Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

••• The Hall effect and the sign of charge carriers. Consider a rectangular conductor of width w and thickness t carrying a current I in a uniform magnetic field B perpendicular to the plane of the conductor, as shown in Fig. 1. The charge carriers, with charge q and drift velocity v, experience a Lorentz force F = qv × B, which tends to push them to one side of the conductor. The resulting steady-state charge buildup causes a voltage difference VH, called the Hall voltage, between the sides of the conductor, (a) Show that the Hall voltage is given by

where n is the concentration of charge carriers. [Hint: In steady state, there is no net force on the charge carriers transverse to the current direction, so the force due to the transverse E field must be canceled by the force from the B field.] (b) Show, with an appropriate diagram, that the sign of the Hall voltage gives the sign of the charge carriers. Thus, the Hall voltage allows one to determine whether the carriers in a doped semiconductor are electrons or holes, (c) Compute the magnitude of the Hall voltage in an n-type semiconductor sample at room temperature with donor concentration Nd = 1022 m−3 and the following parameters: t = 1 mm, I = 0.01 A, B = 1.0 T.

FIGURE 1

The Hall effect.