Oh My! I Was Wrong

Recently, there has been a brouhaha about “Ghost Murmur,” a proposed way to detect the magnetic field of a human heart from miles away. I was interviewed about Ghost Murmur by Deni Béchard, a reporter for Scientific American. He wrote¹

Bradley Roth, a physicist at Oakland University and author of the 2023 review Biomagnetism: The First Sixty Years, agrees. “People have been measuring the magnetic field of the heart for 60 years, and usually it’s done in a lab with shielding, and it’s done just a few centimeters or a couple inches from the heart, and even then you can barely record it.” A helicopter-borne version, he says, “would be not just a small advance, but it’d be a revolutionary advance from the state of the art.”

I stand by that statement, as written. In fact, I’m probably even more skeptical of the reality of Ghost Murmur now than when I spoke to Béchard. I confess, however, that I was wrong about one thing. I based my thinking on the magnetic field falling off as from a current dipole, so with the inverse square of the distance. Current dipoles are the usual way to model biomagnetic fields. I had seen other people online saying that the magnetic field falls off as from a magnetic dipole, so inverse cube, but I presumptuously assumed they were confusing a current dipole and a magnetic dipole.

Then I got to wondering if I was right. I know that for a current dipole in an unbounded conductor, the fall off is indeed one over distance squared. But how about for a bounded conductor, like the human body? Does that change things?

Being a physicist, my first impulse was to model the human body as a sphere. Anyone familiar with biomagnetism knows that a radial dipole in a sphere produces no magnetic field outside it. But how does the magnetic field of a tangential dipole in a sphere fall off with distance?

Homework Problem 21 in Chapter 8 of Intermediate Physics for Medicine and Biology provides the answer. It is a closed form expression for the magnetic field of a dipole in a sphere (originally derived by Jukka Sarvas).² The expression is sort of strange-looking, but it’s exactly what we need.

Let’s assume p = p x̂, r = r ẑ, and r₀ = r₀ẑ, implying that a tangential dipole lies a distance r₀ from the sphere center along the z axis, and the magnetic field is measured a distance r from the sphere center also along the z axis, where r > r₀ ( r₀ is inside the sphere and r is outside). The homework problem defines a = r – r₀. I will take the limit as r >>> r₀. So a = r. In that case F = r (r² + r² – r₀ r) which approaches 2r³. The gradient of F becomes 6r². So the expression for the magnetic field falls with distance as 1/F (first term) or as r∇F/F² (second term). In both cases, the falloff is proportional to the inverse cube.

Ah Ha! I was wrong to say the magnetic field of a current dipole in a conducting sphere falls off as the inverse square. It is the inverse cube. The effect of the sphere boundary changes things from 1/r² to 1/r³. In fact, Flavio Grynszpan and David Geselowitz³ define a magnetic dipole m that is related to the electric, or current, dipole p for the case of a spherical conductor. So the folks who modeled the heart as a magnetic dipole knew what they were talking about. Apparently I didn’t. But at least I learned something, which is always a good thing.

What does this mean for Ghost Murmur? It means I was wrong in saying the magnetic field measured 60 km from the heart is one trillion (10¹²) times smaller than the magnetic field measured 60 mm from the heart. The field would be more like one quintillion (10¹⁸) times smaller. Yikes! 

 References

¹https://www.scientificamerican.com/article/what-is-the-quantum-ghost-murmur-purportedly-used-in-iran-scientists/ 

²Sarvas J (1987) Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem. Phys. Med. Biol. 32:11–22. 

³Grynszpan F, Geselowitz DB (1973) Model studies of the magnetocardiogram. Biophys. J. 13:911–925.