A charged particle moving through a vacuum is deflected sideways, perpendicular to $\vec{v}$, by a magnetic field. In 1879, a graduate student named Edwin Hall showed that the same is true for the charges moving through a conductor as part of a current. This phenomenon—now called the Hall effect—is used to gain information about the charge carriers in a conductor. It is also the basis of a widely used technique for measuring magnetic field strengths.

FIGURE 29.41a shows a magnetic field perpendicular to a flat, current-carrying conductor. You learned in Chapter 27 that the charge carriers move through a conductor at the drift speed $v_{\text{d}}$. Their motion is perpendicular to $\vec{B}$, so each charge carrier experiences a magnetic force $F_{\text{B}}=e{v}_{\text{d}}B$ perpendicular to both $\vec{B}$ and the current I. However, for the first time we have a situation in which it does matter whether the charge carriers are positive or negative.

FIGURE 29.41 In a magnetic field, the charge carriers of a current are deflected to one side.

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FIGURE 29.41b, with the field pointing toward you, shows that positive charge carriers moving in the direction of I are pushed toward the bottom surface of the conductor. This creates an excess positive charge on the bottom surface and leaves an excess negative charge on the top. FIGURE 29.41c, where the electrons in an electron current i move opposite the direction of I, shows that electrons would be pushed toward the bottom surface. (Be sure to use the right-hand rule and the sign of the electron charge to confirm the deflections shown in these figures.) Thus the sign of the excess charge on the bottom surface is the same as the sign of the charge carriers. Experimentally, the bottom surface is negative when the conductor is a metal, and this is one more piece of evidence that the charge carriers in metals are electrons.

Electrons are deflected toward the bottom surface once the current starts flowing, but the process can’t continue indefinitely. As excess charge accumulates on the top and bottom surfaces, it acts like the charge on the plates of a capacitor, creating a potential difference $\text{Δ}V$ between the two surfaces and an electric field $E=\text{Δ}V\text{/}w$ inside the conductor of width w. This electric field increases until the upward electric force $\vec{F}{}_{\text{E}}$ on the charge carriers exactly balances the downward magnetic force $\vec{F}{}_{\text{B}}$. Once the forces are balanced, a steady state is reached in which the charge carriers move in the direction of the current and no additional charge is deflected to the surface.

The steady-state condition, in which $F_{\text{B}}=F_{\text{E}}$, is

Thus the steady-state potential difference between the two surfaces of the conductor, which is called the Hall voltage $\text{Δ}{V}_{\text{H}}$, is

You learned in Chapter 27 that the drift speed is related to the current density J by $J=ne{v}_{\text{d}}$, where n is the charge-carrier density (charge carriers per ${\text{m}}^3$). Thus

where $A=wt$ is the cross-section area of the conductor. If we use this expression for $v_{\text{d}}$ in Equation 29.23, we find that the Hall voltage is

The Hall voltage is very small for metals in laboratory-sized magnetic fields, typically in the microvolt range. Even so, measurements of the Hall voltage in a known magnetic field are used to determine the charge-carrier density n. Interestingly, the Hall voltage is larger for poor conductors that have smaller charge-carrier densities. A laboratory probe for measuring magnetic field strengths, called a Hall probe, measures $\text{Δ}{V}_{\text{H}}$ for a poor conductor whose charge-carrier density is known. The magnetic field is then determined from Equation 29.25.