Through what potential difference must electrons be...

Through what potential difference must electrons be accelerated if they are to have (a) the same wavelength as an x ray of waveIength $0.220 \mathrm{nm}$ and (b) the same energy as the x ray in part (a)?

Key Concepts

Energy and Momentum Relationships

This concept encompasses the interplay between kinetic energy, momentum, and wavelength. For electrons, kinetic energy relates to momentum through the equation KE = p²/(2m), and combining this with the de Broglie wavelength allows one to derive the conditions under which electrons will have a specific wavelength or energy. This framework is crucial for solving problems where the electron’s de Broglie wavelength must match that of a photon or the electron’s kinetic energy must equal the photon’s energy.

Photon Energy Relation

The energy of a photon is given by the relation E = hc/?, where h is Planck’s constant, c is the speed of light, and ? is the wavelength of the photon. This concept is essential for comparing the energy scales of photons (such as x-rays) with those of electrons, enabling a conversion between photon energy and electron kinetic energy when equating the two.

de Broglie Wavelength

This concept states that all matter with momentum has an associated wavelength given by the relation ? = h/p, where h is Planck’s constant and p is momentum. It is central to understanding how particles like electrons can exhibit wave-like properties, and is used to equate the wavelength of an electron with that of a photon, establishing a relationship between the kinetic energy (or momentum) of the electron and its wave characteristics.

Electron Acceleration Through a Potential Difference

When electrons are accelerated by an electric potential difference, they gain kinetic energy equal to the product of the electron charge and the potential difference (KE = eV). This concept links the electric potential to the energy of the accelerated electron and is fundamental in calculating the required voltage to achieve a specific kinetic energy or de Broglie wavelength.

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