In Section 8.1 of the 5th edition of IPMB, Russ Hobbie and I discuss the cyclotron.
One important application of magnetic forces in medicine is the cyclotron. Many hospitals have a cyclotron for the production of radiopharmaceuticals, especially for generating positron-emitting nuclei for use in Positron EmissionTomography (PET) imaging (see Chap. 17).
Consider a particle of charge q and mass m, moving with speed v in a direction perpendicular to a magnetic field B. The magnetic force will bend the path of the particle into a circle. Newton’s second law states that the mass times the centripetal acceleration, v2/r, is equal to the magnetic force
mv2/r = qvB. (8.5)
The speed is equal to [the] circumference of the circle, 2πr, divided by the period of the orbit, T. Substituting this expression for v into Eq. (8.5) and simplifying, we find
T = 2π m/(qB). (8.6)
In a cyclotron particles orbit at the cyclotron frequency, f = 1/T. Because the magnetic force is perpendicular to the motion, it does not increase the particles’ speed or energy. To do that, the particles are subjected periodically to an electric field that changes direction with the cyclotron frequency so that it is always accelerating, not decelerating, the particles. This would be difficult if not for the fortuitous disappearance of both v and r from Eq. (8.6), so that the cyclotron frequency only depends on the charge-to-mass ratio of the particles and the magnetic field, but not on their energy.
This analysis of cyclotron motion works great in a vacuum. The trouble begins when you apply the cyclotron concept to ions in the conducting fluids of the body. The proposed hypothesis says that when an ion is moving about in the presence of the earth’s magnetic field, the resulting magnetic force causes it to orbit about the magnetic field lines, with an orbital period equal to the reciprocal of the cyclotron frequency. If any electric field is present at that same frequency, it could interact with the ion, increasing its energy or causing it to cross the cell membrane.
Below is a draft of the new homework problem, which I hope debunks this erroneous hypothesis.
Section 9.1
Problem 7. One mechanism for how organisms are influenced by extremely low frequency electric fields is the cyclotron resonance hypothesis.(a) The strength of the earth's magnetic field is about 5 × 10–5 T. A calcium ion has a mass of 6.7 × 10–26 kg and a charge of 3.2 × 10–19 C. Calculate the cyclotron frequency of the calcium ion. If an electric field exists in the tissue at that frequency, the calcium ion will be in resonance with the cyclotron frequency, which could magnify any biological effect.
(b) This mechanism seems to provide a way for an extremely low frequency electric field to interact with calcium ions, and calcium influences many cellular processes. But consider this hypothesis in more detail. Use Eq. 4.12 to calculate the root-mean-square speed of a calcium ion at body temperature. Use this speed in Eq. 8.5 to calculate the radius of the orbit. Compare this to the size of a typical cell.
(c) Now make a similar analysis, but assume the radius of the calcium ion orbit is about the size of a cell (since it would have difficulty crossing the cell membrane). Then use this radius in Eq. 8.5 to determine the speed of the calcium ion. If this is the root-mean-square speed, what is the body temperature?
(d) Finally, compare the period of the orbit to the time between collisions of the calcium ion with a water molecule. What does this imply for the orbit?
This analysis should convince you that the cyclotron resonance hypothesis is unlikely to be correct. Although the frequency is reasonable, the orbital radius will be huge unless the ions are traveling extraordinarily slowly. Collisions with water molecules will completely disrupt the orbit.
For those who don't have the 5th edition of IPMB handy, Eq. 4.12 says the root-mean-square speed is equal to the square root of 3 times Boltzmann's constant times the absolute temperature divided by the mass of the particle.
I won’t give away the solution to this problem (once the 6th edition of IPMB is out, instructors can get the solution manual for free by emailing me at roth@oakland.edu). But here are some order-of-magnitude results. The cyclotron frequency is tens of hertz. The root-mean-square (thermal) speed of calcium at body temperature is hundreds of meters per second. The resulting orbital radius is about a meter. That is bigger than the body, and vastly bigger than a cell. To fit the orbit inside a cell, the speed would have to be much slower, on the order of a thousandth of a meter per second, which corresponds to a temperature of about a few nanokelvins. The orbital period is a couple hundredths of a second, but the time between collisions of the ion with a water molecule is one the order of 10–13 seconds, so there are many billions of collisions per orbit. Any circular motion will be destroyed by collisions long before anything like an orbit is established. I’m sorry, but the hypothesis is rubbish.
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| Are Electromagnetic Fields Making Me Ill? |
If you want to learn more about how extremely low frequency electric fields interact with tissue, see my book Are Electromagnetic Fields Making Me Ill?
Finally, for you folks who are really on the ball, you may be wondering why this homework problem is listed as being in Chapter 9 when the discussion of the cyclotron is in Chapter 8 of the 5th edition of IPMB. (In this post I changed the equation numbers in the homework problem to match the 5th edition, so you would have them.) Hmm.. is there a new chapter in the 6th edition? More on that later…
To be fair, I should let my late friend Abraham Liboff tell you his side of the story. In this video, Abe explains how he proposed the cyclotron resonance hypothesis. I liked Abe, but I didn’t like his hypothesis.
https://www.youtube.com/watch?v=YL-wqJ-PMAQ&list=PLCO-VktC6wofkMeEeZknT9Y4WhMnP76Ee&index=6
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