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Topic 10 | The Hall Effect

The reality of the forces acting on the moving charges in a conductor in a mag-

netic field is strikingly demonstrated by the Hall effect, an effect analogous to the

transverse deflection of an electron beam in a magnetic field in vacuum. (The

effect was discovered by the American physicist Edwin Hall in 1879 while he was

still a graduate student.) To describe this effect, lets consider a conductor in the

form of a flat strip, as shown in Fig. T10.1. The current is in the direction of

the +x-axis, and there is a uniform magnetic field Bperpendicular to the plane ofthe strip, in the +y-direction. The drift velocity of the moving charges (charge

magnitude q) has magnitude vd. Figure T10.1a shows the case of negativecharges, such as electrons in a metal, and Fig. T10.1b shows positive charges. In

both cases the magnetic force is upward, just as the magnetic force on a conduc-

tor is the same whether the moving charges are positive or negative. In either case

a moving charge is driven toward the upperedge of the strip by the magnetic force

Fz = qvdB.

If the charge carriers are electrons, as in Fig. T10.1a, an excess negative chargeaccumulates at the upper edge of the strip, leaving an excess positive charge at its

lower edge. This accumulation continues until the resulting transverse electrosta-

tic field Ee becomes large enough to cause a force (magnitude qEe) that is equaland opposite to the magnetic force (magnitude qvdB). After that, there is nolonger any net transverse force to deflect the moving charges. This electric field

causes a transverse potential difference between opposite edges of the strip, called

the Hall voltage or theHall emf. The polarity depends on whether the moving

charges are positive or negative. Experiments show that for metals the upper edge

of the strip in Fig. T10.1a does become negatively charged, showing that the

charge carriers in a metal are indeed negative electrons.

However, if the charge carriers arepositive, as in Fig. T10.1b, then positive

charge accumulates at the upper edge, and the potential difference is opposite to

the situation with negative charges. Soon after the discovery of the Hall effect in1879, it was observed that some materials, particularly some semiconductors,

show a Hall emf opposite to that of the metals, as if their charge carriers were pos-

itively charged. We now know that these materials conduct by a process known as

hole conduction. Within such a material there are locations, called holes, that

would normally be occupied by an electron but are actually empty. Amissing neg-

ative charge is equivalent to a positive charge. When an electron moves in one

direction to fill a hole, it leaves another hole behind it. The hole migrates in the

direction opposite to that of the electron.

In terms of the coordinate axes in Fig. T10.1b, the electrostatic fieldEe for thepositive-q case is in the z-direction; itsz-component Ez is negative. The magnetic

++

++

++

++

++

z

x

a

b

q

By

By

(a)

++

++

++

++

++

y

qEe

Eevd

vd

(b)

z

x

a

b

By

FzFz

Jx

Jx

Jx

Jx

y

By

+

T10.1 Forces on charge carriers in a con-ductor in a magnetic field. (a) Negativecharge carriers (electrons) are pushedtoward the top of the strip, leading to acharge distribution as shown. Point a is ata higher potential than point b. (b) Withpositive charge carriers, the polarity of thepotential difference is opposite to that of

part (a).

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field is in the +y-direction, and we write it as By . The magnetic force (in the +z-

direction) is qvdBy. The current density Jx is in the +x-direction. In the steady

state, when the forces qEz and qvdBy are equal in magnitude and opposite in direc-

tion,

This confirms that when q is positive,Ez is negative. The current densityJx is

Eliminating vd between these equations, we find

(T10.1)

Note that this result (as well as the entire derivation) is valid for both positive and

negative q. When q is negative,Ez is positive, and conversely.

We can measureJx , By , andEz , so we can compute the product nq. In both met-

als and semiconductors, q is equal in magnitude to the electron charge, so the Hall

effect permits a direct measurement ofn, the concentration of current-carrying

charges in the material. The sign of the charges is determined by the polarity of

the Hall emf, as we have described.

The Hall effect can also be used for a direct measurement of electron drift

speed vd in metals. These speeds are very small, often of the order of 1 mm/s or

less. If we move the entire conductor in the opposite direction to the current with

a speed equal to the drift speed, then the electrons are at rest with respect to the

magnetic field, and the Hall emf disappears. Thus the conductor speed needed to

make the Hall emf vanish is equal to the drift speed.

nq 52JxBy

Ez

Jx 5 nqvd

qEz 1 qvdBy 5 0, or Ez 5 2vdBy

Then, from Eq. (T10.1),

The actual value ofn for copper is 8.5 1028 m3, which shows that

the simple model of the Hall effect in this section, ignoring quan-

tum effects and electron interactions with the ions, must be used

with caution. This example also shows that with good conductors,

the Hall emf is very small even with large current densities. Hall-

effect devices for magnetic-field measurements and other purposes

use semiconductor materials, for which moderate current densities

give much larger Hall emfs.

5 11.6 3 1028 m23

n 52JxBy

qEz5

2 1 2.5 3 106 A/m2 2 1 0.40 T 2

121.60 3 10219 C 2 1 5.4 3 1025 V/m 2

Using the Hall effectExampleT10.1

You place a slab of copper, 2.0 mm thick and 1.50 cm wide, in a

uniform magnetic field with magnitude 0.40 T, as shown in Fig.

T10.1a. When you run a 75-A current in the +x-direction, you find

by careful measurement that the potential at the bottom of the slab

is 0.81 mV higher than at the top. From this measurement, deter-

mine the concentration of mobile electrons in copper.

SOLUTION

Well use Eq. (T10.1) to find n. First we find the current densityJxand the electric fieldE

z

:

Ez 5V

d5

0.81 3 1026 V

1.5 3 1022 m5 5.4 3 1025 V/m

Jx 5I

A5

75 A

1 2.0 3 1023 m 2 1 1.50 3 1022 m 25 2.5 3 106 A/m2