveal structures down to the atomic level, making them invaluable in advanced scientific research.

• Two main types of electron microscopes: Transmission Electron Microscopes (TEM) and Scanning Electron Microscopes (SEM).
• They are crucial for observing the ultrastructure of cells and the surfaces of various materials.

By understanding how electron microscopes work, students can comprehend how breakthroughs in imaging technology can lead to fundamental discoveries across various scientific disciplines.

Wavelength calculation

Wavelength calculation is an essential part of understanding wave mechanics and the behavior of particles in motion. In the context of electron microscopes, calculating the wavelength of an electron involves using the de Broglie wavelength formula: \( \lambda = \frac{h}{mv} \). This equation links the wavelength (\( \lambda \)) to Planck's constant (\( h \)), the mass (\( m \)) of a particle, and its velocity (\( v \)).
For an electron with a mass of \( 9.11 \times 10^{-31} \) kilograms moving at a velocity of \( 5.9 \times 10^{6} \) meters per second, substituting these values into the equation provides the electron's wavelength. This calculation results in an extremely small wavelength, about \( 1.29 \times 10^{-10} \) meters, which corresponds to 0.129 nanometers. This tiny wavelength is what grants electron microscopes their extraordinary resolving power.

• Understanding this calculation builds appreciation for how matter can exhibit wave-like properties, a fundamental concept in quantum mechanics.
• This concept is crucial for explaining why electron microscopes can visualize structures that are imperceptible to traditional optical microscopes.

Through mastering wavelength calculations, students can appreciate the intricate balance between particle motion, quantum mechanics, and technological advancements in imaging.