When an electron is accelerated by a voltage difference, the...

When an electron is accelerated by a voltage difference, the kinetic energy acquired by the electron equals the voltage times the charge on the electron. Thus, one volt imparts a kinetic energy of $1.602 \times 10^{-19}$ volt-coulombs, which equals $1.602 \times$ $10^{-19} \mathrm{~J}$. What is the wavelength associated with electrons accelerated by $4.00 \times 10^{3}$ volts?

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Key Concepts

Planck's Constant

Planck's constant (h) is a fundamental constant in quantum mechanics that relates the energy of a photon to its frequency and appears in the de Broglie relation. It sets the scale for quantum effects and is essential for calculating wavelengths associated with particles.

Kinetic Energy and Momentum Relationship

For a non-relativistic particle, the relationship between kinetic energy and momentum is given by KE = p²/(2m). This relation allows one to calculate the momentum from the kinetic energy gained by an electron, which is then used to determine its de Broglie wavelength.

de Broglie Wavelength

The de Broglie hypothesis asserts that any moving particle exhibits wave-like properties, with a wavelength given by ? = h/p, where h is Planck's constant and p is the momentum. This concept is central to understanding the wave-particle duality in quantum mechanics.

Electron Acceleration by Voltage

When an electron is accelerated by a voltage difference, it gains kinetic energy equal to the product of its charge and the voltage. This principle, represented by KE = qV, is key to converting electrical potential energy into kinetic energy in charged particles.

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