The Electric Potential of a Rectangular Sheet of Charge

My idea of a great physics problem is one that is complicated enough so that it is not trivial, yet simple enough that it can be solved analytically. An example can be found in Sec. 6.3 of the 4th edition of Intermediate Physics for Medicine and Biology.

If one considers a rectangular sheet of charge lying in the xy plane of width 2c and length 2b, as shown in Fig. 6.10, it is possible to calculate exactly the E field along the z axis…. The result is

This is plotted in Fig. 6.11 for c = 1 m, b = 100 m. Close to the sheet (z much less than 1) the field is constant, as it is for an infinite sheet of charge. Far away compared to 1 m but close compared to 100 m, the field is proportional to 1/r as with a line charge. Far away compared to 100 m, the field is proportional to 1/r², as from a point charge.

What I like most about this example is that you can take limits of the expression to illustrate the different cases. Russ Hobbie and I leave this as a task for the reader in Problem 8. It is not difficult. All you need is the value of the inverse tangent for a large argument (π/2), and its Taylor’s series, tan^-1(x) = x –x³/3 + ^…. Often expressions like these will show simple behavior in two limits, when some variable is either very large or very small. But this example illustrates intuitive behavior in three limits. How lovely. I wish I could take credit for this example, but it was present in earlier editions of IPMB, on which Russ was the sole author. Nicely done, Russ.

Usually the electric potential, a scalar, is easier to calculate than is the electric field, a vector. This led me to wonder what electric potential is produced by this same rectangle of charge. I imagine the expression for the potential everywhere is extremely complicated, but I would be satisfied with an expression for the potential along the z axis, like in Eq. 6.10 for the electric field. We should be able to find the potential in one of two ways. We could either integrate the electric field along z, or solve for the potential directly by integrating 1/r over the entire sheet. I tried both ways, with no luck. I ground to a halt trying to integrate inverse tangent with a complicated argument. When solving directly, I was able to integrate over y successfully but then got stuck trying to integrate an inverse hyperbolic sine function with an argument that is a complicated function of x. So, I’m left with Eq. 6.10, an elegant expression for the electric field involving an inverse tangent, but no analytical expression for the electric potential.

I was concerned that I might be missing something obvious, so I checked my favorite references: Griffiths’ Introduction to Electrodynamics and Jackson’s infamous Classical Electrodynamics. Neither of these authors solve the problem, even for a square sheet.

As a last resort, I turn to you, dear readers. Does anyone out there—I always assume there is someone out there reading this—know of an analytic expression for the electric potential along the z axis caused by a rectangular sheet of charge, centered at the origin and oriented in the xy plane? If you do, please share it with me. (Warning: I suspect such an expression does not exist.) If you send me one, the first thing I plan to do is to differentiate it with respect to z, and see if I get Eq. 6.10.

This will be fun.

Triplet Production

In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I describe how x rays interact with tissue by pair production.

A photon with energy above 1.02 MeV can produce a particle–antiparticle pair: a negative electron and a positive electron or positron… Since the rest energy (m_ec²) of an electron or positron is 0.51 MeV, pair production is energetically impossible for photons below 2m_ec² = 1.02 MeV.


One can show, using hν₀ = pc for the photon, that momentum [p] is not conserved by the positron and electron if [the conservation of energy] is satisfied. However, pair production always takes place in the Coulomb field of another particle (usually a nucleus) that recoils to conserve momentum. The nucleus has a large mass, so its kinetic energy p²/2m is small…

Then we discuss a related process: triplet production.

Pair production with excitation or ionization of the recoil atom can take place at energies that are only slightly higher than the threshold [2m_ec²]; however, the cross section does not become appreciable until the incident photon energy exceeds 4m_ec² = 2.04 MeV, the threshold for pair production in which a free electron (rather than a nucleus) recoils to conserve momentum. Because ionization and free-electron pair production are (γ, e⁻e⁻e⁺) processes, this is usually called triplet production.” 

Spacetime Physics,
by Taylor and Wheeler.

