Can Induced Electric Fields Explain Biological Effects of Power-Line Magnetic Fields?

Sometimes proponents of pseudoscience embrace nonsense, but other times they propose plausible-sounding ideas that are wrong because the numbers don’t add up. For example, suppose you are discussing with your friend about the biological effects of power-line magnetic fields. Your friend might say something like this:

“You keep claiming that magnetic fields don’t have any biological effects. But suppose it’s not the magnetic field itself, but the electric field induced by the changing magnetic field that causes the effect. We know an electric field can stimulate nerves. Perhaps power-line effects operate like transcranial magnetic stimulation, by inducing electric fields.”

Well, there’s nothing absurd about this hypothesis. Transcranial magnetic stimulation does work by creating electric fields in the body via electromagnetic induction, and these electric fields can stimulate nerves. The qualitative idea is reasonable. But does it work quantitatively? If you do the calculation, the answer is no. The electric field induced by a power line is less than the endogenous electric field associated with the electrocardiogram. You don’t have to perform a difficult, detailed calculation to show this. A back-of-the-envelope estimation suffices. Below is a new homework problem showing you how to make such an estimate.

Section 9.10

Problem 36 ½. Estimate the electric field induced in the body by a power-line magnetic field, and compare it to the endogenous electric field in the body associated with the electrocardiogram.

(a) Use Eq. 8.25 to estimate the induced electric field, E. The magnetic field in the vicinity of a power line can be as strong as 5 μT (Possible Health Effects of Exposure to Residential Electric and Magnetic Fields, 1997, Page 32), and it oscillates at 60 Hz. The radius, a, of the current loop in our body is difficult to estimate, but take it as fairly large (say, half a meter) to ensure you do not underestimate the induced electric field.

(b) Estimate the endogenous electric field in the torso from the electrocardiogram, using Figures 7.19 and 7.23.

(c) Compare the electric fields found in parts (a) and (b). Which is larger? Explain how an induced electric field could have an effect if it is smaller than the electric fields already existing in the body.

Let’s go through the solution to this new problem. First, part (a). The amplitude of the magnetic field is 0.000005 T. The field oscillates with a period of 1/60 Hz, or about 0.017 s. The peak rate of change will occur during only a fraction of this period, and a reasonable approximation is to divide the period by 2π, so the time over which the magnetic field changes is 0.0027 s. Thus, the rate of change dB/dt is 0.000005 T/0.0027 s, or about 0.002 T/s. Now use Eq. 8.25, E = a/2 dB/dt (ignore the minus sign in the equation, which merely indicates the phase), with a = 0.5 m, to get E = 0.0005 V/m.

Now part (b). Figure 7.23 indicates that the QRS complex in the electrocardiogram has a magnitude of about ΔV = 0.001 V (one millivolt). Figure 7.19 shows that the distance between leads is on the order of Δr = 0.5 m. The magnitude of the electric field is approximately ΔV/Δr = 0.002 V/m.

In part (c) you compare the electric field induced by a power line, 0.0005 V/m, to the electric field in the body caused by the electrocardiogram, 0.002 V/m. The field produced by the ECG is four times larger. So, how can the induced electric field have a biological effect if we are constantly exposed to larger electric fields produced by our own body? I don’t know. It seems to me that would be difficult.

Hart and Gandhi (1998)
Phys. Med. Biol.,
43:3083–3099.

But wait! Our calculation in part (b) is really rough. Perhaps we should do a more detailed calculation. Rodney Hart and Om Gandhi did just that (“Comparison of cardiac-induced endogenous fields and power frequency induced exogenous fields in an anatomical model of the human body,”  Physics in Medicine & Biology, Volume 43, Pages 3083–3099, 1998). They found that during the QRS complex the endogenous electric field varied throughout the body, but it is usually larger than what we estimated. It’s giant in the heart itself, about 3 V/m. All through the torso it’s more than ten times what we found; for instance, in the intestines it’s 0.04 V/m. Even in the brain the field strength (0.014 V/m) is seven times larger than our estimate (0.002 V/m).

