What is the estimated path length traveled by a deuteron in a cyclotron?

[gmark]

Homework Statement

Estimate the total path length traveled by a deuteron in a cyclotron of radius 53 cm and operating frequency 12 MHz during the entire acceleration process. Accelerating potential between dees is 80 kV.

mass m = 3.2e-27 kg
charge q = 1.6e-19 C
radius r = 0.53 m
frequency f = 12e6 /s
potential V = 80e3 V

Homework Equations

(1) kinetic energy KE = .5*m*v^2

Consider a charged particle of mass m, charge q, velocity v perpendicular to magnetic field B. The particle will travel in a circle of radius r with frequency f:

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

(3) f = (q*B)/(2*pi*m)

(4) V = J/C

The Attempt at a Solution

My plan is to calculate the KE the particle has on leaving the machine (r = .53m). This total KE results from N crossing of the dees. So divide the final KE by the KE per crossing to get N. The total number of revolutions is half of N. Estimate the total path length by multiplying N/2 by the outer circumference of the machine, 2*pi*r.

1. Final KE

rearrange (2) above: v = (r*q*B)/m

substitute for v in (1):
(5) KE = (m/2)*(r*q*B/m)^2

rearrange (2) B = (2*pi*m*f)/q

substitute for B in (5) and simplify: KE = 2*m*(r*pi*f)^2

Using values given above, KE = 2.67e-12 J (r = .53m).

2. KE per crossing

rearrange (4) J = V*C

Using values given (V=80e3, C=1.6e-19)
KE = 1.28e-14 J

Dividing total KE by per-crossing KE gives 208 crossings, 104 revolutions.

3. Estimated distance is 2*pi*r*104 (r=.53m) = 346 m.

Given answer is 240 m.

??

 

[BvU]

Hi,

Estimate the total path length by multiplying N/2 by the outer circumference of the machine, 2*pi*r.

Particle doesn't start at the outer rim but very close to the center

 

[gmark]

Thanks for your reply. Unfortunately I don't know how to proceed. All I can think of is silly: take the derivative of final KE (step 1 above) wrt the radius, and integrate from r=0 to r=0.53. This accomplishes nothing, of course, and results in the same wrong answer I got before.

 

[BvU]

Calculate the trajectory radius after 1, 2, 3, etc revolution (= 2, 4, 6, ... acceleration steps)

 

[gmark]

Thanks. That works. Yay for Python and brute force.