Where does the factor of four in “4m_ec²” come from? To answer that question, we must know more about special relativity than is presented in IPMB. We know already that the energy of a photon is hν and the momentum is hν/c, where ν is the frequency, h is Planck’s constant, and c is the speed of light. What we need in addition is that the energy of an electron with rest mass m_e is γm_ec², and its momentum is βγm_ec, where β is the ratio of the electron’s speed to the speed of light, β = v/c, and γ = 1/sqrt(1−β²). The factors of β and, especially, γ may look odd, but they are common in special relativity. To learn how they arise, read the marvelous book Spacetime Physics by Edwin Taylor and John Archibald Wheeler. Assume a photon with energy hν interacts with an electron at rest (β = 0, γ = 1). Furthermore (and this is not obvious), assume that after the collision the original electron and the new electron-positron pair all move in the direction of the original photon, and travel at the same speed. The conservation of energy requires

hν + m_ec² = 3γm_ec²,

and conservation of momentum implies

hν/c = 3βγm_ec .

The rest is algebra. Eliminate hν and you find that 3γβ + 1 = 3γ. Then use γ = 1/sqrt(1−β²) to find that β = 4/5 and γ = 5/3. (I love how the Pythagorean triple 3, 4, 5 arises in triplet production). Then conservation of energy or conservation of momentum implies hν = 4m_ec². Now you know the origin of that mysterious factor of four.

The paper “Pair and Triplet Production Revisited for the Radiologist” by Ralph Raymond (American Journal of Roentgenology, Volume 114, Pages 639–644, 1972) provides additional details. To learn about special relativity, I recommend either Spacetime Physics (their Sec. 8.5 analyzes triplet production) or Space and Time in Special Relativity by one of the best writers of physics, N. David Mermin. I hear Mermin’s recent book It’s About Time is also good, but I haven’t read it yet.

Excerpt from the Fifth Edition

Next month, Russ Hobbie and I will receive the page proofs for the 5th edition of Intermediate Physics for Medicine and Biology. I welcome their arrival because I enjoy working on the book with Russ, but also I dread their coming because they will take over my life for weeks. The page proofs are our last chance to rid the book of errors; we will do our best.

I thought that you, dear readers, might like a preview of the 5th edition. We did not add any new chapters, but we did include several new sections such as this one on color vision.

14.15 Color Vision

The eye can detect color because there are three types of cones in the retina, each of which responds to a different wavelength of light (trichromate vision): red, green, and blue, the primary colors. However, the response curve for each type of cone is broad, and there is overlap between them (particularly the green and red cones). The eye responds to yellow light by activating both the red and green cones. Exactly the same response occurs if the eye sees a mixture of red and green light. Thus, we can say that red plus green equals yellow. Similarly, the color cyan corresponds to activation of both the green and blue cones, caused either by a monochromatic beam of cyan light or a mixture of green and blue light. The eye perceives the color magenta when the red and blue cones are activated but the green is not. Interestingly, no single wavelength of light can do this, so there is no such thing as a monochromatic beam of magenta light; it can only be produced my mixing red and blue. Mixing all three colors, red and green and blue, gives white light. Color printers are based on the colors yellow, cyan and magenta, because when we view the printed page, we are looking at the reflection after some light has been absorbed by the ink. For instance, if white light is incident on a page containing ink that absorbs blue light, the reflected light will contain red and green and therefore appear yellow. Human vision is trichromate, but other animals (such as the dog) have only two types of cones (dichromate vision), and still others have more than three types.

Some people suffer from colorblindness. The most common case is when the cones responding to green light are defective, so that red, yellow and green light all activate only the red receptor. Such persons are said to be red-green color blind: they cannot distinguish red, yellow and green, but they can distinguish red from blue.

As with pitch perception, the sensation of color involves both physics and physiology. For instance, one can stare at a blue screen until the cones responding to blue become fatigued, and then immediately stare at a white screen and see a yellow afterimage. Many other optical illusions with color are possible.

You may recognize parts of this excerpt as coming from a previous entry to this blog. In fact, we used the blog as a source of material for the new edition.

A Christmas Carol,
by Charles Dickens.

I will leave you with another excerpt, this one from the conclusion of A Christmas Carol. Every Christmas I read Dickens’s classic story about how three spirits transformed the miser Ebenezer Scrooge. It is my favorite book; I like it better than even IPMB!

I wish you all the happiest of holidays.

Scrooge was better than his word. He did it all, and infinitely more; and to Tiny Tim, who did not die, he was a second father. He became as good a friend, as good a master, and as good a man, as the good old city knew, or any other good old city, town, or borough, in the good old world. Some people laughed to see the alteration in him, but he let them laugh, and little heeded them; for he was wise enough to know that nothing ever happened on this globe, for good, at which some people did not have their fill of laughter in the outset; and knowing that such as these would be blind anyway, he thought it quite as well that they should wrinkle up their eyes in grins, as have the malady in less attractive forms. His own heart laughed: and that was quite enough for him.

