Quark Model of the Neutron. The neutron is a particle with...

Quark Model of the Neutron. The neutron is a particle with zero charge. Nonetheless, it has a nonzero magnetic moment with $z$ -component $9.66 \times$ $10 ^ { - 27 } \mathrm { A } \cdot \mathrm { m } ^ { 2 } .$ This be explained by the internal structure of the neutron. A substantial body of evidence indicates that a neutron is composed of three fundamental particles called of three fundamental particles called quarks: an "up" (u) quark, of charge $+ 2 e / 3 ,$ and two "down" $( d )$ quarks, each of charge $- e / 3 .$ The combination of the three quarks produces a net charge of $2 e / 3 - e / 3 - e / 3 = 0$ . If the quarks are in motion, they can produce a nonzero magnetic moment. As a very simple model, suppose the $u$ quark moves in a counterclockwise circular path and the $d$ quarks move in a clock- wise circular path, all of radius $r$ and all with the same speed $v$ (Fig. P27.84). (a) Determine the current due to the circulation of the $u$ quark. (b) Determine the magnitude of the magnetic moment due to the circulating $u$ quark. (c) Determine the magnitude of the magnetic moment of the three-quark system. (Be careful to use the correct magnetic moment directions.) (d) With what speed $v$ must the quarks move if this model is to reproduce the magnetic moment of the neutron? Use $r = 1.20 \times 10 ^ { - 15 } \mathrm { m }$ (the radius of the neutron) for the radius of the orbits.

Quark Model of the Neutron. The neutron is a particle with zero charge. Nonetheless, it has a nonzero magnetic
moment with $z$ -component $9.66 \times$ $10 ^ { - 27 } \mathrm { A } \cdot \mathrm { m } ^ { 2 } .$ This be explained by the internal structure of
the neutron. A substantial body of
evidence indicates that a neutron
is composed of three fundamental
particles called of three fundamental
particles called quarks: an "up"
(u) quark, of charge $+ 2 e / 3 ,$ and two "down" $( d )$ quarks, each of charge $- e / 3 .$ The combination of
the three quarks produces a net charge of $2 e / 3 - e / 3 - e / 3 = 0$ . If the quarks are in motion, they can produce a nonzero magnetic
moment. As a very simple model, suppose the $u$ quark moves in a
counterclockwise circular path and the $d$ quarks move in a clock-
wise circular path, all of radius $r$ and all with the same speed $v$ (Fig. P27.84). (a) Determine the current due to the circulation of
the $u$ quark. (b) Determine the magnitude of the magnetic moment
due to the circulating $u$ quark. (c) Determine the magnitude of the
magnetic moment of the three-quark system. (Be careful to use the correct magnetic moment directions.) (d) With what speed $v$ must
the quarks move if this model is to reproduce the magnetic
moment of the neutron? Use $r = 1.20 \times 10 ^ { - 15 } \mathrm { m }$ (the radius of
the neutron) for the radius of the orbits.

Key Concepts

Quark Model

The quark model is a framework in particle physics that describes hadrons, such as neutrons and protons, as being composed of fundamental particles called quarks. These quarks possess properties like fractional electric charge, color charge, and spin, and their combinations determine the overall quantum numbers and characteristics of the composite particle.

Fractional Electric Charge

Quarks carry fractional electric charges (for example, some carry +2/3 and others -1/3 of the elementary charge). The way these fractional charges add up within a composite particle explains how particles like the neutron can have a net zero charge despite being made of charged constituents.

Magnetic Moment

The magnetic moment is a measure of the magnetic strength and orientation of a particle's magnetic field. In composite systems, this can arise from the intrinsic spin of the constituents as well as from their orbital motion. Even if the net charge of a particle is zero, the motion of its internal charged constituents can result in a nonzero net magnetic moment.

Current from Moving Charges

A moving charged particle in a closed-loop path creates an effective electric current. This current can be used in conjunction with the area enclosed by the motion to calculate the magnetic moment, linking the concepts of charge motion and magnetic effects.

Magnetic Moment of a Current Loop

A current loop generates a magnetic dipole moment proportional to the product of the current and the area of the loop. The direction of this magnetic moment is determined by the right-hand rule, which connects classical electromagnetism to the behavior of subatomic particles in simplified models.

Semiclassical Modeling in Particle Physics

Semiclassical models apply classical physics concepts, such as circular motion and current loops, to approximate and understand quantum mechanical phenomena. These models provide qualitative insights into the internal structure and magnetic properties of particles like the neutron despite the underlying quantum complexity.

Transcript

00:01 This is chapter 27 problem number 76.

00:04 We are using a model to determine the total magnetic moment of a neutron.

00:09 Even though it has zero net charge, it does not have zero magnetic moment due to the individual moments of the quarks that the neutron is consisting of.

00:22 So in part a, we are asked to determine the current due to the circulation of a u cork.

00:27 So in order to figure this out, as you know, the current is going to be q over t, t being the period for one revolution.

00:35 Q for u cork is 2 plus 2e over 3.

00:42 Then remember t is 2 pi r over b.

00:46 Then plugging it in here, 2e over 3 times 2 .5r over b.

00:52 And we have two's canceling b over 3 pi r as the current of an up, excuse me, up quark.

01:06 Now, in part b, we were asked to calculate the magnetic moment due to this current of an up quark.

01:13 So, you use going to be iu times the area, area being pi r squared.

01:19 So, substituting for iu, the current of a few cork, b .v over 3 pi r times pi r squared.

01:28 Pies are going to cancel, r's are going to cancel, evr over 3 is going to be our magnetic moment for an up quark.

01:38 In part c, though, let's separate this part.

01:44 In part c, we're asked to calculate the total magnetic moment of the system, since no.

01:51 We have two down corks and one up cork.

01:54 Mew down plus mu down plus mu up is going to give us to total lignmatic moment of the system, which means we need to figure out what the lignetive moment is of a down quark.

02:09 Using pretty much the same logic that we used in part a.

02:14 First we can calculate i down, which is q over t, which i'm calculating the magnitude...

Please give Ace some feedback