A beam of electrons is accelerated from rest through a...

A beam of electrons is accelerated from rest through a potential difference of 0.100 $\mathrm{kV}$ and then passes through a thin slit. The diffracted beam shows its first diffraction minima at $\pm 11.5^{\circ}$ from the original direction of the beam when viewed far from the slit. (a) Do we need to use relativity formulas? How do you know? (b) How wide is the slit?

Key Concepts

de Broglie Wavelength

The de Broglie wavelength expresses the wave-like nature of particles and is given by ? = h/p, where h is Planck’s constant and p is the momentum of the particle. This concept is essential in analyzing phenomena such as diffraction, as the wavelength determines how much the particle will spread out when passing through narrow apertures.

Single-Slit Diffraction

Single-slit diffraction occurs when waves pass through a narrow aperture and interfere, creating a pattern of maxima and minima. The position of the first diffraction minima is determined by the relationship a sin? = ?, where a is the slit width, ? is the diffraction angle, and ? is the wavelength of the particles or waves involved.

Energy Conversion from Potential Difference

An electron accelerated from rest through a potential difference gains kinetic energy equal to the product of its charge and the potential (KE = eV). This fundamental energy conversion allows us to compute the electron’s velocity by equating this kinetic energy to its classical expression (½ mv²) in the non-relativistic regime.

Non-Relativistic Approximation

Since the accelerating voltage (0.100 kV) is relatively low, the electron’s resulting speed is much less than the speed of light, meaning that classical (non-relativistic) formulas are sufficient for accurate calculations. Relativistic effects become significant only when speeds approach a considerable fraction of the speed of light.

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