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 290 V.
(a) By how much does its kinetic energy increase with each passage through the gap?
290 eV

(b) What is its kinetic energy as it completes 100 passes through the gap?
28.961 keV

(c) Let r100 be the radius of its circular path in the dee that it enters as it completes those 100 passes, and let r101 be its next radius, as it enters a dee the next time. By what percentage does the radius increase when it changes from r100 to r101? (r101 - r100) / r100%)
?????%

Explanation / Answer

we have that Bqv=mv^2/r Bq=mv/r so r=mv/Bq where v=sqrt(2K/m) so the percentage. (r101-r100)/r100=(v101-v100)/v100=(sqrt(K101)-sqrt(K100))/sqrt(K100)=(sqrt(101)-sqrt(100))/sqrt(100)=0.5e-2=0.5%

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