Moreover, the heart isn’t the only source of endogenous fields (although it’s the strongest). The brain, peripheral nerves, skeletal muscle, and the gut all produce electric fields. In addition, our calculation of the induced electric field is evaluated at the edge of the body, where the current loop is largest. Deeper within the torso, the field will be less. Finally, our value of 5 μT is extreme. Magnetic fields associated with power lines are usually about one tenth of this. In other words, in all our estimates we took values that favor the induced electric field over the endogenous electric field, and the endogenous electric field is still four times larger.

What do we conclude? The qualitative mechanism proposed by your friend is not ridiculous, but it doesn’t work when you do the calculation. The induced electric field would be swamped by the endogenous electric field.

The moral of the story is that proposed mechanisms must work both qualitatively and quantitatively. Doing the math is not an optional step to refine your hypothesis and make it more precise. You have to do at least an approximate calculation to decide if your idea is reasonable. That’s why Russ Hobbie and I emphasize solving toy problems and estimation in Intermediate Physics for Medicine and Biology. Without estimating how big effects are, you may go around saying things that sound reasonable but just aren’t true.

The Central Slice Theorem: An Example

The central slice theorem is key to understanding tomography. In Intermediate Physics for Medicine and Biology, Russ Hobbie and I ask the reader to prove the central slice theorem in a homework problem. Proofs are useful for their generality, but I often understand a theorem better by working an example. In this post, I present a new homework problem that guides you through every step needed to verify the central slice theorem. This example contains a lot of math, but once you get past the calculation details you will find it provides much insight.

The central slice theorem states that taking a one-dimensional Fourier transform of a projection is equivalent to taking the two-dimensional Fourier transform and evaluating it along one direction in frequency space. Our “object” will be a mathematical function (representing, say, the x-ray attenuation coefficient as a function of position). Here is a summary of the process, cast as a homework problem.

Section 12.4 

Problem 21½. Verify the central slice theorem for the object

(a) Calculate the projection of the object using Eq. 12.29,

Then take a one-dimensional Fourier transform of the projection using Eq. 11.59,

 

(b) Calculate the two-dimensional Fourier transform of the object using Eq. 12.11a,

Then transform F̂(k_x,k_y ) to F̂(θ,k) by converting from Cartesian to polar coordinates in frequency space.

(c) Compare your answers to parts (a) and (b). Are they the same?

I’ll outline the solution to this problem, and leave it to the reader to fill in the missing steps. 

 

Fig. 12.12 from IPMB, showing how to do a projection.

The Projection 

Figure 12.12 shows that the projection is an integral of the object along various lines in the direction θ, as a function of displacement perpendicular to each line, x'. The integral becomes

Note that you must replace x and y by the rotated coordinates x' and y'

You can verify that x² + y²= x'² + y'².

After some algebra, you’re left with integrals involving e^−by'2 (Gaussian integrals) such as those analyzed in Appendix K of IPMB. The three you’ll need are

The resulting projection is

Think of the projection as a function of x', with the angle θ being a parameter.

 

The One-Dimensional Fourier Transform

The next step is to evaluate the one-dimensional Fourier transform of the projection

The variable k is the spatial frequency. This integral isn’t as difficult as it appears. The trick is to complete the square of the exponent

Then make a variable substitution u = x' + ik⁄2b. Finally, use those Gaussian integrals again. You get

This is our big result: the one-dimensional Fourier transform of the projection. Our next goal is to show that it’s equal to the two-dimensional Fourier transform of the object evaluated in the direction θ.

Two-Dimensional Fourier Transform

To calculate the two-dimensional Fourier transform, we must evaluate the double integral

The variables k_x and k_y are again spatial frequencies, and they make up a two-dimensional domain we call frequency space.

You can separate this double integral into the product of an integral over x and an integral over y. Solving these requires—guess what—a lot of algebra, completing the square, and Gaussian integrals. But the process is straightforward, and you get

Select One Direction in Frequency Space

If we want to focus on one direction in frequency space, we must convert to polar coordinates: k_x = k cosθ and k_y = k sinθ. The result is 

This is exactly the result we found before! In other words, we can take the one-dimensional Fourier transform of the projection, or the two-dimensional Fourier transform of the object evaluated in the direction θ in frequency space, and we get the same result. The central slice theorem works.

