Learning Goal: To learn the basic physics and applications of cyclotrons. Particle accelerators are used to create well-controlled beams of high-energy particles. Such beams have many uses, both in research and industry. One common type of accelerator is the cyclotron, as shown in the figure. In a cyclotron, a magnetic field confines charged particles to circular paths while an oscillating electric field accelerates them. It is useful to understand the details of this process. Consider a cyclotron in which a beam of particles of positive charge q and mass m is moving along a circular path restricted by the magnetic field (which is perpendicular to the velocity of the particles). <img src="//img.wizedu.com/questions/2755a4d0-cf81-11ec-8911-d114dd09e363.jpg" alt="111599.jpg"/> Part A) Before entering the cyclotron, the particles are accelerated by a potential difference V. Find the speed v with which the particles enter the cyclotron.Express your answer in terms of V, m, and q.Part B)Find the radius r of the circular path followed by the particles. The magnitude of the magnetic field is B.Express your answer in terms of v, m, B, and q.Part C)Find the period of revolution T for the particles.Express your answer in terms of m, B, and q. Part D)Find the angular frequency w of the particles.Express your answer in terms of m, B, and q.Part E)Your goal is to accelerate the particles to kinetic energy K. What minimum radius R of the cyclotron is required?Express your answer in terms of m, q, B, and K.

A)work done on particles = K.E stored = .. W = (1/2)*m*v^2... q*V = (1/2)*m*v^2... speed v = sqrt[2*q*V/m]...B) m*v^2/r = q*v*b..r = m*v/(q*B)...C)T = 2*pi*m/(q*B) D) angular frequency w = 2*pi/T = 2*pi/[2*pi*m/(q*B)]=q*B/m...E)r = m*v/(q*B).. But momentum p = m*v = sqrt(2*m*K).. r = sqrt(2*m*k)/(q*B)

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Particle accelerators are used to create well-controlled beams of high-energy particles.

Learning Goal: To learn the basic physics and applications of cyclotrons.

Particle accelerators are used to create well-controlled beams of high-energy particles. Such beams have many uses, both in research and industry.

One common type of accelerator is the cyclotron, as shown in the figure. In a cyclotron, a magnetic field confines charged particles to circular paths while an oscillating electric field accelerates them. It is useful to understand the details of this process.

Consider a cyclotron in which a beam of particles of positive charge q and mass m is moving along a circular path restricted by the magnetic field (which is perpendicular to the velocity of the particles).

Part A)
Before entering the cyclotron, the particles are accelerated by a potential difference V. Find the speed v with which the particles enter the cyclotron.

Express your answer in terms of V, m, and q.

Part B)

Find the radius r of the circular path followed by the particles. The magnitude of the magnetic field is B.

Express your answer in terms of v, m, B, and q.

Part C)

Find the period of revolution T for the particles.

Express your answer in terms of m, B, and q.
Part D)

Find the angular frequency w of the particles.

Express your answer in terms of m, B, and q.

Part E)

Your goal is to accelerate the particles to kinetic energy K. What minimum radius R of the cyclotron is required?

Express your answer in terms of m, q, B, and K.

Expert Solution

A)work done on particles = K.E stored = ..
W = (1/2)*m*v^2...
q*V = (1/2)*m*v^2...
speed v = sqrt[2*q*V/m]...

B) m*v^2/r = q*v*b..

r = m*v/(q*B)...

C)T = 2*pi*m/(q*B)
D) angular frequency w = 2*pi/T = 2*pi/[2*pi*m/(q*B)]=q*B/m...

E)r = m*v/(q*B)..
But momentum p = m*v = sqrt(2*m*K)..
r = sqrt(2*m*k)/(q*B)

Amanda K answered 2 years ago

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