A proton circulates in a cyclotron, beginning approximately at rest at the cente

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 140 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 127 passes through the gap? (c) Let r127 be the radius of the proton's circular path as it completes those 127 passes and enters a dee, and let r128be its next radius, as it enters a dee the next time. By what percentage does the radius increase when it changes from r127 to r128? That is, what is percentage increase-(r128-r127)/r127*100%?

Explanation / Answer

given

V = 140 V

a. increase inKE with each passage = qV

q = 1.6*10^-19 C

hence

dE = 1.6*10^-19 * 140 = 2.24*10^-17 J

b. after 127 passes, KE = 127*dE = 284.48 *10^-17 = 2.8448*10^-15 J

c. v127 = sqroot(2KE/m) = 1.8857359*10^6 m/s

v128 = sqroot(2(128dE)/m) = 1.89314553*10^6 m/s

hence

qvB = mv^2/r

r = mv/qB

hence

(r128 - r127)*100/(r127) = (v128/v127 - 1)*100 = 0.3929304518 %

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