A Present From Santa

Santa arrived last night and left you, dear reader, a present in your stocking: two new homework problems for the 4^th edition of Intermediate Physics for Medicine and Biology. The problems belong to Chapter 8 on Biomagnetism (one of my favorite chapters), and specifically to Section 8.6 on Electromagnetic Induction. They both explore the idea of skin depth, but from somewhat different perspectives. Please forgive Santa for being a bit long-winded; he got carried away.

Enjoy.

Section 8.6

Problem 25.1 The concept of “skin depth” plays a role in some biomagnetic applications.
(a) Write Ampere’s law (Eq. 8.22) for the case when the displacement current is negligible.
(b) Use Ohm’s law (Eq. 6.26) to write the result from (a) in terms of the electric field.
(c) Take the curl of both sides of the equation you found in (b) (Assume the conductivity σ is homogeneous and isotropic).
(d) Use Faraday’s law (Eq. 8.20), ∇·B=0 (Eq. 8.7), and the vector identity ∇×(∇×B)=∇(∇·B)-∇²B to simplify the result from (c).
(e) Your answer to (d) should be the familiar diffusion equation (Eq. 4.24). Express the diffusion constant D in terms of electric and magnetic parameters.
(f) In Chapter 4, we found that diffusion over a distance L takes a time T equal to L²/2D. During transcranial magnetic stimulation, L=0.1 m, σ=0.1 S/m and μ_o=4π × 10⁻⁷ T m/A. How long does the magnetic field take to diffuse into the head? Is this time much longer than or much shorter than the rise time of the magnetic field for the stimulator designed by Barker et al. (1985)?
(g) Solve T= L²/2D for L, using the expression for D found in (e). Calculate L for T=0.1 ms. Is L much larger than or much smaller than the size of your head? L is closely related to the “skin depth” defined in electromagnetic theory.
(h) During magnetic resonance imaging (see Chapter 18), an 85 MHz radio-frequency magnetic field is applied to the body. Calculate L using half a period for T. How does L compare to the size of the head? The frequency of the RF field is proportional to the strength of the static magnetic field in an MRI device, and 85 MHz corresponds to 2 T. If the static field is 7 T (common in modern high-field MRI), calculate L. Is it safe to ignore skin depth during high-field MRI?

Problem 25.2 During magnetic stimulation, a changing magnetic field B induces eddy currents in the body that produce their own magnetic field B'. The goal of this problem is to compare B' and B. We can estimate B' using the following approximations. First, ignore the vector nature of all fields and do not distinguish between components. Second, ignore all negative signs. Third, replace all time derivatives with multiplication by 1/T, where T is a characteristic time. Fourth, replace all space derivatives (such as the curl) by multiplication with 1/L, where L is a characteristic length.
(a) Use Faraday’s law (Eq. 8.20) to estimate the induced electric field E from B.
(b) Use Ohm’s law (Eq. 6.26) to estimate the current density J from E.
(c) Use Ampere’s law (Eq. 8.22, but ignore displacement currents) to estimate B' from J.
(d) Combine parts (a), (b), and (c) to determine an expression for the ratio B'/B in terms of the conductivity σ, the permeability μ_o, L, and T.
(e) In magnetic stimulation, L=0.1 m, T=0.1 ms, σ=0.1 S/m and μ_o=4π × 10⁻⁷ T m/A. Calculate B'/B. Is it safe to ignore B' compared to B during magnetic stimulation?

Where's Albert?

Albert Einstein is considered one of the greatest physicists of the 20^th century, and perhaps of all time. He certainly is one of the best-known physicists, being selected by TIME Magazine as their Person of the Century in 1999. Yet, Einstein is curiously absent in the 4^th edition of Intermediate Physics for Medicine and Biology. If you look in the index under Einstein, you find only one entry: on page 393, where Russ Hobbie and I introduce the unit of an einstein (a mole of photons) in a homework problem.

Does Einstein’s work appear anywhere else in Intermediate Physics for Medicine and Biology? Certainly his masterpiece, the general theory of relativity, has little or no direct impact on biology or medicine. I don’t believe we even refer indirectly to this monumental description of gravity. However, Einstein’s earlier theory, special relativity, does appear occasionally in our book. In Chapter 8 on Biomagnetism, we write “the appearance of the magnetic force is a consequence of special relativity,” a topic we explore further in Homework Problems 5 and 23. Yet, the relationship between electrodynamics and relativity is mentioned as an aside, and is not a central feature of our analysis of magnetism. We could have left out mention of relativity from Chapter 8 altogether, and the rest of the chapter would be unaffected.

Special relativity enters in a more profound way in Chapter 15, on the Interaction of Photons and Charged Particles with Matter. There, we analyze Compton Scattering, and need the relationship between photon energy E and momentum p, given by special relativity as E = pc, where c is the speed of light. Moreover, the concept of rest mass m is introduced in this chapter, and we use Einstein’s most famous equation E = mc², relating energy and mass. Rest mass appears again in the discussion of pair production, where enough photon energy must be present to produce an electron-positron pair. The equation appears one more time in Chapter 17 on Nuclear Physics and Nuclear Medicine, where mass can be converted into energy in nuclear reactions.

