The de-Broglie wavelength of an electron accelerated by an...

The de-Broglie wavelength of an electron accelerated by an electric field of $V$ volts is given by:
(a) $\lambda=\frac{1.23}{\sqrt{m}}$
(b) $\lambda=\frac{1.23}{\sqrt{h}} m$
(c) $\lambda=\frac{1.23}{\sqrt{V}} \mathrm{~nm}$
(d) $\lambda=\frac{1.23}{V}$

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Key Concepts

de Broglie Wavelength

This is the concept that every moving particle has an associated wavelength given by ? = h/p, where h is Planck's constant and p is the momentum of the particle. It is fundamental in quantum mechanics, illustrating the wave-particle duality of matter.

Electron Acceleration by Electric Potential

When an electron is accelerated through a potential difference V, it gains kinetic energy equal to the product of its charge and the voltage (eV). This energy gain converts directly into its motion, facilitating the calculation of its momentum and thereby its de Broglie wavelength.

Relationship between Kinetic Energy and Momentum

The kinetic energy of a particle is related to its momentum through the equation K = p²/(2m), where m is the mass of the particle. By equating the kinetic energy acquired from an accelerating voltage (eV) to this expression, one can derive the momentum and, consequently, the de Broglie wavelength.

Dimensional Analysis and Approximated Formulas

Using dimensional analysis and substituting known constants into the de Broglie equation, one can derive approximate formulas for specific conditions, such as electrons accelerated by a voltage. Such approximations, like ? = 1.23/?V (in nm) for electrons, simplify calculations and are valuable in experimental physics.

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