A beam of electrons passes through a single slit 40.0 nm wide. The width of the central fringe

1. First, we need to find the wavelength of the electrons. We can use the formula for single-sli

A beam of electrons passes through a single slit 40.0 nm...

A beam of electrons passes through a single slit $40.0 \mathrm{~nm}$ wide. The width of the central fringe of a diffraction pattern formed on a screen $1.0 \mathrm{~m}$ away is $6.2 \mathrm{~cm} .$ What is the kinetic energy of the electrons passing through the slit?

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Key Concepts

de Broglie Wavelength

The de Broglie wavelength is a fundamental quantum mechanical concept that associates a wavelength with any moving particle, given by ? = h/p, where h is Planck’s constant and p is the particle's momentum. This idea is essential to understand the wave-like behavior of particles, such as electrons, which is evident in phenomena like diffraction and interference.

Single-Slit Diffraction

Single-slit diffraction is the phenomenon where waves, including matter waves like electrons, spread out after passing through a narrow aperture. The resulting diffraction pattern, characterized by a central maximum and subsidiary minima and maxima, directly relates the slit width and the wavelength of the wave, often using the approximation for small angles that the angular position of the first minimum is given by sin? ? ?/a.

Relationship Between Kinetic Energy and Momentum

The kinetic energy of a particle is directly related to its momentum through the equation K = p^2/(2m), where m is the mass of the particle. By determining the momentum (for instance, through the de Broglie wavelength), one can compute the kinetic energy, linking measurable wave-like phenomena to classical energy considerations.

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