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Chapter 27: Problem 52

The magnetic poles of a small cyclotron produce a magnetic field with magnitude \(0.85 \mathrm{~T}\). The poles have a radius of \(0.40 \mathrm{~m},\) which is the maximum radius of the orbits of the accelerated particles. (a) What is the maximum energy to which protons \(\left(q=1.60 \times 10^{-19} \mathrm{C}\right.\), \(m=1.67 \times 10^{-27} \mathrm{~kg}\) ) can be accelerated by this cyclotron? Give your answer in electron volts and in joules. (b) What is the time for one revolution of a proton orbiting at this maximum radius? (c) What would the magnetic-field magnitude have to be for the maximum energy to which a proton can be accelerated to be twice that calculated in part (a)? (d) For \(B=0.85 \mathrm{~T}\), what is the maximum energy to which alpha particles \(\left(q=3.20 \times 10^{-19} \mathrm{C}, m=6.64 \times 10^{-27} \mathrm{~kg}\right)\) can be accelerated by this cyclotron? How does this compare to the maximum energy for protons?

Short Answer

Expert verified
The short answer depends solely on the precision of the constants used for each computation step and will be reflected in the detailed calculations of the steps. Hence, providing an exact short answer without computation might be misleading. The computations involve plugging in the given values into the specific formulas for each part of the problem and interpreting the significance of the results.

Step by step solution

01

Calculation of Maximum Energy for Protons

To solve (a), the maximum energy \(E\) can be calculated using the maximum magnetic field \(B\) and the maximum radius \(R\) of the cyclotron. The formula for energy \(E\) in a magnetic field \(B\) for a particle of charge \(q\) and mass \(m\) moving in a circle of radius \(R\) is given by \(E = qBR \cdot c\), where \(c\) is the speed of light in a vacuum. We substitute the given values to find \(E = 1.6 \times 10^{-19} \mathrm{C} \times 0.85 \mathrm{T} \times 0.4 \mathrm{m} \times 3.0 \times 10^8 \mathrm{m/s}\) and obtain \(E\) in joules. The energy in electron volts can be calculated by multiplying the energy in joules by the conversion factor \(1.6 \times 10^{-19} \mathrm{J/eV}\).
02

Calculation of Time for One Revolution of a Proton

To determine (b), the formula for the time period \(T\) of revolution of a charged particle in a magnetic field is \(T = 2\pi m / qB\). The time for one revolution can be calculated by substituting \(m = 1.67 \times 10^{-27} \mathrm{kg}\), \(q = 1.60 \times 10^{-19} \mathrm{C}\), and \(B = 0.85 \mathrm{T}\) into the formula.
03

New Magnetic-Field Magnitude to Double the Maximum Energy

The new magnetic field strength required to double the maximum energy calculated in (a) can be obtained using the formula \(B_{\text{new}} = 2E_{\text{new}} / (qRc)\). To find the \(B_{\text{new}}\), replace \(E_{\text{new}}\) with twice the original maximum energy obtained from (a), assuming \(q\), \(R\), and \(c\) to be constant.
04

Maximum Energy for Alpha Particles and Comparison

To answer part (d), we find the maximum energy for alpha particles using the formula with substitutes \(q = 3.20 \times 10^{-19} \mathrm{C}\) and \(m = 6.64 \times 10^{-27} \mathrm{kg}\), but retain \(B = 0.85 \mathrm{T}\) and \(R = 0.4 \mathrm{m}\). The result in joules and electron volts is compared to the maximum energy for protons calculated in (a).

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Key Concepts

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

Understanding the Magnetic Field
In the context of cyclotron particle acceleration, the magnetic field plays a crucial role. It's essentially the driving force that enables the cyclotron to accelerate charged particles, like protons. A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. When charged particles enter a magnetic field, they experience a force called the Lorentz force, which is perpendicular both to their velocity and the magnetic field. This force causes them to spiral and accelerate within the cyclotron until they reach the outer edge.

For the given cyclotron exercise, the magnetic field has a magnitude of 0.85 Tesla (T). Tesla is the SI unit of magnetic field strength or magnetic flux density, and it is a measure of how much force a magnetic field exerts on moving charges. The radius of the cyclotron's magnetic poles is 0.40 meters, determining the path that the protons will follow as they accelerate. The stronger the magnetic field or the larger the radius, the greater the maximum energy to which the particle can be accelerated.
Proton Energy Calculation in a Cyclotron
Calculating the maximum energy a proton can achieve in a cyclotron is fundamental to understanding the capabilities of the accelerator. Energy in physics is broadly a measure of the ability to do work. When it comes to protons in a magnetic field, their kinetic energy increases as they are accelerated by the cyclotron's magnetic field.