A Theoretical Physicist’s Journey into Biology

Many physicists have shifted their research to biology, but rarely do we learn how they make this transition or, more importantly, why. But the recent article “A Theoretical Physicist’s Journey into Biology: From Quarks and Strings to Cells and Whales” by Geoffrey West (Physical Biology, Volume 11, Article number 053013, 2014) lets us see what is involved when changing fields and the motivation for doing it. Readers of the 4th edition of Intermediate Physics for Medicine and Biology will remember West from Chapter 2, where Russ Hobbie and I discuss his work on Kleber’s law. West writes

Biology will almost certainly be the predominant science of the twenty-first century but, for it to become successfully so, it will need to embrace some of the quantitative, analytic, predictive culture that has made physics so successful. This includes the search for underlying principles, systemic thinking at all scales, the development of coarse-grained models, and closer ongoing collaboration between theorists and experimentalists. This article presents a personal, slightly provocative, perspective of a theoretical physicist working in close collaboration with biologists at the interface between the physical and biological sciences.

On Growth and Form,
by D'Arcy Thompson.

West describes his own path to biology, which included reading some classic texts such as D’Arcy Thompson’s On Growth and Form. He learned biology during intense free-for-all discussions with his collaborator James Brown and Brown’s student Brian Enquist.

The collaboration, begun in 1995, has been enormously productive, extraordinarily exciting and tremendous fun. But, like all excellent and fulfilling relationships, it has also been a huge challenge, sometimes frustrating and sometimes maddening. Jim, Brian and I met every Friday beginning around 9:00 am and finishing around 3:00 pm with only short breaks for necessities. This was a huge commitment since we both ran large groups elsewhere. Once the ice was broken and some of the cultural barriers crossed, we created a refreshingly open atmosphere where all questions and comments, no matter how “elementary,” speculative or “stupid,” were encouraged, welcomed and treated with respect. There were lots of arguments, speculations and explanations, struggles with big questions and small details, lots of blind alleys and an occasional aha moment, all against a backdrop of a board covered with equations and hand-drawn graphs and illustrations. Jim and Brian generously and patiently acted as my biology tutors, exposing me to the conceptual world of natural selection, evolution and adaptation, fitness, physiology and anatomy, all of which were embarrassingly foreign to me. Like many physicists, however, I was horrified to learn that there were serious scientists who put Darwin on a pedestal above Newton and Einstein.

West’s story reminds me of the collaboration between physicist Joe Redish and biologist Todd Cook that I discussed previously in this blog, or Jane Kondev’s transition from basic physics to biological physics when an assistant professor at Brandeis (an awkward time in your career to make such a dramatic change).

I made my own shift from physics to biology much earlier in my career—in graduate school. Changing fields is not such a big deal when you are young, but I think all of us who make this transition have to cross that cultural barrier and make that huge commitment to learning a new field. I remember spending much of my first summer at Vanderbilt University reading papers by Hodgkin, Huxley, Rushton, and others, slowly learning how nerves work. Certainly my years at the National Institutes of Health provided a liberal education in biology.

I will give West the last word. He concludes by writing

Many of us recognize that there is a cultural divide between biology and physics, sometimes even extending to what constitutes a scientific explanation as encapsulated, for example, in the hegemony of statistical regression analyses in biology versus quantitative mechanistic explanations characteristic of physics. Nevertheless, we are witnessing an enormously exciting period as the two fields become more closely integrated, leading to new inter-disciplinary sub-fields such as biological physics and systems biology. The time seems right for revisiting D’Arcy Thompson’s challenge: “How far even then mathematics will suffice to describe, and physics to explain, the fabric of the body, no man can foresee. It may be that all the laws of energy, and all the properties of matter, all… chemistry… are as powerless to explain the body as they are impotent to comprehend the soul. For my part, I think it is not so.” Many would agree with the spirit of this remark, though new tools and concepts including closer collaboration may well be needed to accomplish his lofty goal.

