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Double-slit interference is a fascinating phenomenon that occurs when waves, such as light or in this case, electron waves, pass through two closely spaced slits. The result is an interference pattern of alternating light and dark fringes. This happens because the waves emerging from each slit overlap and combine.

• Constructive interference occurs when the waves align, creating bright fringes.
• Destructive interference occurs when the waves are out of sync, resulting in dark fringes.

Electron diffraction experiments, like the one described, use this principle to show that electrons have wave-like properties. The distance between the slits and the observing screen impacts where fringes are seen, as it influences the path difference between the waves from each slit.
In the given exercise, the first dark fringe is noted at \(\pm18.0^{\circ}\), highlighting the destructive interference between the electron waves.

The de Broglie wavelength is a concept introduced by physicist Louis de Broglie. It proposes that all matter exhibits wave-like properties, not just light. This wavelength is inversely proportional to the momentum of a particle, given by the formula:\[\lambda = \frac{h}{p}\]where \(h\) is the Planck's constant and \(p\) is the momentum of the particle. For electrons, this concept is crucial to understanding their behavior in quantum mechanics.

• The exercise assumes a de Broglie wavelength of \(2.43 \times 10^{-12}\ m\) for the electrons, indicating their wave nature.
• This wavelength plays a part in determining the position of interference fringes in the double-slit experiment.

By using the de Broglie wavelength, we can calculate various aspects of an electron's motion, including its velocity, when related to double-slit interference patterns.

To understand if electrons are relativistic, we need to evaluate their speed relative to the speed of light. Electrons are considered relativistic when their speed approaches a significant fraction (generally more than 10%) of the speed of light (\(3.00 \times 10^8\ m/s\)). This requires using relativistic physics to accurately describe their behavior.
In the problem, we calculate the speed of the electrons once they have passed through a potential difference and compare that speed to the speed of light.

• If the calculated speed is a significant percentage of the speed of light, relativistic effects cannot be ignored.
• Non-relativistic calculations might yield incorrect results at these high speeds.

By checking the calculated velocity against this relativistic threshold, we can determine the necessary physics models to apply when evaluating the behavior of electrons in experiments.