A Present From Santa

Santa arrived last night and left you, dear reader, a present in your stocking: two new homework problems for the 4^th edition of Intermediate Physics for Medicine and Biology. The problems belong to Chapter 8 on Biomagnetism (one of my favorite chapters), and specifically to Section 8.6 on Electromagnetic Induction. They both explore the idea of skin depth, but from somewhat different perspectives. Please forgive Santa for being a bit long-winded; he got carried away.

Enjoy.

Section 8.6

Problem 25.1 The concept of “skin depth” plays a role in some biomagnetic applications.
(a) Write Ampere’s law (Eq. 8.22) for the case when the displacement current is negligible.
(b) Use Ohm’s law (Eq. 6.26) to write the result from (a) in terms of the electric field.
(c) Take the curl of both sides of the equation you found in (b) (Assume the conductivity σ is homogeneous and isotropic).
(d) Use Faraday’s law (Eq. 8.20), ∇·B=0 (Eq. 8.7), and the vector identity ∇×(∇×B)=∇(∇·B)-∇²B to simplify the result from (c).
(e) Your answer to (d) should be the familiar diffusion equation (Eq. 4.24). Express the diffusion constant D in terms of electric and magnetic parameters.
(f) In Chapter 4, we found that diffusion over a distance L takes a time T equal to L²/2D. During transcranial magnetic stimulation, L=0.1 m, σ=0.1 S/m and μ_o=4π × 10⁻⁷ T m/A. How long does the magnetic field take to diffuse into the head? Is this time much longer than or much shorter than the rise time of the magnetic field for the stimulator designed by Barker et al. (1985)?
(g) Solve T= L²/2D for L, using the expression for D found in (e). Calculate L for T=0.1 ms. Is L much larger than or much smaller than the size of your head? L is closely related to the “skin depth” defined in electromagnetic theory.
(h) During magnetic resonance imaging (see Chapter 18), an 85 MHz radio-frequency magnetic field is applied to the body. Calculate L using half a period for T. How does L compare to the size of the head? The frequency of the RF field is proportional to the strength of the static magnetic field in an MRI device, and 85 MHz corresponds to 2 T. If the static field is 7 T (common in modern high-field MRI), calculate L. Is it safe to ignore skin depth during high-field MRI?

Problem 25.2 During magnetic stimulation, a changing magnetic field B induces eddy currents in the body that produce their own magnetic field B'. The goal of this problem is to compare B' and B. We can estimate B' using the following approximations. First, ignore the vector nature of all fields and do not distinguish between components. Second, ignore all negative signs. Third, replace all time derivatives with multiplication by 1/T, where T is a characteristic time. Fourth, replace all space derivatives (such as the curl) by multiplication with 1/L, where L is a characteristic length.
(a) Use Faraday’s law (Eq. 8.20) to estimate the induced electric field E from B.
(b) Use Ohm’s law (Eq. 6.26) to estimate the current density J from E.
(c) Use Ampere’s law (Eq. 8.22, but ignore displacement currents) to estimate B' from J.
(d) Combine parts (a), (b), and (c) to determine an expression for the ratio B'/B in terms of the conductivity σ, the permeability μ_o, L, and T.
(e) In magnetic stimulation, L=0.1 m, T=0.1 ms, σ=0.1 S/m and μ_o=4π × 10⁻⁷ T m/A. Calculate B'/B. Is it safe to ignore B' compared to B during magnetic stimulation?