A beam of electrons is accelerated across a potential of 15 kV before passing through two slits.

A beam of electrons is accelerated across a potential of 15...

Key Concepts

Interference Pattern Geometry

The geometry of the interference pattern in a double-slit experiment relates the slit separation, the distance to the screen, and the observed fringe positions through trigonometric relationships. By using the distance between the central maximum and the first-order maximum along with the known separation between the slits and the screen, one can derive important parameters about the system, such as the slit separation or the particle wavelength.

Double-Slit Interference

In a double-slit experiment, waves passing through two closely spaced slits interfere to form a pattern of alternating bright and dark fringes on a detection screen. This phenomenon, initially demonstrated with light, is a key demonstration of the wave-particle duality in quantum mechanics. The position and spacing of the interference maxima are related to the wavelength of the particles and the geometry of the setup.

de Broglie Wavelength

The de Broglie wavelength is a fundamental concept in quantum mechanics that describes the wave nature of particles. It is given by the relationship ? = h/p, where h is Planck’s constant and p is the momentum of the particle. This concept explains how particles such as electrons can exhibit interference and diffraction, similar to light.

Electron Acceleration and Energy Conversion

When electrons are accelerated by an electric potential, their potential energy is converted into kinetic energy. This process determines the velocity and therefore the momentum of the electrons, which is crucial for calculating their de Broglie wavelength. The relationship between the accelerating voltage and the kinetic energy is used to derive the momentum, connecting classical electromagnetism and quantum behavior.

Transcript

00:01 In this exercise, we have electrons that are accelerated through a potential difference of 15 kilo volts, and after they are accelerated, they pass through two slits, and 2 .5 meters after the slits, we have a screen where an interference pattern is recorded.

00:21 In the interference pattern, we have that the first order maximum is 8 .3 centimeters away from the central maximum.

00:36 With these informations, we want to find what is the separation d between the two slits.

00:51 So here in red, i have highlighted some important information, and one of them is the formula for the maxima of the interference pattern created by a double slit experiment.

01:07 So notice that the elements that enter into the formula are d, that is the separation between the two slits, the sign of theta, which is just, theta is just the angle that.

01:24 That the region in the screen we're looking at makes with the horizontal.

01:30 M is the order of the maximum and lambda is the wavelength of the electron.

01:40 In this case, we have the position of the first order maximum, that is, m equals 1.

01:48 For m equals 1, our equation reduces to d times sine theta equals lambda.

01:55 And our goal is to find d, so we can isolate d in the formula, so we have lambda divided by the sign of theta.

02:04 Okay? and now, what we have to do is to find both lambda and the sign of theta, and then we can find what is the separation between the two slits.

02:17 First of all, i'm going to find lambda.

02:21 Notice that the electrons were accelerated by a potential difference of 15 kilo -volt, and this and we know that the kinetic energy of electrons that are accelerated through a potential difference delta v is e, which is the charge of the electron times delta v.

02:43 But we also know that the kinetic energy of electrons can be written as the momentum squared divided by 2m.

02:52 M is the mass of the electron.

02:54 And for their reason, we can isolate p and find the momentum as a function of the kinetic energy.

03:02 Actually as a function of a function of the potential difference delta v.

03:09 But we also know that from the debreleif formula, we have that the momentum is equal to the plunks constant divided by the wavelength lambda.

03:23 So lambda is equal to h divided by the square root of 2m .e...

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