A New Homework Problem

While in Minneapolis last week, attending the 31st Annual International Conference of the IEEE Engineering in Medicine and Biology Society, I had the pleasure of co-chairing a session with Professor Michael Joy of the University of Toronto. Joy has done some fascinating work on measuring current and conductivity in biological tissue using magnetic resonance imaging. His research inspired me to write a new homework problem for Chapter 8 of the 4th edition of Intermediate Physics for Medicine and Biology.

Problem 21.5 The differential form of Ampere’s law, derived in Problem 21, provides a relationship between the current density J and the magnetic field B that allows you to measure biological current with magnetic resonance imaging [see, for example, Scott, G. C., M. L. G. Joy, R. L. Armstrong, and R. M. Henkelman (1991) “Measurement of Nonuniform Current Density by Magnetic Resonance,” IEEE Transactions on Medical Imaging, Volume 10, Pages 362–374]. Suppose you use MRI and find the distribution of magnetic field to be

B_x = C (y z² – y x²)
B_y = C (x z² – x y²)
B_z = C 4 x y z

where C is a constant with the units of T/m³. Determine the current density. Assume the current varies slowly enough that the displacement current can be neglected.

To solve this problem, you need the result of Problem 21 in Chapter 8, which asks the reader to derive the differential form of Ampere’s law from the integral form given in the book by Eq. 8.11. If I were teaching a class from the book, I would assign both Problems 21 and 21.5, and expect the student to solve them both. But for readers of this blog, I will tell you the answer to Problem 21 (ignoring displacement current), so you will have the relationship needed to solve the new Problem 21.5: curl B = μ₀ J. The curl is introduced in Section 8.6. If you don’t have the 4th edition of Intermediate Physics for Medicine and Biology handy, take a look in a math handbook for information about how to calculate the curl (or see Schey’s book Div, Grad, Curl, and All That).

The story of how you measure B using MRI is interesting, but a bit too complicated to describe in detail here. In brief, a magnetic resonance imaging device has a strong static magnetic field about which nuclear spins (such as those of hydrogen) precess. The magnetic field produced by the current density modifies the static magnetic field, causing a phase shift in this precession. This phase shift is detected, and the magnetic field can be deduced from it. Technically, this method allows one to determine the component of the magnetic field that is parallel to the static field. Obtaining the other components requires rotating the object and repeating the procedure. See Chapter 18 for more about MRI.

Send me an email (roth@oakland.edu) if you would like the answer to the new Problem 21.5.

Enjoy.