Chapter 27, Problem 47P

(II) A Hall probe used to measure magnetic field strengths consists of a rectangular slab of material (free-electron density n) with width d and thickness t, carrying a current I along its length l. The slab is immersed in a magnetic field of magnitude B oriented perpendicular to its rectangular face (of area ld), so that a Hall emf $E_{\text{H}}$ is produced across its width d. The probe’s magnetic sensitivity, defined as $K_{\text{H}}=E_{\text{H}}/IB$, indicates the magnitude of the Hall emf achieved for a given applied magnetic field and current. A slab with a large K_H is a good candidate for use as a Hall probe. (a) Show that K_H = 1/ent. Thus, a good Hall probe has small values for both n and t. (b) As possible candidates for the material used in a Hall probe, consider (i) a typical metal $\left(n\approx 1\times {10}^{29}/{\text{m}}^3\right)$ and (ii) a (doped) semiconductor $\left(n\approx 3\times {10}^{22}/{\text{m}}^3\right)$. Given that a semiconductor slab can be manufactured with a thickness of 0.15 mm, how thin (nm) should a metal slab be to yield a K_H value equal to that of the semiconductor slab? Compare this metal slab thickness with the 0.3-nm size of a typical metal atom, (c) For the typical semiconductor slab described in part (b), what is the expected value for $E_{\text{H}}$ when I = 100 mA and B = 0.1 T?