I admit, the steps I left out involve a lot of calculations, and not everyone enjoys math (why not?!). But in the end you verify the central slice theorem for a specific example. I hope this helps clarify the process, and provides insight into what the central slice theorem is telling us.

John Schenck and the First Brain Selfie

Schenck, J. F. (2005) 
Prog. Biophys. Mol. Biol.
87:185–204.

In Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss biomagnetism, magnetic resonance imaging, and the biological effects of electromagnetic fields. We don’t, however, talk about the safety of static magnetic fields. If you want to learn more about that topic, I suggest an article by John Schenck:

Schenck, J. F. (2005) “Physical interactions of static magnetic fields with living tissues,” Prog. Biophys. Mol. Biol. Volume 87, Pages 185–204.

This paper appeared in a special issue of the journal Progress in Biophysics and Molecular Biology analyzing the health effects of magnetic fields. The abstract states:

Clinical magnetic resonance imaging (MRI) was introduced in the early 1980s and has become a widely accepted and heavily utilized medical technology. This technique requires that the patients being studied be exposed to an intense magnetic field of a strength not previously encountered on a wide scale by humans. Nonetheless, the technique has proved to be very safe and the vast majority of the scans have been performed without any evidence of injury to the patient. In this article the history of proposed interactions of magnetic fields with human tissues is briefly reviewed and the predictions of electromagnetic theory on the nature and strength of these interactions are described. The physical basis of the relative weakness of these interactions is attributed to the very low magnetic susceptibility of human tissues and the lack of any substantial amount of ferromagnetic material normally occurring in these tissues. The presence of ferromagnetic foreign bodies within patients, or in the vicinity of the scanner, represents a very great hazard that must be scrupulously avoided. As technology and experience advance, ever stronger magnetic field strengths are being brought into service to improve the capabilities of this imaging technology and the benefits to patients. It is imperative that vigilance be maintained as these higher field strengths are introduced into clinical practice to assure that the high degree of patient safety that has been associated with MRI is maintained.

The article discusses magnetic forces due to tissue susceptibility differences, magnetic torques caused by anisotropic susceptibilities, flow or motion-induced currents, magnetohydrodynamic pressure, and magnetic excitation of sensory receptors.

On the lighter side, below are excerpts from a 2015 General Electric press report that describes one of Schenck’s claims to fame: his brain was the first one imaged using a clinical 1.5 T MRI scanner.

Heady Times: This Scientist Took the First Brain Selfie and Helped Revolutionize Medical Imaging

Early one October morning 30 years ago, GE scientist John Schenck was lying on a makeshift platform inside a GE lab in upstate New York. The [lab itself] was put together with special non-magnetic nails because surrounding his body was a large magnet, 30,000 times stronger than the Earth’s magnetic field. Standing at his side were a handful of colleagues and a nurse. They were there to peer inside Schenck’s head and take the first magnetic resonance scan (MRI) of the brain…

[In the 1970s] GE imaging pioneer Rowland “Red” Redington… hired Schenck, a bright young medical doctor with a PhD in physics [to work on MRI]... Schenck spent days inside Redington’s lab researching giant magnets and nights and weekends tending to emergency room patients. “This was an exciting time,” Schenck remembers….

It took Schenck and the team two years to obtain a magnet strong enough to… achieve useful high-resolution images. The magnet... arrived in Schenck’s lab in the spring of 1982. Since there was very little research about the effects of such [a] strong magnetic field on humans, Schenck turned it on, asked a nurse to monitor his vitals, and went inside it for ten minutes.

The field did Schenck no harm and the team spent that summer building the first MRI prototype using [a] high-strength magnetic field. By October 1982 they were ready to image Schenck’s brain.

Many scientists at the time thought that at 1.5 tesla, signals from deep tissue would be absorbed by the body before they could be detected. “We worried that there would only be a big black hole in the center” of the image, Schenck says. But the first MRI imaging test was a success. “We got to see my whole brain,” Schenck says. “It was kind of exciting.”…

Schenck, now 76, still works at his GE lab and works on improving the machine. He’s been scanning his brain every year and looking for changes… “When we started, we didn’t know whether there would be a future,” he says. “Now there is an MRI machine in every hospital.”

