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

CP 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

Electron Acceleration via Potential Difference

This principle explains that electrons accelerated from rest gain kinetic energy equal to the charge multiplied by the potential difference (K = eV). This energy conversion underlies the determination of the electron's momentum, allowing one to relate the potential difference used for acceleration to the subsequent behavior of the electrons in experiments, such as exhibiting interference patterns.

Relativistic Effects

This concept pertains to determining whether the electrons' kinetic energy is significant compared to their rest energy. When particles are accelerated through a potential difference, if their kinetic energy is a small fraction of their rest energy, non-relativistic formulas are sufficient. In contrast, if the energy becomes comparable to the rest energy, relativistic corrections must be applied. Thus, understanding the energy scale relative to the electron’s rest mass energy (approximately 511 keV) is crucial in deciding the appropriate physics framework.

de Broglie Wavelength

The de Broglie hypothesis links a particle's momentum to its wavelength through the relation ? = h/p, where h is Planck’s constant. This concept is essential in quantum mechanics for understanding how particles such as electrons exhibit wave-like properties, which is directly observable in interference experiments like the double-slit setup.

Double-Slit Interference

Double-slit interference involves the superposition of waves emanating from two closely spaced slits, leading to a pattern of constructive and destructive interference on a distant screen. The positions of interference fringes depend on the slit separation, the wavelength of the electrons (as given by the de Broglie relation), and the angles at which the fringes appear. Analyzing these patterns allows one to deduce the electrons’ wavelength and, indirectly, their momentum and energy.

Transcript

00:01 Okay, so here we have a question about double -sit interference.

00:05 We have electrons that are passing through these two splits, which are 1 .25 nanometers apart.

00:12 And the first angle at which we see dark lines is at 18 degrees.

00:19 And in question a, they ask us, are the electrons relativistic? so typically relativistic speeds are greater than are equal to 1 .10.

00:31 Of the speed of light.

00:35 So we need to find the velocity that the electrons must be traveling at such that the first interference is at 18 degrees.

00:44 So let's start at our formula, which relates d and theta.

00:50 So sine of theta times d is equal to lambda.

00:58 So this is going to be useful formula because it takes the two things which we already know and related to lambda, which we know how to relate to velocity.

01:06 You can use the deboile formula.

01:08 Lambda is equal to h over mb, and it's been solved for b in terms of lambda.

01:16 So first what we need to do is just plug in d and theta, and we get that lambda is equal to 3 .863 times 10 to minus 10 meters.

01:33 All right, so now we need to find the velocity.

01:35 Velocity, and then we can check if it's well as realistic.

01:38 So first, let's get velocity in terms of lambda.

01:42 So multiply velocity to the other side, and apply lambda to the other side.

01:46 So v is equal to h over m lambda.

01:50 So now we just solve for lambda...