In Vitro Evaluation of a 4-leaf Coil Design for Magnetic Stimulation of Peripheral Nerve

In the comments to last week’s blog entry, Frankie asks if there is a way to “safely, reversibly block nerve conduction (first in the lab, then in the clinic) with an exogenously applied E and M signal?” This is a fascinating question, and I may have an answer.

When working at the National Institutes of Health in the early 1990’s, Peter Basser and I analyzed magnetic stimulation of a peripheral nerve. The mechanism of excitation is similar to the one Frank Rattay developed for stimulating a nerve axon with an extracellular electrode. You can find Rattay’s method described in Problems 38–41 of Chapter 7 in the 4th edition of Intermediate Physics for Medicine and Biology. The bottom line is that excitation occurs where the spatial derivative of the electric field is largest. I have already recounted how Peter and I derived and tested our model, so I won’t repeat it today.

If you accept the hypothesis that excitation occurs where the electric field derivative is large, then the traditional coil design for magnetic stimulation—a figure-of-eight coil—has a problem: the axon is not excited directly under the center of the coil (where the electric field is largest), but a few centimeters from the center (where the electric field gradient is largest). What a nuisance. Doctors want a simple design like a crosshair: excitation should occur under the center. X marks the spot.

As I pondered this problem, I realized that we could build a coil just like the doctor ordered. It wouldn’t have a figure-of-eight design. Rather, it would be two figure-of-eights side by side. I called this the four leaf coil. With this design, excitation occurs directly under the center.

A four-leaf-coil used for
magnetic stimulation of nerves.

John Cadwell of Cadwell Labs built a prototype of this coil; an x ray of it is shown above. We wanted to test the coil in a well-controlled animal experiment, so we sent it to Paul Maccabee at the State University of New York Health Science Center in Brooklyn. Paul did the experiments, and we published the results in the journal Electroencephalography and clinical Neurophysiology (Volume 93, Pages 68–74, 1994). The paper begins

Magnetic stimulation is used extensively for non-invasive activation of human brain, but is not used as widely for exciting limb peripheral nerves because of both the uncertainty about the site of stimulation and the difficulty in obtaining maximal responses. Recently, however, mathematical models have provided insight into one mechanism of peripheral nerve stimulation: peak depolarization occurs where the negative derivative of the component of the induced electric field parallel to nerve fibers is largest (Durand et al. 1989; Roth and Basser 1990). Both in vitro (Maccabee et al. 1993) and in vivo (Nilsson et al. 1992) experiments support this hypothesis for uniform, straight nerves. Based on these results, a 4-leaf magnetic coil (MC) design has been suggested that would provide a well defined site of stimulation directly under the center of the coil (Roth et al. 1990). In this note, we perform in vitro studies which test the performance of this new coil design during magnetic stimulation of a mammalian peripheral nerve.

Maccabee’s experiments showed that the coil worked as advertised. In the discussion of the paper we concluded that “the 4-leaf coil design provides a well defined stimulus site directly below the center of the coil.”

This is a nice story, but it’s all about exciting an action potential. What does it have to do with Frankie’s goal of blocking an action potential? Well, if you flip the polarity of the coil current, instead of depolarizing the nerve under the coil center, you hyperpolarize it. A strong enough hyperpolarization should block propagation. We wrote

In a final type of experiment, performed on 3 nerves, the action potential was elicited electrically, and a hyperpolarizing magnetic stimulus was applied between the stimulus and recording sites at various times. The goal was to determine if a precisely timed stimulus could affect action potential propagation. Using induced hyperpolarizing current at the coil center, with a strength that was approximately 3 times greater than that needed to excite by depolarization at that location, we never observed a block of the action potential. Moreover, no significant effect on the latency of the action potential propagating to the recording site was observed… Our magnetic stimulator was able to deliver stimuli with strengths up to only 2 or 3 times the threshold strength, and therefore the magnetic stimuli were probably too weak to block propagation. It is possible that such phenomena might be observed using a more powerful stimulator.

Frankie, I have good news and bad news. The good news is that you should be able to reversibly block nerve conduction with magnetic stimulation using a four-leaf coil. The bad news is that it didn’t work with Paul’s stimulator; perhaps a stronger stimulator would do the trick. Give it a try.

