Question 18.6: Transcranial magnetic stimulation The magnitude of the B fie......
Transcranial magnetic stimulation
The magnitude of the \vec{B} field from a TMS coil increases from 0 T to 0.2 T in 0.002 s. The \vec{B} field lines pass through the scalp into a small region of the brain, inducing a small circular current in the conductive brain tissue in the plane perpendicular to the field lines. Assume that the radius of the circular current in the brain is 0.0030 m and that the tissue in this circular region has an equivalent resistance of 0.010 Ω. What are the direction and magnitude of the induced electric current around this circular region of brain tissue?
Sketch and translate We first sketch the situation: a small coil on the top of the scalp and a small circular disk region inside the brain tissue through which the changing magnetic field passes (see the figure below). The change in magnetic flux through this disk is caused by the increasing \vec{B} field produced by the TMS coil \left(\text { called } \vec{B}_{e x}\right) . This change in flux causes an induced emf, which produces an induced current in the brain tissue. The direction of this current can be determined using Lenz’s law.
Simplify and diagram Assume that \vec{B}_{\mathrm{ex}} throughout the disk of brain tissue is uniform and increases at a constant rate. Model the disk-like region as a singleturn coil. Viewed from above, \vec{B}_{\mathrm{ex}} points into the page (shown as X’s in the figure top right). Since the number of \vec{B} field lines is increasing into the page, the \vec{B}_{\text {in }} field produced by the induced current will point out of the page (shown as dots). Using the right-hand rule for the \vec{B}_{\text {in }} field, we find that the direction of the induced current is counterclockwise.
Represent mathematically To find the magnitude of the induced emf, use Faraday’s law:
\varepsilon_{\text {in }}=N\left|\frac{\Phi_{\mathrm{f}}-\Phi_{\mathrm{i}}}{t_{\mathrm{f}}-t_{\mathrm{i}}}\right|
where the magnetic flux through the loop at a specific clock reading is \Phi=B_{e x} A \cos \theta . The area A of the loop and the orientation angle θ between the loop’s normal vector and the \vec{B}_{\mathrm{ex}} field are constant, so
\varepsilon_{\text {in }}=N\left|\frac{B_{\text {ex }} A \cos \theta-B_{\text {exi }} A \cos \theta}{t_{\mathrm{f}}-t_{\mathrm{i}}}\right|=N A \cos \theta\left|\frac{B_{\mathrm{ex}}-B_{\mathrm{exi}}}{t_{\mathrm{f}}-t_{\mathrm{i}}}\right|
Using our understanding of electric circuits, we relate this induced emf to the resulting induced current:
I_{\text {in }}=\frac{\varepsilon_{\text {in }}}{R}
Solve and evaluate Combine these two equations and solve for the induced current:
I_{\text {in }}=\frac{1}{R} \varepsilon_{\text {in }}=\frac{1}{R}\left(N A \cos \theta\left|\frac{B_{ex\text{ }\text {f }}-B_{ex\text{ }\text {i }}}{t_{\mathrm{f}}-t_{\mathrm{i}}}\right|\right)
=\frac{N \pi r^2 \cos \theta}{R}\left|\frac{B_{\mathrm{ex\text{ }f}}-B_{\mathrm{ex\text{ }i}}}{t_{\mathrm{f}}-t_{\mathrm{i}}}\right|
=\frac{(1) \pi(0.0030 \mathrm{~m})^2(1)}{0.010 \Omega}\left|\frac{0.2 \mathrm{~T}-0}{0.002 \mathrm{~s}-0}\right|=0.28 \mathrm{~A}
Note that the normal vector to the loop’s area is parallel to the magnetic field; therefore, cos θ = cos(0°) = 1.
This is a significant current and could affect brain function in that region of the brain.
Try it yourself: A circular coil of radius 0.020 m with 200 turns lies so that its area is parallel to this page. A bar magnet above the coil is oriented perpendicular to the coil’s area, its north pole facing toward the coil. You quickly (in 0.050 s) move the bar magnet sideways away from the coil to a location far away. During this 0.050 s, the magnitude of the \vec{B} field throughout the coil’s area changes from 0.40 T to nearly 0 T. Determine the average magnitude of the induced emf around the coil while the bar magnet is being moved away.
Answer: 2.0 V.