Problem 84 - University Physics with Modern Physics

Question

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.

Step-by-Step Solution

Answer

The speed of quarks must be calculated to match the observed magnetic moment.

Step 1: Find Current Due to u Quark

To find the current associated with a circulating charge, use the formula for current, defined as charge per unit time. If a charge moves in a circular path, the current \( I \) is given by \( I = \frac{Q}{T} \), where \( Q \) is the charge and \( T \) is the period of revolution. For the \( u \) quark: the charge \( Q_u = +\frac{2}{3}e \). If \( v \) is the speed, the period \( T \) is \( \frac{2\pi r}{v} \). Therefore, \( I_u = \frac{Q_u}{T} = \frac{2}{3}e\cdot\frac{v}{2\pi r} = \frac{e v}{3 \pi r} \).

Step 2: Calculate Magnetic Moment of u Quark

Magnetic moment \( \mu \) for a charge \( Q \) moving in a circle of radius \( r \) with speed \( v \) is \( \mu = \frac{1}{2} Q r v \). Substituting for the \( u \) quark: \( \mu_u = \frac{1}{2}\cdot\frac{2}{3}e\cdot r\cdot v = \frac{e r v}{3} \).

Step 3: Calculate Magnetic Moment of d Quarks

The \( d \) quarks each have charge \( Q_d = -\frac{1}{3}e \) and rotate clockwise. Their combined magnetic moment will have the same magnitude as the \( u \) quark but opposite direction. Since there are two \( d \) quarks: combined \( \mu_d = 2 \times \frac{1}{2}\cdot\left( -\frac{1}{3}e \right)\cdot r\cdot v = -\frac{e r v}{3} \).

Step 4: Net Magnetic Moment of the Neutron

The net magnetic moment \( \mu_{net} \) is calculated by summing the contributions from all quarks. Since the \( d \) quarks' moment is opposite and equal to the \( u \) quark's moment, the net is \( \mu_{net} = \mu_u + \mu_d = \frac{e r v}{3} + \left( -\frac{e r v}{3} \right) = 0 \). However, in reality, this balance is not perfect, leading to the observed nonzero magnetic moment.

Step 5: Calculate Speed for Observation Alignment

Given the neutron has a magnetic moment \( \mu_n = 9.66 \times 10^{-27} \mathrm{A}\cdot\mathrm{m}^2 \), and using \( r = 1.20 \times 10^{-15} \mathrm{m} \), equate the model moment to the observed moment: \( \frac{e r v}{3} = 9.66 \times 10^{-27} \). Solving for \( v \), \( v = \frac{3 \times 9.66 \times 10^{-27}}{e \times 1.20 \times 10^{-15}} \) where \( e = 1.6 \times 10^{-19} \mathrm{C} \). Calculating gives \( v \).

Key Concepts

Neutron Structure

The neutron is a fascinating particle that seems simple on the surface, but it is quite complex. Even though it has no charge, its inner workings reveal a hidden complexity. The structure of a neutron is explained by the quark model, wherein quarks are the building blocks. A neutron is made of three quarks - one "up" (u) quark and two "down" (d) quarks. Each quark carries a specific charge, which adds up to zero, explaining why the neutron itself is neutral.
By examining how these quarks arrange themselves within the neutron, scientists can explain the neutron’s properties, like its magnetic moment. Understanding the neutron structure lets us delve into fundamental particle physics, helping us explore the very fabric of the universe. It's like discovering the secret ingredients in a recipe that result in its unique flavor.

Magnetic Moment

Magnetic moment is an intriguing characteristic of particles, which explains how they respond to magnetic fields. Despite being an uncharged particle, the neutron exhibits a nonzero magnetic moment, which might seem puzzling at first. This phenomenon can be understood by analyzing the movement and distribution of its internal quarks.
The up quark, with a charge of \( rac{2}{3}e \), follows a counterclockwise path, while the two down quarks with charges of \( -rac{1}{3}e \) each, follow a clockwise path. The motion of these charged particles generates tiny currents, much like a loop of wire, contributing to the particle’s magnetic moment. Think of it as a little whirlpool inside the neutron, contributing to its overall magnetic properties.
However, balancing forces and direction leads to phenomena that might appear magic but are just physics playing with spins and charges, resulting in the observed magnetic moment.

Particle Physics

Particle physics is like the detective science of physics, delving into the smallest known particles to understand the essential building blocks of matter. In this realm, the quark model plays a crucial role, especially in explaining the structure of composite particles like protons and neutrons.
Beyond just quarks, particle physics explores a zoo of particles, including the fundamental forces, elucidating how they interact with each other. The quark model helps us gain insight into how protons and neutrons form the core of atomic nuclei. By piecing together how quarks and forces work within particles like those, we get a more comprehensive picture of atomic behavior.
With particle physics, we're discovering a world where everything is ruled by interactions often governed by exchange particles, allowing larger structures to come to life by these simple yet brilliant interactions.

Up and Down Quarks

In the quirky world of particle physics, up and down quarks are fundamental particles making up many of the stable matter we know. They belong to a group called "fermions" that include other particles like electrons and are subject to specific rules called Pauli’s exclusion principle.

Up quarks carry a positive charge \( +rac{2}{3}e \), while down quarks have a negative charge of \( -rac{1}{3}e \). These fractional charges seem odd but are exactly what’s needed for quarks to combine into particles like protons, with a net charge, and neutrons, with no charge.
Due to their properties, especially their electrical charges and interactions, up and down quarks play a significant role in how matter forms and behaves. Without their specific balance and interactions, the universe as we know it wouldn’t exist. They’re like tiny puzzle pieces essential for making up the big picture of matter surrounding us.