The Bubble Experiment

When I was a graduate student, my mentor John Wikswo assigned to me the job of measuring the magnetic field of a nerve axon. This experiment required me to dissect the ventral nerve cord out of a crayfish, thread it through a wire-wound ferrite-core toroid, immerse the nerve and toroid in saline, stimulate one end of the nerve, and record the magnetic field produced by the propagating action currents. One day as I was lowering the instrument into the saline bath, a bubble got stuck in the gap between the nerve and the inner surface of the toroid. “Drat” I thought as I searched for a needle to remove it. But before I could poke it out I wondered “how will the bubble affect the magnetic signal?”

A wire-wound, ferrite-core toroid,
used to measure the magnetic field of a nerve.

To answer this question, we need to review some magnetism. Ampere’s law states that the line integral of the magnetic field around a closed path is proportional to the net current passing through a surface bounded by that path. For my experiment, that meant the magnetic signal depended on the net current passing through the toroid. The net current is the sum of the current inside the nerve axon and that fraction of the current in the saline bath that threads the toroid—the return current. In general, these currents flow in opposite directions and partially cancel. One of the difficulties I faced when interpreting my data was determining how much of the signal was from intracellular current and how much was from return current.

I struggled with this question for months. I calculated the return current with a mathematical model involving Fourier transforms and Bessel functions, but the calculation was based on many assumptions and required values for several parameters. Could I trust it? I wanted a simpler way to find the return current.

Then along came the bubble, plugging the toroid like Pooh stuck in Rabbit’s front door. The bubble blocked the return current, so the magnetic signal arose from only the intracellular current. I recorded the magnetic signal with the bubble, and then—as gently as possible—I removed the bubble and recorded the signal again. This was not easy, because surface tension makes a small bubble in water sticky, so it stuck to the toroid as if glued in place. But I eventually got rid of it without stabbing the nerve and ending the experiment.

To my delight, the magnetic field with the bubble was much larger than when it was absent. The problem of estimating the return current was solved; it’s the difference of the signal with and without the bubble. I reported this result in one of my first publications (Roth, B. J., J. K. Woosley and J. P. Wikswo, Jr., 1985, “An Experimental and Theoretical Analysis of the Magnetic Field of a Single Axon,” In: Biomagnetism: Applications and Theory, Weinberg, Stroink and Katila, Eds., Pergamon Press, New York, pp. 78–82.).

When taking data from a crayfish nerve, the toroid and axon were lifted out of the bath for a short time. […] When again placed in the bath an air bubble was trapped in the center of the toroid, filling the space between the axon and the toroid inner surface. […] Taking advantage of this fortunate occurrence, data were taken with and without the bubble present. […] The magnetic field with the bubble present […] is narrower and larger than the field with the toroid filled with saline.

The magnetic field of a nerve axon
with and without a bubble trapped
between the nerve and toroid.

On the day of the bubble experiment I was lucky. I didn’t plan the experiment. I wasn’t wise enough or thoughtful enough to realize in advance that a bubble was the ideal way to eliminate the return current. But when I looked through the dissecting microscope and saw the bubble stuck there, I was bright enough to appreciate my opportunity. “Chance favors the prepared mind.”

I have a habit of turning all my stories into homework problems. You will find the bubble story in the 4th edition of Intermediate Physics for Medicine and Biology, Problem 39 of Chapter 8. Focus on part (b).

Problem 39 A coil on a magnetic toroid as in Problem 38 is being used to measure the magnetic field of a nerve axon.

(a) If the axon is suspended in air, with only a thin layer of extracellular fluid clinging to its surface, use Ampere’s law to determine the magnetic field, B, recorded by the toroid.

(b) If the axon is immersed in a large conductor such as a saline bath, B is proportional to the sum of the intracellular current plus that fraction of the extracellular current that passes through the toroid (see Problem 13). Suppose that during an experiment an air bubble is trapped between the axon and the inner radius of the toroid? How is the magnetic signal affected by the bubble? See Roth et al. (1985).

The Bowling Ball and the Feather

Dropping a feather and a ball in a vacuum to show that they fall at the same rate is a classic physics demonstration. We have a version of this demo at Oakland University, but it is not very effective. A small ball and a feather are in a tube about 1 meter long and a few centimeters in diameter. We have vacuum pump to remove the air, but it is difficult to see the objects from the back of the room, and often they bump into the wall of the tube, slowing them down. I have never found it useful. Yet, the physical principle being demonstrated is fundamental. The gravitational mass in Newton’s universal law of gravity and the inertial mass in Newton’s second law of motion cancel out, so that all objects fall downward with acceleration g = 9.8 m/s².

