Since a changing magnetic field generates an induced electric
field, it is possible to stimulate nerve or muscle cells
without using electrodes. The advantage is that for a given
induced current deep within the brain, the currents in the
scalp that are induced by the magnetic field are far less than
the currents that would be required for electrical stimulation.
Therefore transcranial magnetic stimulation (TMS) is
relatively painless...
One of the earliest investigations was reported by Barker, Jalinous and Freeston (1985). They used a solenoid in which
the magnetic field changed by 2 T in 110 μs to apply a stimulus
to different points on a subject’s arm and skull. The
stimulus made a subject’s finger twitch after the delay required
for the nerve impulse to travel to the muscle.
The story of how Tony Barker invented transcranial magnetic stimulation is fascinating. You can hear about it in the video below, where John Rothwell—another early magnetic stimulation researcher—reminisces with Barker about his invention. The most interesting part of the video is when Barker describes a crucial trip he made from Sheffield (he worked at the Royal Hallamshire Hospital in Sheffield, England) to London (The National Hospital, Queen’s Square), so he could demonstrate his device to leading neurophysiologist Pat Merton. Rothwell, also at Queen’s Square, had his brain stimulated that day, and the next day he wrote Barker asking to get a stimulator of his own. Barker’s 1985 paper in The Lancet (cited in IPMB) was the first publication about magnetic stimulation of the brain. As Barker says, “like all the best papers it was one page long.”
The 15-minute video is well worth your time. I’ll stop writing so you can listen. Enjoy!
Anthony Barker reminiscing with John Rothwell about the invention of transcranial magnetic stimulation.
In Table 13.1 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I list the approximate intensity levels of various sounds, in decibels. The minimum perceptible sound is 0 dB, a typical office has a sound level of 50 dB, a jack hammer is 100 dB, and the loudest sound listed is a rocket launch pad at 170 dB.
Can there be even louder sounds? Yes, there can! This new homework problem lets you calculate the loudest possible sound.
Section 13.4
Problem 17 ½. Let us calculate the loudest possible sound in air.
(a) Use Eq. 13.29 to calculate the intensity of a sound in Wm−2, using 428 Pa s m−1 for the acoustic impedance of air and one atmosphere (1.01 × 105 Pa) for the pressure. This pressure is the largest that can exist for a sinusoidally varying sound wave, as an even louder sound would create a minimum pressure below zero (less than a vacuum).
(b) Use the result from part (a) to calculate the intensity in decibels using Eq. 13.34.
For those of you who don’t have a copy of IPMB at your side, here are the two equations you need
I = ½ p2/Z(13.29)
Intensity level = 10 log10(I/I0)
(13.34)
where I is the intensity, Z is the acoustic impedance, p is the pressure, I0 is the minimum perceptible intensity (10−12 W m−2), and log10 is the common logarithm.
I’ll let you do the calculation, but you should find that the loudest sound is about 191 dB. Is this really an upper limit? No, you could have a peak pressure larger than one atmosphere, but in that case you wouldn’t be dealing with a traditional sound wave (with pressure ranging symmetrically above and below the ambient pressure) but more of a nonlinear acoustic shock wave.
Krakatoa, by Simon Winchester.
Has there ever been a sound that loud? Or, more interestingly, what is the loudest sound ever heard on earth? That’s hard to say for sure, but one possibility is the 1883 eruption of the Krakatoa volcano. We know this sound was loud, because people heard it so far from where the eruption occurred.
In August 1883 the chief of police on Rodriguez was a man named James Wallis, and in his official report… for the month he noted:
On Sunday the 26th the weather was stormy, with heavy rain and squalls; the wind was from SE, blowing with a force of 7 to 10, Beaufort scale. Several times during the night (26th–27th) reports were heard coming from the eastward, like the distant roar of heavy guns. These reports continued at intervals of between three and four hours, until 3 pm on the 27th, and the last two were heard in the directions of Oyster Bay and Port Mathurie [sic].
This was not the roar of heavy guns, however. It was the sound of Krakatoa—busily destroying itself fully 2,968 miles away to the east. By hearing it that night and day, and by noting it down as any good public servant should, Chief Wallis was unknowingly making for himself two quite separate entries in the record books of the future. For Rodriguez Island was the place furthest from Krakatoa where its eruptions could be clearly heard. And the 2,968-mile span that separates Krakatoa and Rodriguez remains to this day the most prodigious distance recorded between the place where unamplified and electrically unenhanced natural sound was heard and the place where that same sound originated.
Winchester concludes
The sound that was generated by the explosion of Krakatoa was enormous, almost certainly the greatest sound ever experienced by man on the face of the earth. No manmade explosion, certainly, can begin to rival the sound of Krakatoa—not even those made at the height of the Cold War’s atomic testing years.
