Ch 44: Particle Physics and Cosmology

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 44: Particle Physics and Cosmology

Problem 10a

Chapter 44, Problem 10a

The magnetic field in a cyclotron that accelerates protons is $1.70$ T. How many times per second should the potential across the dees reverse? (This is twice the frequency of the circulating protons.)

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Determine the relationship between the cyclotron frequency and the magnetic field. The cyclotron frequency (also called the angular frequency) is given by the formula:

f = \frac{q}{2πm} B

, where

q

is the charge of the proton,

m

is the mass of the proton, and

B

is the magnetic field strength.

Substitute the known values into the formula. The charge of a proton is

q = 1.60 \times 10 ⁻ 19

C, the mass of a proton is

m = 1.67 \times 10 ⁻ 27

kg, and the magnetic field strength is

B = 1.70

Calculate the cyclotron frequency using the formula:

f = \frac{q}{2πm} B

. This will give the frequency of the circulating protons in hertz (Hz).

Since the potential across the dees reverses twice per cycle of the proton's motion, multiply the cyclotron frequency by 2 to find the required frequency of the potential reversal.

Express the final result in hertz (Hz) after performing the calculations, ensuring the units are consistent throughout the process.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Cyclotron Frequency

Cyclotron frequency refers to the frequency at which a charged particle orbits in a magnetic field. It is determined by the charge of the particle, the mass of the particle, and the strength of the magnetic field. For protons in a magnetic field, the frequency can be calculated using the formula f = (qB)/(2πm), where q is the charge, B is the magnetic field strength, and m is the mass of the proton.

Dees in a Cyclotron

In a cyclotron, the 'dees' are the two D-shaped electrodes that create an oscillating electric field to accelerate charged particles. As protons move between the dees, they gain energy from the electric field, which reverses direction at a specific frequency. The frequency of the electric field must match the cyclotron frequency of the protons to ensure efficient acceleration.

Magnetic Field Strength

Magnetic field strength, measured in teslas (T), is a measure of the intensity of the magnetic field. In a cyclotron, a stronger magnetic field allows for higher energy acceleration of particles. The magnetic field influences the radius of the particle's path and its frequency of rotation, which is crucial for determining how often the electric potential across the dees must reverse to maintain synchronization with the protons.

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Related Practice

Textbook Question

Calculate the minimum beam energy in a proton-proton collider to initiate the $p + p \to p + p + η^{0}$ reaction. The rest energy of the $eta to the 0 power$ is $547.3$ MeV (see Table $44.3$ ).

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Textbook Question

Deuterons in a cyclotron travel in a circle with radius $32.0$ cm just before emerging from the dees. The frequency of the applied alternating voltage is $9.00$ MHz. Find the magnetic field.

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Textbook Question

The starship Enterprise, of television and movie fame, is powered by combining matter and antimatter. If the entire $400$ -kg antimatter fuel supply of the Enterprise combines with matter, how much energy is released? How does this compare to the U.S. yearly energy use, which is roughly $1.0 \times 10^{20}$ J?

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Textbook Question

A high-energy beam of alpha particles collides with a stationary helium gas target. What must the total energy of a beam particle be if the available energy in the collision is $16.0$ GeV?

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Textbook Question

What is the speed of a proton that has total energy $1000$ GeV?

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Textbook Question

An electron with a total energy of $30.0$ GeV collides with a stationary positron. What is the available energy?

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The magnetic field in a cyclotron that accelerates protons is 1.701.70 T. How many times per second should the potential across the dees reverse? (This is twice the frequency of the circulating protons.)

Verified video answer for a similar problem:

Key Concepts

Cyclotron Frequency

Dees in a Cyclotron

Magnetic Field Strength

The magnetic field in a cyclotron that accelerates protons is $1.70$ T. How many times per second should the potential across the dees reverse? (This is twice the frequency of the circulating protons.)