Besides relativity, Einstein also played a leading role in the development of quantum mechanics, especially as related to the quantization of light and the idea of photons. This idea is first presented in Chapter 9, in a section on the Possible Effects of Weak External Electric and Magnetic Fields, where we compare the photon energy (equal to Planck’s constant times the frequency of the radiation) to the thermal energy. The idea is developed in more detail in Chapter 14, in a section about The Nature of Light: Waves versus Photons. The idea of photons is central to Chapter 15, and particularly Sec. 15.2 on Photon Interactions. There, we discuss the photoelectric effect—one mechanism by which x rays interact with tissue—which is the research that won Einstein the Nobel Prize.

One final place where Einstein’s research impacts Intermediate Physics for Medicine and Biology is in the study of diffusion (Chapter 4). Einstein did fundamental work on diffusion in his doctoral thesis, and derived a relationship between the diffusion constant and the viscosity that we give as Eq. 4.23.

Subtle is the Lord,
by Abraham Pais.

In summary, we rarely mention Einstein by name in our book, but his influence is present throughout, and most fundamentally when we discuss the idea of a photon. For readers interested in Einstein’s life and work, I recommend the brilliant biography Subtle is the Lord by Abraham Pais. I have heard good things about Isaacson’s more recent biography, Einstein: His Life and Universe, although I haven’t read it. You might also enjoy the American Institute of Physics website about Einstein prepared by the AIP Center for the History of Physics. Einstein published most of the ideas I have discussed in one miraculous year, 1905. John Rigden describes these publications and their impact in his book Einstein 1905: The Standard for Greatness (I have not read this book either, but I understand it is good). Finally, the equation E = mc² has received a lot of press recently, including a NOVA special and Bodanis’s book E=mc²: A Biography of the World’s Most Famous Equation.

Error Function

In the November 6th entry to this blog, I mentioned one special function introduced in the 4^th edition of Intermediate Physics for Medicine and Biology: the Bessel function. Another special function Russ Hobbie and I discuss briefly is the error function, which arises naturally when solving the one-dimensional cable equation (Chapters 6 and 7) or the diffusion equation (Chapter 4). The error function is the integral of the familiar Gaussian function, and has a sigmoidal shape, being minus one for large negative values of its argument and one for large positive values.

Handbook of Mathematical Functions
with Formulas, Graphs, and
Mathematical Tables,
by Abramowitz and Stegun.

To learn more about the error function, see the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables by Milton Abramowitz and Irene Stegun (1972). This classic math handbook is available online at http://www.math.ucla.edu/~cbm/aands//. Also, Wikipedia has a very thorough article about the error function, including beautiful plots of the error function in the complex plane.
I’m not sure how the error function got its name. Perhaps it has something to do with experimental errors often being Gaussianly distributed. If anyone knows, please let me know.


P.S. Speaking of errors: For any students or instructors preparing to use the 4^th edition of Intermediate Physics for Medicine and Biology next semester, I recommend you download the errata, which can be found at https://sites.google.com/view/hobbieroth. In it, Russ Hobbie and I list all known errors in our book. The number of errors has grown, and in particular some are present in homework problems. Generally I frown on writing in my books, but in this case do yourself a favor: download the errata and mark the corrections in your copy of the text. And as always, let us know if you find additional errors. The only thing worse than finding errors in a book you wrote is having errors in a book you wrote that you are not even aware of.

P.P.S. I have written in this blog about Steven Strogatz, a mathematician and author, and about Kleber’s law, which relates metabolic rate to body mass. Here is an article by Strogatz about Kleber’s law. It doesn’t get much better than that!

Hot Tubs and Heat Stroke

In Chapter 10 (about Feedback and Control) of the 4^th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss hot tubs and heat stroke.

The body perspires in order to prevent increases in body temperature. At the same time blood flows through vessels near the surface of the skin, giving the flushed appearance of an overheated person. The cooling comes from the evaporation of the perspiration from the skin. If the perspiration cannot evaporate or is wiped off, the feedback loop is broken ad the cooling does not occur. If a subject in a hot tub overheats, the same blood flow pattern and perspiration occur, but now heat flows into the body from the hot water in the tub. The feedback has become positive instead of negative, and heat stroke and possibly death occurs.

Were we overly alarming about hot tubes? Not according to an article by Nicholas Bakalar in the November 23rd issue of the New York Times, which indicates hot tub accidents are a growing problem.

A hot tub might not seem an especially dangerous place, but over a period of 18 years, 1990–2007, more than 80,000 people were injured in hot tubs or whirlpools seriously enough to wind up in an emergency room. Almost 74 percent of the injuries occurred at home… About half the injuries were caused by slipping or falling, but heat overexposure was the problem in 10 percent of the accidents, and near-drowning in about 2.5 percent. Almost 7 percent of the injuries were serious enough to require hospitalization… The Consumer Products Safety Commission reported more than 800 deaths associated with hot tubs since 1990, nearly 90 percent of them in children under age 3.

This means that about 8000 people suffered from heat stroke accidents in hot tubs over 18 years, or over one per day. Perhaps a better understanding of biological thermodynamics and feedback loops has more than merely academic value.

To learn more, see “Death in a Hot Tub: The Physics of Heat Stroke,” by Albert Bartlett and Thomas Braun (American Journal of Physics, Volume 51, Pages 127–132, 1983).