This result is unexpected because in everyday life we experience air friction. When you include air friction, objects do not all fall at the same rate. Russ Hobbie and I illustrate this point in Problem 28 of Chapter 2 in the 4th edition of Intermediate Physics for Medicine and Biology.

Problem 28 When an animal of mass m falls in air, two forces act on it: gravity, mg, and a force due to air friction. Assume that the frictional force is proportional to the speed v.

(a) Write a differential equation for v based on Newton’s second law, F = m(dv/dt).

(b) Solve this differential equation. (Hint: Compare your equation with Eq. 2.24.)

(c) Assume that the animal is spherical, with radius a and density ρ. Also, assume that the frictional force is proportional to the surface area of the animal. Determine the terminal speed (speed of descent in steady state) as a function of a.

(d) Use your result in part (c) to interpret the following quote by J. B. S. Haldane [1985]: “You can drop a mouse down a thousand-yard mine shaft; and arriving at the bottom, it gets a slight shock and walks away. A rat is killed, a man is broken, a horse splashes.”

If we ignore air fraction, v = gt; the acceleration is g and does not depend on mass. With air friction, objects reach a terminal velocity that depends on their mass. We are all so used to seeing a feather float downward with its motion dominated by air friction that it is difficult to believe it could ever fall as fast as a ball. To persuade students that this behavior does indeed happen, to convince them that in a vacuum a feather drops like a rock, we need a powerful demonstration. The result is so significant, and so nonintuitive, that the demo must be dramatic and memorable.

Now we have it. Watch this amazing video with British Physics Professor Brian Cox. He found the biggest vacuum chamber in the world—a large room used by NASA to test space vehicles—and inside it he dropped a bowling ball and a feather simultaneously from the same height. When the room was filled with air, the feather slowly fluttered to the ground. When the room was evacuated, the feather stayed right beside the bowling ball all the way down. The visual effect is stunning. Cox has a fine sense of drama, building the tension until the final sensational experiment. The video is less than five minutes long. You’ve gotta see it.

The MCAT and IPMB

The Medical College Admission Test, famously known as the MCAT, is an exam taken by students applying to medical school. The Association of American Medical Colleges will introduce a new version of the MCAT next year, focusing on competencies rather than on prerequisite classes. How well does the 4th edition of Intermediate Physics for Medicine and Biology prepare premed students for the MCAT?

The new MCAT will be divided into four sections, and the one most closely related to IPMB deals with the chemical and physical foundations of biological systems. Within that section are two foundational concepts, of which one is about how “complex living organisms transport materials, sense their environment, process signals, and respond to changes that can be understood in terms of physical principles.” This concept is further subdivided into five categories. Below, I review the topics included in these categories and indicate what chapter in IPMB addresses each.

MCAT: Translational motion, forces, work, energy, and equilibrium in living systems

IPMB: Chapter 1 discusses mechanics, including forces and torques, with applications to biomechanics. Work and energy are introduced in Chapter 1, and analyzed in more detail in Chapter 3 on statistical mechanics and thermodynamics (parts of thermodynamics are included under another foundational concept dealing mostly with chemistry). Periodic motion is covered in Chapter 11, which discusses the amplitude, frequency and phase of an oscillator. Waves are analyzed in Chapter 13 about sound and ultrasound.

MCAT: Importance of fluids for the circulation of blood, gas movement, and gas exchange

IPMB: Chapter 1 analyzes fluids, including buoyancy, hydrostatic pressure, viscosity, Poiseuille flow, turbulence, and the circulatory system. Much of this material is not covered in a typical introductory physics class. Chapter 3 introduces absolute temperature, the ideal gas law, heat capacity, and Boltzmann’s constant.

MCAT: Electrochemistry and electrical circuits and their elements

IPMB: Chapters 6 and 7 cover electrostatics, including charge, the electric field, current, voltage, Ohm’s law, resistors, capacitors, and nerve conduction. Chapter 8 discusses the magnetic field and magnetic forces.

MCAT: How light and sound interact with matter

IPMB: Sound is analyzed in Chapter 13, including the speed of sound, the decibel, attenuation, reflection, the Doppler effect, ultrasound, and the ear. Chapter 14 covers light, photon energy, color, interference, and the eye. This chapter also describes absorption of light in the infrared, visible, and ultraviolet. Chapter 18 analyzes nuclear magnetic resonance.

