A cyclotron in a magnetic field of 8.01 T is used to accelerate protons to 46.1

Question

A cyclotron in a magnetic field of 8.01 T is used to accelerate protons to 46.1 % of the speed of light.

a) What is the cyclotron frequency of these protons?

b) What is the radius of their trajectory in the cyclotron?

c) What is the cyclotron frequency of the same protons in the Earth's magnetic field? Assume that the Earth's magnetic field is about 0.453 G.

d) What is the trajectory radius of the same protons in the Earth's magnetic field? Assume that the Earth's magnetic field is about 0.453 G.

Explanation / Answer

a) cyclotron frequency = Bq/2m where B=magnetic field, q=charge on particle, m=mass of particle

thus cyclotron frequency = (8.01 x 1.602 x 10-19)/(2 x 1.67262158 x 10-27)

= 1.221 x 108 per second

b)radius = mv/Bq where v= velocity

radius = 1.67262158 x 10-27x 0.461 x 3 x 108/(8.01 x 1.602 x 10-19) = 0.18027m

c) now B=0.453G=0.453 x 6.673 x 10-11= 3.022869 x 10-11T

new cyclotron frequency = (3.022869 x 10-11 x 1.602 x 10-19)/(2 x 1.67262158 x 10-27)

= 0.46079 x 10-3 per second

d)new trajectory radius = 1.67262158 x 10-27x 0.461 x 3 x 108/(3.022869 x 10-11 x 1.602 x 10-19) = 0.47768 x 1011m

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