Question

A patient with a pacemaker is mistakenly being scanned for an MRI image. A 10.0-cm-long section of pacemaker wire moves at a speed of 10.0 cm/s perpendicular to the MRI unit’s magnetic field and a 20.0-mV Hall voltage is induced What is the magnetic field strength?

Short answer

The required magnetic field is${}^{\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{T}}$

Step-by-step solution

Given information

The diameter of the pacemaker wire is ${}^{\mathbf{L}\mathbf{=}\mathbf{10}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{cm}\left(\frac{1m}{100\mathrm{cm}}\right)\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{1}\mathbf{}\mathbf{m}}$

The speed of the moving wire is${}^{\mathbf{v}\mathbf{=}\mathbf{10}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{cm}\mathbf{/}\mathbf{s}\mathbf{\left(}\frac{\mathbf{1}\mathbf{}\mathbf{m}}{\mathbf{100}\mathbf{}\mathbf{cm}}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{1}\mathbf{}\mathbf{m}\mathbf{/}\mathbf{s}}$

Induced Hall voltage is${}^{\mathbf{E}\mathbf{=}\mathbf{20}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{mV}\mathbf{\left(}\frac{{\mathbf{10}}^{\mathbf{3}}\mathbf{}\mathbf{V}}{\mathbf{1}\mathbf{}\mathbf{mV}}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{2}}\mathbf{}\mathbf{V}}$

Define the Hall effect

The Hall effect defines the generation of induced voltage due to the motion of charge carriers in a perpendicular applied magnetic field and can be defined as,

$\mathbf{E}\mathbf{=}\mathbf{BLv}$…………………(1)

Evaluating the magnetic field

Substitute the given data in equation (1), which will result,

$$\begin{gathered} \mathbf{B}\mathbf{=}\frac{\mathbf{E}}{\mathbf{Lv}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{2}}\mathbf{}\mathbf{V}}{\left(0.1m\right)\mathbf{}\mathbf{\times }\left(0.1m/s\right)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{T} \end{gathered}$$

Therefore, the required magnetic field is${}^{\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{T}}$