We have a cyclotron. The radius of the poles, R0, is 10 meters. The Magnetic fie

Question

We have a cyclotron. The radius of the poles, R0, is 10 meters. The Magnetic field, B, is 1 Tesla. The alternating, accelerating potential across the gap is V =100 volts.

We have a cyclotron. The radius of the poles, R0, is 10 meters. The Magnetic field, B, is 1 Tesla. The alternating, accelerating potential across the gap is V =100 volts. If a proton starts from rest, at what kinetic energy will the radius of its cyclotron orbit be equal to the radius of the pole pieces? How many acceleration cycles will it take to get to that energy? How long will it take to get to that energy?

Explanation / Answer

1)

Magnetic force on the proton due to magnetic field is

Bq v = m v^2 / r

v = Bq r / m = ( 1 T ) ( 1.6*10^ -19 C ) ( 10 m ) / 1.67*10^ -27 kg )

=9.58*10^ 8 m/s

kinetic energy of proton is

KE = ( 1/2) m v^2

= ( 1/2) ( 1.67*10^ -27 kg ) ( 9.58*10^ 8 m/s ) ^2

= 7.66*10^ -10 J

b)

Time taken to complete the orbit by proton is

T = 2 pi r / v

Number of cycles completed by proton is

f = v / 2 pi r

= ( 9.58*10^ 8 ) / 2 pi ( 10 m ) = 1.525*10^ 7 cycles / sec

c)

T = 1/ f = 6.55*10^ -8 s

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