(a) In an electron microscope, what accelerating voltage is...

(a) In an electron microscope, what accelerating voltage is needed to produce electrons with wavelength 0.0600 nm? (b) If protons are used instead of electrons, what accelerating voltage is needed to produce protons with wavelength 0.0600 $\mathrm{nm}$ ? (Hint: In each case the initial kinetic energy is negligible.)

Key Concepts

De Broglie Wavelength

This concept relates to the wave-particle duality of matter, where any moving particle is associated with a wavelength given by the ratio of Planck's constant to its momentum. In problems involving microscopic particles, such as electrons or protons, the de Broglie wavelength is essential for predicting and understanding their interference and diffraction behavior.

Kinetic Energy and Accelerating Voltage

A charged particle accelerated from rest through an electric potential difference gains kinetic energy equal to the product of its charge and the voltage (KE = qV). This relationship is fundamental in determining the speed (and hence momentum) of the particle, which is then related to its de Broglie wavelength.

Non-relativistic Approximation

In many cases involving moderate accelerating voltages, the kinetic energy acquired by particles is small compared to their rest mass energy. This allows for a non-relativistic treatment where the kinetic energy is expressed as KE = (1/2)mv². This approximation simplifies the relation between kinetic energy, momentum, and wavelength, though for very high voltages or very light particles (like electrons) relativistic corrections might become necessary.

Particle Mass and Charge

The properties of a particle, notably its mass and charge, play a critical role in how it responds to an accelerating electric field. When applying the same accelerating voltage to different particles (e.g., electrons versus protons), the resulting kinetic energies, momenta, and hence de Broglie wavelengths differ due to the large differences in their masses and the common charge magnitude they carry. This concept is key when comparing the behavior of different types of particles under similar experimental conditions.

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