To calculate the energy of a proton, we use the formula:
\[E = qBR \times c\]
where \(q\) is the charge of the proton, \(B\) is the strength of the magnetic field, \(R\) is the radius of the cyclotron, and \(c\) is the speed of light. When substituting the respective values into this formula, we can determine the energy in joules. Since scientists and engineers often work with electron volts (eV), the exercise also requires converting this energy into eV using the conversion factor \(1.6 \times 10^{-19} \mathrm{J/eV}\). Understanding how to convert between these units helps when comparing energies across different systems or experiments, and is particularly important for applications in particle physics.
Charged Particle Revolution Time
To fully grasp the operation of cyclotrons, we must comprehend the charged particle revolution time. This is the time it takes for a charged particle, like a proton, to make one full orbit within the cyclotron. The revolution time is important because it affects how often a particle is accelerated and consequently its final energy when ejected.

The formula to calculate the time for one complete revolution is: \[T = \frac{2\pi m}{qB}\]
Here, \(T\) represents the revolution time, \(m\) is the mass of the charged particle (proton, in this case), \(q\) is the charge of the particle, and \(B\) is the magnetic field strength. By substituting in the known values for a proton in a magnetic field of 0.85 Tesla, we can determine the time it takes to complete one orbit at the maximum radius. This time period is critical as it represents one full cycle of acceleration and is a factor considered in the design and operation of cyclotrons for a variety of applications, including medical isotope production and fundamental research in particle physics.

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Most popular questions from this chapter

An electron is moving in the \(x y\) -plane. If at time \(t\) a magnetic field \(B=0.200 \mathrm{~T}\) in the \(+z\) -direction exerts a force on the electron equal to \(F=5.50 \times 10^{-18} \mathrm{~N}\) in the \(-y\) -direction, what is the velocity (magnitude and direction) of the electron at this instant?

An electron in the beam of a cathode-ray tube is accelerated by a potential difference of \(2.00 \mathrm{kV}\). Then it passes through a region of transverse magnetic field, where it moves in a circular arc with radius \(0.180 \mathrm{~m} .\) What is the magnitude of the field?

A mass spectrograph is used to measure the masses of ions, or to separate ions of different masses (see Section 27.5 ). In one design for such an instrument, ions with mass \(m\) and charge \(q\) are accelerated through a potential difference \(V\). They then enter a uniform magnetic field that is perpendicular to their velocity, and they are deflected in a semicircular path of radius \(R .\) A detector measures where the ions complete the semicircle and from this it is easy to calculate \(R\). (a) Derive the equation for calculating the mass of the ion from measurements of \(B, V, R,\) and \(q\). (b) What potential difference \(V\) is needed so that singly ionized \({ }^{12} \mathrm{C}\) atoms will have \(R=50.0 \mathrm{~cm}\) in a 0.150 T magnetic field? (c) Suppose the beam consists of a mixture of \({ }^{12} \mathrm{C}\) and \({ }^{14} \mathrm{C}\) ions. If \(v\) and \(B\) have the same values as in part \((\mathrm{b}),\) calculate the separation of these two isotopes at the detector. Do you think that this beam separation is sufficient for the two ions to be distinguished? (Make the assumption described in Problem 27.53 for the masses of the ions.)

A group of particles is traveling in a magnetic field of unknown magnitude and direction. You observe that a proton moving at \(1.50 \mathrm{~km} / \mathrm{s}\) in the \(+x\) -direction experiences a force of \(2.25 \times 10^{-16} \mathrm{~N}\) in the \(+y\) -direction, and an electron moving at \(4.75 \mathrm{~km} / \mathrm{s}\) in the \(\begin{array}{llll}-z \text { -direction } & \text { experiences } & \text { a } & \text { force } & \text { of } & 8.50 \times 10^{-16} \mathrm{~N} & \text { in the }\end{array}\) \(+y\) -direction. (a) What are the magnitude and direction of the magnetic field? (b) What are the magnitude and direction of the magnetic force on an electron moving in the \(-y\) -direction at \(3.20 \mathrm{~km} / \mathrm{s} ?\)

One method for determining the amount of corn in early Native American diets is the stable isotope ratio analysis (SIRA) technique. As corn photosynthesizes, it concentrates the isotope carbon-13, whereas most other plants concentrate carbon-12. Overreliance on corn consumption can then be correlated with certain diseases, because corn lacks the essential amino acid lysine. Archaeologists use a mass spectrometer to separate the \({ }^{12} \mathrm{C}\) and \({ }^{13} \mathrm{C}\) isotopes in samples of human remains. Suppose you use a velocity selector to obtain singly ionized (missing one electron) atoms of speed \(8.50 \mathrm{~km} / \mathrm{s}\), and you want to bend them within a uniform magnetic field in a semicircle of diameter \(25.0 \mathrm{~cm}\) for the \({ }^{12} \mathrm{C}\). The measured masses of these isotopes are \(1.99 \times 10^{-26} \mathrm{~kg}\left({ }^{12} \mathrm{C}\right)\) and \(2.16 \times 10^{-26} \mathrm{~kg}\left({ }^{13} \mathrm{C}\right) .\) (a) What strength of magnetic field is required? (b) What is the diameter of the \({ }^{13} \mathrm{C}\) semicircle? (c) What is the separation of the \({ }^{12} \mathrm{C}\) and \({ }^{13} \mathrm{C}\) ions at the detector at the end of the semicircle? Is this distance large enough to be easily observed?
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