Intermediate Physics for Medicine and Biology: Why Does Inductance Not Play a Bigger Role in Biology?

In this blog, I talk a lot about topics discussed in Intermediate Physics for Medicine and Biology. Almost as interesting is what topics are NOT discussed in IPMB. One example is inductance.

It’s odd that inductance is not examined in more detail in IPMB, because it is one of my favorite physics topics. To be fair, Russ Hobbie and I do discuss electromagnetic induction: how a changing magnetic field induces an electric field and consequently creates eddy currents. That process underlies transcranial magnetic stimulation, and is analyzed extensively in Chapter 8. However, what I want to focus on today is inductance: the constant of proportionality relating a changing current (I) and an induced electromotive force (

E

; it’s similar to a voltage, although there are subtle differences). The self-inductance of a circuit element is usually denoted L, as in the equation

{\mathcal {E}}

= - L dI/dt .

The word “inductance” appears only twice in IPMB. When deriving the cable equation of a nerve axon, Russ and I write

This rather formidable looking equation is called the cable equation or telegrapher’s equation. It was once familiar to physicists and electrical engineers as the equation for a long cable, such as a submarine cable, with capacitance and leakage resistance but negligible inductance.

Joseph Henry
(1797–1878)

Then, in Homework Problem 44 of Chapter 8, Russ and I ask the reader to calculate the mutual inductance between a nerve axon and a small, toroidal pickup coil. The mutual inductance between two circuit elements can be found by calculating the magnetic flux threading one element divided by the current in the other element. This means the units of inductance are tesla meter squared (flux) over ampere (current), which is given the nickname the henry (H), after American physicist Joseph Henry.

The inductance plays a key role in some biomedical devices. For example, during transcranial magnetic stimulation a magnetic stimulator passes a current pulse through a coil held near the head, inducing an eddy current in the brain. The self-inductance of the coil determines the rate of rise of the current pulse. Another example is the toroidal pickup coil mentioned earlier, where the mutual inductance is the magnetic flux induced in the coil divided by the current in an axon.

Interestingly, the magnetic permeability, μ₀, is related to the inductance. In fact, the units of μ₀ can be expressed in henries per meter (H/m, an inductance per unit length). If you are using a coaxial cable in an electrical circuit to make electrophysiological measurements, the inductance introduced by the cable is equal to μ₀ times the length of the cable times a dimensionless factor that depends on things like the geometry of the cable.

In a circuit, the inductance will induce an electromotive force that opposes a change in the current; It’s a conservative process that acts to keep the current from easily changing. It’s the electrical analogue to mechanical inertia. An inductor sometimes acts like a “choke,” preventing high frequency current from passing through a circuit (say, a few microsecond long spike caused by a nearby lighting strike) while having little effect on the low frequency current (say, the 60 Hz current associated with our power distribution system). You can use inductors to create high- and low-pass filters (although capacitors are more commonly used nowadays).

Why do inductors play such a small role in biology? The self-inductance of a circuit is typically equal to μ₀ times ℓ, where ℓ is a characteristic distance, so L = μ₀ℓ. What can you do to make the inductance larger? First, you could use iron or some other material with a large magnetic permeability, so instead of the magnetic permeability being μ₀ (the permeability of free space) it is μ (which can be many thousands of times larger than μ₀). Another way to increase the inductance is to wind a conductor with many (N) turns of wire. The self-inductance generally increases as N². Finally, you can just make the circuit larger (increase ℓ). However, biological materials contain little or no iron or other ferromagnetic materials, so the magnetic permeability is just μ₀. Rarely do you find lots of turns of wire (some would say the myelin wrapping around a nerve axon is a biological example with large N, but there is little evidence that current flows around the axon within the myelin sheath). And most electrical circuits are small (say, on the order of millimeters or centimeters). If we take the permeability of biological tissue (4π × 10^-7 H/m) times a size of 10 cm (0.1 m) you get an inductance of about 10^-7 H. That’s a pretty small inductance.

Why do I say that 10^-7 H is small? Let’s calculate the induced electromotive force by a current changing in a circuit. Most biological currents are small (John Wikswo and I measured currents of a microamp in a large crayfish nerve axon, and rarely are biological currents larger than this). They also don’t change too rapidly. Nerves work on a time scale on the order of a millisecond. So the magnitude of the induced electromotive force is

${\mathcal {E}}$ = L dI/dt = (10^-7 H) (10^-6 A)/(10^-3 s) = 10^-10 V.

Nerves work using voltages on the order of tens or hundreds of millivolts. So, the induced electromotive force is a thousand million times too small to affect nerve conduction. Sure, some of my assumptions might be too conservative, but even if you find a trick to make ${\mathcal {E}}$ a thousand times larger, it is still a million times too small to be important.

There is one more issue. An electrical circuit with inductance L and resistance R will typically have a time constant of L/R. Regardless of the inductance, if the resistance is large the time constant will be small and inductive effects will happen so quickly that they won’t really matter. If you want small resistance use copper wires, whose conductivity is a million times greater than saltwater. If you’re stuck with saline or other body fluids, the resistance will be high and the time constant will be short.

In summary, the reason why inductance is unimportant in biology is that there is no iron to increase the magnetic field, no copper to lower the resistance, no large number of turns of wire, the circuits are small, and the current changes too slowly. Inductive effects are tiny in biology, which is why we rarely discuss them in Intermediate Physics for Medicine and Biology.