A beam of electrons is accelerated from rest and then passes...

A beam of electrons is accelerated from rest and then passes through a pair of identical thin slits that are 1.25 $\mathrm{nm}$ apart. You observe that the first double-slit interference dark fringe occurs at $\pm 18.0^{\circ}$ from the original direction of the beam when viewed on a distant screen. (a) Are these electrons relativistic? How do you know? (b) Through what potential difference were the electrons accelerated?

Key Concepts

Double-Slit Interference

This concept relates to the phenomenon where waves passing through two closely spaced slits interfere, producing a pattern of bright and dark fringes on a screen. The geometry of the slit spacing and the de Broglie wavelength of the electrons determine the locations of these interference fringes, allowing one to extract physical information such as the wavelength and energy of the electrons.

Relativistic Effects in Electron Dynamics

This concept addresses the need to consider relativistic mechanics when the speeds of electrons become a significant fraction of the speed of light. Relativistic corrections modify the momentum and kinetic energy relationships, so determining whether electrons are relativistic involves comparing their speed (or energy) to the scale of c (the speed of light).

de Broglie Wavelength

This concept describes the wave nature of particles, where every particle has an associated wavelength given by the de Broglie formula (? = h/p). It is crucial in understanding electron diffraction and interference phenomena, as the wave behavior explains the formation of interference patterns when particles such as electrons pass through slits.

Electron Acceleration and Kinetic Energy

This concept explains that when electrons are accelerated from rest through a potential difference, they gain kinetic energy equal to the product of the electron’s charge and the potential difference (K.E. = eV). This relationship is instrumental in relating the potential difference to the momentum and therefore to the de Broglie wavelength of the electrons.

Transcript

00:01 In this question we're looking at a double slit experiment.

00:04 So for the double slit experiment, the equation that we're going to use is d -saitre equals to n lambda.

00:19 Well, in this case, d is the distance apart between the slits.

00:29 The angle is the angle made for the minimal or maximum from the central maximum depending on your n value lambda is the wavelength of light or the electron that we're looking at and is the order of the diffraction so n if it is a integer value this is constructive interference, this is where you get a maxima.

01:10 But if they are half integers, this is where you get destructive interference or minimus.

01:29 So we get the minimus over here and we are interested in the minimal because we're looking at the dark fringe.

01:36 So the first dark fringe will be when n equals to half.

01:41 And we are given that this is at 18 degrees.

01:46 We can use that knowledge to find what is our wavelength of our electron.

01:53 And from this wavelength of electron, we can use the de broccoli's wavelength formula, where p equals to h over lambda to find the momentum and has the velocity to determine whether these electrons are relativistic or not.

02:08 So we start off.

02:14 Substituting in the slit difference which is 1 .25 nanometers.

02:24 Sign of 18 degrees goes to half times the wavelength.

02:31 So the wavelength over here will be 0 .7725.

02:38 So it was just 7 .725...

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