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

Question

1) We have a cyclotron. The radius of the poles, R0, is 10 meters. The Magnetic field, B, is 2 Tesla, into the page as drawn. The alternating, accelerating potential across the gap is V =50 volts.

a) 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?

b) How many acceleration cycles will it take to get to that energy?

c) How long will it take to get to that energy?

Please give a step by step solution, more details and formulas please.

Explanation / Answer

cyclotron pole radius r=10m

B=2 Tesla

v=50 volts

and

mass of the proton mp=1.672*10^-27 kg

charge of the proton q=1.602*10^-19 C

a)

magnetic force=centipetal force

qvB=mv^2/r

v=qBr/m

kinetic energy K.E=1/2*m*V^2

=(B*q*r)^2/2m

=(2*1.602*10^-19*10)^2/2*1.672*10^-27

=306.9861*10^-11 J

b)

v=qBr/m

and v=2pi*r/T

qBr/m=2pi*r/T

T=2pi*m/qB

and f=1/T

frequency f=qB/2pi*mp

=1.602*10^-19*2/2*3.14*1.672*10^-27

=15256910.37 Hz

=15.25691 MHz

c)

Time perod T=1/f=6.554*10^-8 sec

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