Two-Semester Intermediate Course Sequence in Physics for the Life Sciences

This week I spoke at the American Association of Physics Teachers 2021 Summer Meeting. Getting to the meeting was easy; I just logged onto a website. Because of the Covid-19 pandemic, the entire conference was virtual and all the talks were prerecorded. A video of my talk—“Two-Semester Intermediate Course Sequence in Physics for the Life Sciences”—is posted below. If you want a powerpoint of the slides, you can find it here. As readers of this blog might suspect, the courses I describe are based on the textbook Intermediate Physics for Medicine and Biology. 

“Two-Semester Intermediate Course Sequence in Physics for the Life Sciences,” delivered at the AAPT 2021 Virtual Summer Meeting on August 2, 2021. https://www.youtube.com/watch?v=_1b9OdQktrI

Redish, E. F. (2021)
“Using Math in Physics: Overview,”
The Physics Teacher, 59:314–318.

In my lecture, I emphasize the role of toy models in developing insight, and the importance of connecting math to physics and biology. After the talk, I had a chat with Ed Redish (who I’ve mentioned in this blog before), and he referred me to a series of articles he’s publishing in The Physics Teacher. The first is titled “Using Math in Physics: Overview” (Volume 59, Pages 314–318, 2021). Redish and I seem to be singing the same song, although his lyrics are better. What he says about math in physics describes what Russ Hobbie and I try to do in IPMB. Redish begins

The key difference between math as math and math in science is that in science we blend our physical knowledge with our knowledge of math. This blending changes the way we put meaning to math and even the way we interpret mathematical equations. Learning to think about physics with math instead of just calculating involves a number of general scientific thinking skills that are often taken for granted [my italics] (and rarely taught) in physics classes. In this paper, I give an overview of my analysis of these additional skills. I propose specific tools for helping students develop these skills in subsequent papers.

He makes other good points, such as

• Math in math classes tends to be about numbers. Math in science is not. Math in science blends physics conceptual knowledge with mathematical symbols

and my favorite

• In introductory math, equations are almost always about solving and calculating. In physics [they’re] often about explaining! [his italics, my exclamation point].

The Art of Insight
in Science and Engineering
by Sanjoy Mahajan.

I like to paraphrase Richard Hamming and say “the purpose of equations is insight, not numbers.” Redish’s article reminds me of Sanjoy Mahajan’s book The Art of Insight in Science and Engineering. Both are superb.

In subsequent articles in The Physics Teacher (some already published, some in the works), Redish discusses skills every student needs to master.

• Dimensional Analysis 
• Estimation 
• Anchor Equations 
• Toy Models 
• Functional Dependence 
• Reading the Physics in a Graph 
• Telling the Story
  

I like to think that IPMB reinforces these skills. They certainly are ones that I try to emphasize in my “Biological Physics” and “Medical Physics” classes, and that Russ and I attempt to reinforce in our homework problems.

Screenshot of the
Living Physics Portal.

Finally, a valuable resource for teachers of physics-for-the-life-sciences was noted during the Q&A: the Living Physics Portal.

The Living Physics Portal is an online environment for physics faculty to share and discuss free curricular resources for teaching introductory physics for life sciences (IPLS). The objective of the Portal is to improve the education of the next generation of medical professionals and biologists by making physics classes more relevant for life sciences students. We do this by supporting physics instructors in finding and creating curricular materials and engaging in community discussions with other instructors to improve their courses.

Although IPMB is not intended to be used in an introductory course, I believe many materials on the Living Physics Portal would be useful to instructors teaching from IPMB. Conversely, much of the information you find in IPMB, and on this blog, could be helpful to introductory teachers. 

 

If you’re preparing to teach a class based on Intermediate Physics for Medicine and Biology, I suggest first looking at the materials on the book’s website, then scanning through the book’s blog (especially those posts marked “useful for instructors”), next reading Redish’s The Physics Teacher articles, and finally browsing the Living Physics Portal. Then you’ll be ready to teach physics for the life sciences at any level.