A proton circulates in a cyclotron, beginning approximately at rest at the cente
ID: 1422801 • Letter: A
Question
A proton circulates in a cyclotron, beginning approximately at rest at the center. Whenever it passes through the gap between dees, the electric potential difference between the dees is 190 V. (a) By how much does its kinetic energy increase with each passage through the gap? (b) What is its kinetic energy as it completes 144 passes through the gap? (c) Let r144 be the radius of the proton's circular path as it completes those 144 passes and enters a dee, and let r145be its next radius, as it enters a dee the next time. By what percentage does the radius increase when it changes from r144 to r145? That is, what is percentage increase = (r145 - r144)/r144*100%?
Explanation / Answer
a) Increase in kinetic energy for every paas = workdone
= q*delta_V
= 1.6*10^-19*190
= 3.04*10^-17 J
b) after completing 144 passes, KE = 144*3.04*10^-17
= 4.38*10^-15 J
c) we know, radius of the path, r = m*v/(B*q)
= p/(B*q)
= sqrt(2*m*KE)/(B*q)
so, (r145 - r144)*100/(r144) = sqrt(KE_144) - sqrt(KE144))*100/sqrt(Ke_144)
= ( sqrt(145) - sqrt(144))*100/sqrt(144)
= 0.347%
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