No one knows how many decibels Krakatoa’s eruption caused on the island itself. The sound was almost certainly in the nonlinear regime, and probably had an intensity of over 200 dB.
I was particularly fascinated by Rhodes’s tale of Wilson and James Watson as competing assistant professors at Harvard in the late 1950s. Watson advocated for molecular biology, while Wilson favored evolutionary biology. It was a battle between the unity and diversity of life. Wilson, with a job offer from Stanford in hand, was offered tenure if he would remain at Harvard. Watson—already famous for discovering the structure of DNA with Francis Crick—was livid that Wilson was to be tenured before he was. In the end, Harvard gave them both tenure (a wise decision). Decades later Wilson and Watson become friends. Listen to them discuss their rivalry in the video at the end of this post.
Readers of Intermediate Physics for Medicine and Biology will be interested in Wilson’s online high-school biology textbook Life on Earth. Physicists, mathematicians, and engineers who want to apply their field to biology or medicine always face the obstacle of learning biology. Sometimes they don’t need a deep knowledge of biology, but merely must know enough to collaborate with a biologist. Life on Earth is an excellent introduction to the field. It is free, available online, is written by a giant in the field of biology, and contains beautiful photographs and engaging videos. The only problem: it was written to be used on a Mac. I am a Mac guy, so this is not a problem for me. I don’t know if it works on a PC. Life on Earth should provide you with enough biology to understand IPMB.
Problem 37. The consumption of a finite resource is
often modeled using the logistic equation. Let y(t) be the
cumulative amount of a resource consumed and y∞ be
the total amount that was initially available at t = −∞.
Model the rate of consumption [I wish Russ and I had written “amount consumed” instead of “rate of consumption”] using Eq. 2.29 over the range
−∞ < t < ∞.
(a) Set y0 = y∞/2, so that the zero of the time axis
corresponds to when half the resource has been used.
Show that this simplifies Eq. 2.29.
(b) Differentiatey(t) to find an expression for the rate
of consumption. Sketch plots of dy/dt versus t on linear
and semilog graph paper. When does the peak rate of
consumption occur?
The answer to this exercise can be found in the IPMB solution manual. (The solution manual is available free of charge to instructors. If you need a copy, email me at roth@oakland.edu.) All exercises in the solution manual have a brief preamble, explaining the goal of the exercise and why it’s important.
2.37¶ This is not a biological example, except in the sense that if we ignore this example we humans may all end up dead. Students use a variation of the logistic equation to analyze the consumption of a finite resource (e.g., oil).
I won’t solve the entire problem in this blog post, but I will show the semilog plot from the solution manual.
A semilog plot of amount consumed (solid) and the rate of consumption (dashed) for a finite resource modeled using the logistic equation. This plot is part of the solution to Problem 37b.
The rate of consumption of the resource (dy/dt) first rises exponentially, reaches a peak, and then falls exponentially. (Remember, a straight line on a semilog plot corresponds to exponential growth or decay.) For the mathematically inclined, the dy/dt curve corresponds to a hyperbolic secant squared.
Why do I bring up this topic? Recently I read Energy: A Human History, by Richard Rhodes, a sweeping account of energy transitions that changed our world. Rhodes includes a figure that looks a little bit like this:
My rendition of a figure from the final chapter of Energy: A Human History showing the historical evolution of the world energy mix.
What a wonderful plot! It both summarizes Rhodes’s book and illustrates the power and ubiquity of Hubbert’s peak. That semilog plot from Homework Problem 37 appears over and over as one finite resource replaces another.
I should add a few qualifiers.
Historical data is noisy and the curves pictured above merely approximate a complicated behavior.
The plot begins at about the time of the industrial revolution. The population of humans was probably too small, and our technology too primitive, to apply this model before that time.
All future data (say, after 2016, the year Energy was published) is extrapolation or prediction.
Let’s hope that the Renewables curve corresponds to an infinite resource, not a finite one, so it will never reach a peak and then fall. Is that wishful thinking? I don’t know, but the figure encourages us to ask such questions.
Nuclear energy shot up much faster than would be expected right after World War II, but then the curve flattened prematurely because of fears about radiation.
Natural gas appears to be with us for the foreseeable future, unless we can wean ourselves off of it to address global warming. The use of coal is almost done (regardless of what a certain senator from West Virginia thinks), and the use of oil has reached its peak and is on its way down (now might be a good time to buy an electric car).
Climate change is the critical issue looming over the right side of the plot. We must leave many of those fossil fuels (coal, oil, gas) in the ground to prevent an environmental disaster.
Perhaps I need to add extra parts to that homework problem.
(c) Suppose at time t you discover that pollution from this finite resource is killing people, and you stop consuming it immediately. How would that change the plots you made in part (b)?
(d) What would happen if the resource is killing people but people continue to consume it nevertheless?
