CP Coherent light is passed through two narrow slits whose...

CP Coherent light is passed through two narrow slits whose separation is $20.0 \mu \mathrm{m}$. The second-order bright fringe in the interference pattern is located at an angle of 0.0300 rad. If electrons are used instead of light, what must the kinetic energy (in electron volts) of the electrons be if they are to produce an interference pattern for which the second-order maximum is also at 0.0300 rad?

Key Concepts

Relationship Between Momentum and Kinetic Energy

For non-relativistic particles, the kinetic energy (K) and momentum (p) are connected through the formula K = p²/(2m), with m being the mass of the particle. This relation allows one to determine the kinetic energy from the de Broglie wavelength by first finding the momentum from ? = h/p and then using the kinetic energy formula.

Energy Unit Conversion

In many physics problems, especially ones involving quantum phenomena, it is necessary to convert energy values from joules to electron volts (eV). One electron volt is defined as the energy gained by an electron when it is accelerated through an electric potential difference of one volt, and is numerically equivalent to 1.602 × 10?¹? joules.

Double-slit Interference

Double-slit interference refers to the phenomenon where coherent waves passing through two narrow, closely spaced apertures produce a pattern of alternating bright and dark fringes due to constructive and destructive interference. The condition for the occurrence of a bright fringe is given by d sin ? = m?, where d is the distance between the slits, ? is the angle of the fringe relative to the central maximum, m is an integer (order of the maximum), and ? is the wavelength.

de Broglie Wavelength

The de Broglie wavelength is a fundamental concept in quantum mechanics that assigns a wavelength to any particle based on its momentum. It is given by the relation ? = h/p, where h is Planck’s constant and p is the momentum of the particle. This idea is essential for explaining wave-particle duality and phenomena such as electron diffraction, where particles display wave-like interference effects.

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