Electrons are accelerated by a 1000-V potential drop. (a) Calculate the de Broglie wavelength. (

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Electrons are accelerated by a 1000-V potential drop. (a)...

Electrons are accelerated by a $1000-\mathrm{V}$ potential drop. $(a)$ Calculate the de Broglie wavelength. (b) Calculate the wavelength of the $X$ -rays that could be produced when these electrons strike a solid.

Key Concepts

de Broglie Wavelength

This concept explains that every moving particle has a wave nature, characterized by a wavelength that is inversely proportional to its momentum. The de Broglie wavelength is calculated using Planck’s constant divided by the momentum of the particle, establishing a fundamental connection between matter and waves in quantum mechanics.

Acceleration of Charged Particles

When charged particles, such as electrons, are accelerated by a voltage (or electric potential drop), they gain kinetic energy equal to the product of the charge and the potential difference. This increase in energy translates into an increase in momentum, which is critical for calculating the de Broglie wavelength.

Energy-Momentum Relationship

In both classical and quantum contexts, there is a direct relationship between the kinetic energy of a particle and its momentum. By converting a known amount of kinetic energy (gained from acceleration through a potential difference) into momentum, one can apply the de Broglie relation to determine the particle’s wavelength.

X-ray Production via Electron Impact

When high-energy electrons strike a target, the rapid deceleration or collision causes emission of X-rays, a process often involving Bremsstrahlung radiation. The shortest possible wavelength (or cutoff wavelength) of the emitted X-rays is determined by the complete conversion of the electron’s kinetic energy into the energy of a photon, as dictated by the energy–wavelength relationship of photons.

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