A beam of protons and a beam of alpha particles (of mass 6.64 ×10^-27 kg and charge +2 e ) are a

A beam of protons and a beam of alpha particles (of mass...

A beam of protons and a beam of alpha particles (of mass $6.64 \times 10^{-27} \mathrm{kg}$ and charge $+2 e )$ are accelerated from rest through the same potential difference and pass through identical circular holes in a very thin, opaque film. When viewed far from the hole, the diffracted proton beam forms its first dark ring at $15^{\circ}$ with respect to its original direction. When viewed similarly, at what angle will the alpha particle form its first dark ring?

Key Concepts

De Broglie Wavelength

This concept states that every moving particle exhibits wave-like behavior with an associated wavelength, determined by dividing Planck's constant by the particle's momentum. It is crucial in understanding how particles, like protons and alpha particles, can display diffraction effects when they pass through apertures.

Particle Kinetic Energy in an Electric Potential

When a charged particle is accelerated through a potential difference, it gains kinetic energy equal to the charge multiplied by the potential difference. This kinetic energy determines the particle's velocity, which in turn affects its momentum and its de Broglie wavelength.

Diffraction Through a Circular Aperture

Diffraction is the spreading of waves when they encounter an obstacle or pass through an opening. For a circular aperture, the resulting diffraction pattern, known as an Airy pattern, features a central bright spot surrounded by concentric dark and bright rings. The positions of these rings depend on the wavelength of the incident beam and the geometry of the aperture.

Relationship Between Wavelength and Diffraction Angle

In the diffraction pattern from a circular aperture, the angles at which the dark rings occur are related to the particle's wavelength. A change in wavelength, resulting from changes in kinetic energy and momentum, directly scales the diffraction angles, making it possible to predict how the pattern shifts for particles with different masses and charges.

Transcript

00:01 In this question we are looking at a diffraction of the proton and the alpha particle through a circular aperture.

00:11 So for a circular aperture, the diffraction limit or the first, the angle in which we get the first duck ring, it's given as sine theta equals to 1 .22 lambda over d, where it is often referred to as the diffraction limit, limit for a circular aperture, right? where is the diameter of the ring, of the hole? right? but of course, we don't know what is the diameter of this aperture.

00:48 So we will need to somehow get rid of it, right? and we can do that because we are using the same hole.

00:56 So taking the angle of the...

01:01 Proton and we divide this from the angle from the alpha particle.

01:14 What we get is 1 .22 lambda of the alpha right the wavelength of the alpha particle divided by 1 .22 wavelength of the proton.

01:30 So now we just have a simple ratio between the wavelengths and the the diameter of the hole is gone, right? so very simple to get the wavelength.

01:43 This is related to the broglis wavelength, right? goes to h over the momentum, right, where momentum is related to the kinetic energy, right? p -square -2m is given as kinetic energy.

02:06 Now furthermore, the kinetic energy is related to the acceleration of, our particle because it depends on the charge and the potential difference.

02:23 And we're given that this potential difference is applied to be the same for both these particles.

02:35 So basically, relating everything together, right, p square over 2m is equal to q times b...

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