A very thin rectangular strip of zinc has been deposited on an insulating substrate. Let the length, width, and thickness of the strip be x, y, and z, respectively. The length and width are measured to be 10.0 cm and 2.0 cm. When a potential difference of 20 mV is applied as shown in Figure 11.9a (page 404), the current through the strip is 400 mA.
(a) Use the fact that the resistivity of zinc is (at room temperature) \rho=5.92 \times 10^{-8} \Omega \cdot m to find the thickness of the strip.
(b) Now a magnetic field of 0.25 T is applied perpendicular to the strip as shown in Figure 11.9b. A Hall voltage V_{ H }=+0.56 \mu V appears when the voltmeter leads (+ and -) are connected as shown. Determine the sign of the charge carriers and their number density.
Strategy (a) The resistance of a wire with uniform crosssectional area is related to the resistivity by R=\rho x / A, where A is the cross-sectional area and x is the length of the wire. In this case A=y z. This allows us to find the thickness of a wire with known resistivity.
(b) In equilibrium the magnetic force on a charge carrier is equal to the electric force due to the Hall voltage. Thus e E=e v B. From basic electrical conductivity, the electron drift speed is v=I / n e A. These two relations allow us to solve for the charge carrier density n.