/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 19 In a proton accelerator used in ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 30: Problem 19

In a proton accelerator used in clementary particle physics experiments, the trajectories of protons are controlled by bending magnets that produce a magnetic field of \(4.80 \mathrm{~T}\). What is the magnetic-field cnergy in a \(10.0 \mathrm{~cm}^{3}\) volume of space where \(B=4.80 \mathrm{~T} ?\)

Short Answer

Expert verified
The magnetic field energy in a \(10.0 \, cm^{3}\) volume of space where \(B = 4.80 \, T\) is calculated by first determining the energy density and then multiplying the result by the volume of the space.

Step by step solution

01

Calculate energy density

To compute the energy density (\(u\)) in the magnetic field, the formula \(u = \frac{B^{2}}{2\mu}\) should be applied, where \(B\) is the magnetic field strength and \(\mu\) is the permeability of free space. In this case, \(B = 4.80 \, T\) and \(\mu = 4\pi \times 10^{-7} \, T.m/A\). Insert these values into the equation and compute the energy density.
02

Calculate total energy in the volume

The total energy within a certain volume can be obtained by multiplying the energy density by the volume itself. For this problem, the volume given is \(10.0 \, cm^{3}\), which needs to be converted to \(m^{3}\) (since \(\mu\) is in terms of meters) before performing the multiplication.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Energy Density
Energy density is a measure that reflects how much energy is stored in a given system or region of space per unit volume. In the context of a magnetic field, the energy density tells us how much energy the magnetic field holds in a specific space. This is crucial in applications like proton accelerators, where knowing the energy stored in a field can aid in controlling particle trajectories.

To compute the energy density of a magnetic field, we use the formula \[u = \frac{B^2}{2\mu}\] where \(B\) is the magnetic field strength, and \(\mu\) is the permeability of free space.

In the exercise, a magnetic field strength \(B = 4.80 \, T\) is stated, and the permeability of free space is \(\mu = 4\pi \times 10^{-7} \, T.m/A\). Plugging these values into the energy density formula helps calculate how much energy is stored per cubic meter of space in the magnetic field.

Understanding this concept is key in fields where magnetic fields are crucial for the operation of equipment, such as in MRI machines or electric generators.
Role of Proton Accelerators
Proton accelerators are devices used in particle physics to accelerate protons to high speeds, often close to the speed of light. These machines play an essential role in research, especially in studying the fundamental properties of matter and forces of the universe.

In a proton accelerator, bending magnets are critical components. They generate strong magnetic fields to steer and focus the protons along predefined paths. The effectiveness of an accelerator largely depends on how well these magnetic fields are controlled.

The question posed in the exercise is typical in examining the efficiency of these magnetic fields, as the energy density helps determine how much magnetic "influence" is being exerted on the protons in a given volume of space.

Such accelerators have real-world applications not only in theoretical physics experiments but also in medical therapies, such as cancer treatment, where precise proton beams are used to target tumors.
Explaining Permeability of Free Space
The permeability of free space, denoted by \(\mu\), is a fundamental physical constant that describes how a magnetic field affects and propagates through a vacuum. It is an essential factor in determining the strength of electromagnetic interactions.

This constant plays a crucial role in various electromagnetic calculations, as it relates magnetic field strength to magnetic flux density in free space. The value of the permeability of free space is precisely defined as \(4\pi \times 10^{-7} \, T.m/A\). This allows scientists and engineers to predict how materials will interact with magnetic fields.

In our given problem, this constant is used to calculate the energy density of a magnetic field. Understanding \(\mu\) is fundamental to advance knowledge in electromagnetic theory and is vital in designing equipment that relies on magnetic fields, from radio transmitters to electric motors and beyond.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A charged capacitor with \(C=590 \mu \mathrm{F}\) is connected in series to an inductor that has \(L=0.330 \mathrm{H}\) and negligible resistance. At an instant when the current in the inductor is \(i=2.50 \mathrm{~A},\) the current is increasing at a ratc of \(d i / d t=73.0 \mathrm{~A} / \mathrm{s}\). During the current oscillations. what is the maximum voltage across the capacitor?

DATA To investigate the properties of a large industrial solenoid, you connect the solenoid and a resistor in series with a battery. Switches allow the battery to be replaced by a short circuit across the solenoid and resistor. Therefore Fig. 30.11 applies, with \(R=R_{\text {ext }}+R_{L}\). where \(R_{L}\) is the resistance of the solenoid and \(R_{\text {ext }}\) is the resistance of the series resistor. With switch \(S_{2}\) opcn, you close switch \(S_{1}\) and kecp it closed until the current \(i\) in the solenoid is constant (Fig. 30.11 ). Then you close \(S_{2}\) and open \(S_{1}\) simultaneously, using a rapid- response switching mechanism. With high-speed clectronics you measure the time \(t_{\text {balf }}\) that it takes for the current to decrease to half of its initial value. You repeat this measurement for several values of \(R_{\text {ext }}\) and obtain these results: \begin{tabular}{l|lllllll} \(R_{\text {ext }}(\Omega)\) & 3.0 & 4.0 & 5.0 & 6.0 & 7.0 & 8.0 & 10.0 & 12.0 \\\ \hline\(t_{\text {half }}(\mathrm{s})\) & 0.735 & 0.654 & 0.589 & 0.536 & 0.491 & 0.453 & 0.393 & 0.347 \end{tabular} (a) Graph your data in the form of \(1 / t_{\text {halt }}\) versus \(R_{e x l}\). Explain why the data points plotted this way fall close to a straight line. (b) Use your graph from part (a) to calculate the resistance \(R_{L}\) and inductance \(L\) of the solenoid. (c) If the current in the solenoid is \(20.0 \mathrm{~A}\), how much energy is stored there? At what rate is electrical energy being dissipated in the resistance of the solenoid?

A long. straight solenoid has 800 turns. When the current in the solenoid is \(2.90 \mathrm{~A}\), the average flux through each turn of the solenoid is \(3.25 \times 10^{-3} \mathrm{~Wb}\). What must be the magnitude of the rate of change of the current in order for the sclf-induced cmf to cqual \(6.20 \mathrm{mV} ?\)

An Wectromagnetic Car Alarm. Your latest invention is a car alarm that produces sound at a particularly annoying frequency of \(3500 \mathrm{~Hz}\). To do this, the car-alarm circuitry must produce an alternating electric current of the same frequency. That's why your design includes an inductor and a capacitor in series. The maximum voltage across the capacitor is to be \(12.0 \mathrm{~V}\). To produce a sufticiently loud sound, the capacitor must store \(0.0160 \mathrm{~J}\) of energy. What values of capacitance and inductance should you choose for your car-alarm circuit?

. When the current in a long. straight, air-filled solenoid is changing at the rate of \(2000 \mathrm{~A} / \mathrm{s},\) the voltage across the solcnoid is \(0.600 \mathrm{~V}\). The solenoid has 1200 turns and uniform cross-sectional area \(25.0 \mathrm{~mm}^{2}\). Assume that the magnetic field is uniform inside the solenoid and zero outside, so the result \(L=\mu_{0} A N^{2} / l\) (sec Exercise 30.11 ) applics. What is the magnitude \(B\) of the magnctic ficld in the interior of the solenoid when the current in the solenoid is 3,00 A?
See all solutions

Recommended explanations on Physics Textbooks

Mechanics and Materials

Read Explanation

Electrostatics

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Physical Quantities And Units

Read Explanation

Fundamentals of Physics

Read Explanation

Work Energy and Power

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.