MCAT: Atoms, nuclear decay, electronic structure, and atomic chemical behavior

IPMB: Chapter 17 is about nuclear physics and nuclear medicine, covering isotopes, radioactive decay, and half life. Atoms and atomic energy levels are explained in Chapter 14.

MCAT: General mathematical concepts and techniques

IPMB: Chapter 1 and many other chapters require students to estimate numerically. Chapter 2 covers linear, semilog, and log-log plots, and exponential growth. Metric units and dimensional analysis are used everywhere. Probability concepts are discussed in Chapter 3 and other chapters. Basic math skills such as exponentials, logarithms, scientific notation, trigonometry, and vectors are reinforced throughout the book and in the homework problems, and are reviewed in the Appendices.

The MCAT section about biological and biochemical foundations of living systems includes diffusion and osmosis (discussed in Chapters 4 and 5 of IPMB), membrane ion channels (covered in Chapter 9), and feedback regulation (analyzed in Chapter 10).

Overall, Intermediate Physics for Medicine and Biology covers many of the topics tested on the MCAT. A biological or medical physics class based on IPMB would prepare a student for the exam, and would reinforce problem solving skills and teach the physical principles underlying medicine, resulting in better physicians.

I’m a realist, however. I know premed students take lots of classes, and they don’t want to take more physics beyond a two-semester introduction, especially if the class might lower their grade point average. I have tried to recruit premed students into my Biological Physics (PHY 325) and Medical Physics (PHY 326) classes here at Oakland University, with little success. Perhaps if they realized how closely the topics and skills required for the MCAT correspond to those covered by Intermediate Physics for Medicine and Biology they would reconsider.

To learn more about how to prepare for the physics competencies on the MCAT, see Robert Hilborn’s article “Physics and the Revised Medical College Admission Test,” published in the American Journal of Physics last summer (Volume 82, Pages 428–433, 2014).

Faraday, Maxwell, and the Electromagnetic Field

Faraday, Maxwell, and the
Electromagnetic Field,
by Nancy Forbes and Basil Mahon.

Michael Faraday and James Clerk Maxwell are two of my scientific heroes. So, when I saw the book Faraday, Maxwell, and the Electromagnetic Field displayed in the new book section of the Rochester Hills Public Library, I had to check it out. In their introduction, Nancy Forbes and Basil Mahon write

It is almost impossible to overstate the scale of Faraday and Maxwell’s achievement in bringing the concept of the electromagnetic field into human thought. It united electricity, magnetism, and light into a single, compact theory; changed our way of life by bringing us radio, television, radar, satellite navigation, and mobile phones; inspired Einstein’s special theory of relativity; and introduced the idea of field equations, which became the standard form used by today’s physicists to model what goes on in the vastness of space and inside atoms.

I have read previous biographies of both Faraday and Maxwell, so their story was familiar to me. But one anecdote about Faraday I had never heard before.

The Royal Institution’s Friday Evening Discourses had by now become an institution in their own right. The lecture on April 3, 1846, turned out to be a historic occasion, although none of the audience recognized it as such and the whole thing happened by chance in a rather bizarre fashion. Charles Wheatstone was to have been the latest in a long line of distinguished speakers, but he panicked and ran away just as he was due to make his entrance[!]. Although amply confident in his professional dealings as a scientist, inventor, and businessman, Wheatstone was notoriously shy of speaking in public, and Faraday had taken a gamble when engaging him to talk about his latest invention, the electromagnetic chronoscope—a device for measuring small time intervals, like the duration of a spark. The gamble had failed, and Faraday was left with the choice of sending disappointed customers home or giving the talk himself. He chose to talk, but he ran out of things to say on the advertised topic well before the allotted hour was up.

Caught off-guard, he did what he had never done before and gave the audience a glimpse into his private meditations on matter, lines of force, and light. In doing so, he draw an extraordinary prescient outline of the electromagnetic theory of light, as it would be developed over the next sixty years….

Writing about Maxwell’s electromagnetic theory, Forbes and Mahon say

The theory’s construction had been an immense creative effort, sustained over a decade and inspired, from first to last, by the work of Michael Faraday. Thanks to Faraday’s meticulous recording of his findings and thoughts in his Experimental Researches in Electricity, Maxwell had been able to see the world as Faraday did, and, by bringing together Faraday’s vision with the power of Newtonian mathematics, to give us a new concept of physical reality, using the power of mathematics. But mathematics would not have been enough without Maxwell’s own near-miraculous intuition; witness the displacement current, which gave the theory its wonderful completeness. The theory belongs to both Maxwell and Faraday.

