# Does a Nerve Axon Have an Inductance?

When I was measuring the magnetic field of a nerve axon in graduate school, I wondered if I should worry about a nerve’s inductance. Put another way, I asked if the electric field induced by the axon’s changing magnetic field is large enough to affect the propagation of the action potential.

Here is a new homework problem that will take you through the analysis that John Wikswo and I published in our paper “ The Magnetic Field of a Single Axon” (*Biophysical Journal*, Volume 48, Pages 93–109, 1985). Not only does it answer the question about induction, but also it provides practice in back-of-the-envelope estimation. To learn more about biomagnetism and magnetic induction, see Chapter 8 of *Intermediate Physics for Medicine and Biology*.

Section 8.6

Problem 29½. Consider an action potential propagating down a nerve axon. An electric fieldE, having a rise timeTand extended over a lengthL, is associated with the upstroke of the action potential.

(a) UseOhm’s lawto relateEto the current densityJand the electrical conductivityσ.

(b) UseAmpere’s law(Eq. 8.24, but ignore thedisplacement current) to estimate the magnetic fieldBfromJand thepermeability of free space, . To estimate the derivative, replace thecurl operatorwith 1/L.

(c) UseFaraday’s law(Eq. 8.22, ignoring the minus sign) to estimate the induced electric fieldE* fromB. Replace the time derivative by 1/T.

(d) Write your result as the dimensionless ratioE*/E.

(f) Check that the units in your calculation in part (e) are consistent withE*/Ebeing dimensionless.

(g) Draw a picture of the axon showing *E*, *J*, *B*, *E**, and *L*.

(h) What does your result in part (e) imply about the need to consider inductance when analyzing action potential propagation along a nerve axon.

For those of you who don’t have *IPMB* handy, Equation 8.24 (Ampere’s law, ignoring the displacement current) is

∇×**B** = *μ0* **J**

and Eq. 8.22 (Faraday’s law) is

∇×**E** = *−∂***B***/∂t .*

I’ll leave it to you to solve this problem. However, I’ll show you my picture for part (g).

Also, for part (d) I get a small value, on the order of ten parts per billion (10^–8). The induction of a nerve axon is negligible. We don’t need an inductor when modeling a nerve axon.

*Originally published at **http://hobbieroth.blogspot.com**.*