Please work out using original revolutions formula and use algebra to convert it

Question

Please work out using original revolutions formula and use algebra to convert it to what is being used here.

I've worked out the max Kinetic energy: 5.5883e-13 J

A cyclotron (figure) designed to accelerate protons has an outer radius of 0.375 m. The protons are emitted nearly at rest from a source at the center and are accelerated through 546 V each time they cross the gap between the dees. The dees are between the poles of an electromagnet where the field is 0.720 T. The black, dashed curved lines represent the path of the particles Alternating V D2 After being accelerated, the particles exit here North pole of magnet

Explanation / Answer

KE = 5.5883*10^-13 J

q = 1.6*10^-19 C

delta_V = 546 volts

workdone on protons in one revolution, W = 2*q*delta_V (2 times per cycle)

let N is the no of revolutions made by the protons before emerging out, total wordne = N*2*q*delta_V

we know, total workdone = gain in kinetic energy

N*2*q*delta_V = KE

N = KE/(2*q*delta_V)

= 5.5883*10^-13/(2*1.6*10^-19*546)

= 3198 revolutions <<<<<<--------ANswer

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