Russ Hobbie and I discuss electricity and magnetism in the 4th edition of Intermediate Physics for Medicine and Biology. Chapters 6 and 7 show how electrostatics can be used to describe how nerves and muscles behave. Chapter 8 discusses magnetism and electromagnetism. Chapter 9 examines in more detail how electromagnetic fields interact with the body, and Chapter 18 describes how magnetism leads to magnetic resonance imaging. So, it’s safe to say that IPMB has Maxwell and Faraday’s influence throughout.

If you want to learn more about Maxwell’s work, I suggest Maxwell on the Electromagnetic Field: A Guided Study by Thomas K. Simpson. He reproduces Maxwell’s three landmark papers, and provides the necessary context and background to understand them. Forbes and Mahon talk briefly at the end of their book about the scientists who came after Maxwell and firmly established his theory. For more on this topic, read The Maxwellians, one of the best histories of science I know. I enjoyed Faraday, Maxwell, and the Electromagnetic Field. It provides a great introduction to a fascinating story in the history of science.

Low Reynolds Number Flows

In Chapter 1 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the Reynolds number.

The importance of turbulence (nonlaminar) flow is determined by a dimensionless number characteristic of the system called the Reynolds number N_R. It is defined by N_R = LVρ/η where L is a length characteristic of the problem, V a velocity characteristic of the problem, ρ the density, and η the viscosity of the fluid. When N_R is greater than a few thousand, turbulence usually occurs….

When N_R is large, inertial effects are important. External forces accelerate the fluid. This happens when the mass is large and the viscosity is small. As the viscosity increases (for fixed L, V , and ρ) the Reynolds number decreases. When the Reynolds number is small, viscous effects are important. The fluid is not accelerated, and external forces that cause the flow are balanced by viscous forces. Since viscosity is a form of internal friction in the fluid, work done on the system by the external forces is transformed into thermal energy. The low-Reynolds-number regime is so different from our everyday experience that the effects often seem counterintuitive. They are nicely described by Purcell (1977).

“Life at Low Reynolds Number,”
by Edward Purcell.

Edward Purcell’s 1977 paper in the American Journal of Physics provides much insight into low Reynolds number flow, and is a classic. But to learn from this paper you have to read it. Nowadays, students often want to learn from videos rather than reading text (don’t get me started...). Fortunately, a good video exists to explain low-Reynolds-number flow, and it has been around for many years. Click here to watch G. I. Taylor illustrate low Reynolds flow. Sir Geoffrey Ingram Taylor (1886–1975) was an English physicist and an expert in fluid dynamics. He contributed to the Manhattan Project by analyzing the hydrodynamics of implosion needed to develop a plutonium bomb. Among his many contributions is the description of Taylor-Couette flow between two rotating cylinders.

The video shows a beautiful example of reversibility of low Reynolds number flow. A blob of dye is placed into the fluid between the cylinders, one of the cylinders is rotated, and the dye spreads throughout the fluid. They rotation is then reversed, and the dye eventually returns to its original localized blob. This demonstration always reminds me of the formation of a spin echo during magnetic resonance imaging (see Chapter 18 of IPMB), where all spins begin in phase after a 90 degree radio-frequency magnetic field pulse. Then, because of slight heterogeneities in the static magnetic field, the spins dephase as they all rotate at slightly different Larmor frequencies. If you reverse their positions using a 180 degree RF pulse, the spins eventually return to their original configuration, all in phase (the echo). When you think about it, the formation of spin echoes during MRI is nearly as “magical” as the reformation of the dye blob in Taylor’s cylinder.

The video also analyzes how small machines can “swim” at low Reynolds numbers. He even has built small devices, one a machine that zooms through water but just sits there in a viscous fluid, and another that has a helix for a tail and that swims—slowly but steadily—through the viscous fluid. This example reminds me of both Purcell’s article and the research of Howard Berg, who studies how E coli bacteria swim.

To learn more about Taylor’s life and work, watch Katepalli Sreenivasan’s lecture, or read The Life and Legacy of G. I. Taylor by George Batchelor.