For any microscope, the size of the smallest observable...

For any microscope, the size of the smallest observable object is one-half the wavelength of the radiation used. For example, the smallest object observable with 400 -nm light is $2 \times 10^{-7} \mathrm{~m}$. (a) What is the smallest observable object for an electron microscope using electrons moving at $5.5 \times 10^{4} \mathrm{~m} / \mathrm{s} ?$ (b) At $3.0 \times 10^{7} \mathrm{~m} / \mathrm{s} ?$

For any microscope, the size of the smallest observable object is one-half the wavelength of the radiation used. For example, the smallest object observable with 400 -nm light is $2 \times 10^{-7} \mathrm{~m}$. (a) What is the smallest observable object for an electron microscope using electrons moving at $5.5 \times 10^{4} \mathrm{~m} / \mathrm{s} ?$
(b) At $3.0 \times 10^{7} \mathrm{~m} / \mathrm{s} ?$

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Key Concepts

Diffraction Limit

The diffraction limit is a fundamental concept in microscopy which states that the resolution of an imaging system is limited by the wavelength of the radiation it uses. Specifically, the smallest observable object is approximately half the wavelength of the radiation, setting a physical boundary on the smallest detail that can be resolved.

de Broglie Wavelength

The de Broglie wavelength concept reveals that all matter exhibits wave-like behavior, and its wavelength is given by the formula ? = h / (mv), where h represents Planck's constant, m is the particle's mass, and v its velocity. This principle is critical for electron microscopes, as it explains how electrons can be used to achieve much shorter wavelengths than visible light.

Relationship between Particle Velocity and Wavelength

For particles such as electrons, the wavelength is inversely proportional to their momentum. This means that as the electron’s speed increases, its momentum increases, resulting in a shorter de Broglie wavelength. A shorter wavelength leads to a higher resolution in electron microscopes, allowing for the observation of smaller objects.

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