Topic 10 | The Hall Effect
The reality of the forces acting on the moving charges in a conductor in a magnetic 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, let’s 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 B perpendicular to the plane of
the strip, in the +y-direction. The drift velocity of the moving charges (charge
magnitude 兩q兩) has magnitude v d . Figure T10.1a shows the case of negative
charges, 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 conductor is the same whether the moving charges are positive or negative. In either case
a moving charge is driven toward the upper edge of the strip by the magnetic force
Fz = 兩q兩vd B.
If the charge carriers are electrons, as in Fig. T10.1a, an excess negative charge
accumulates at the upper edge of the strip, leaving an excess positive charge at its
lower edge. This accumulation continues until the resulting transverse electrostatic field Ee becomes large enough to cause a force (magnitude 兩q兩Ee) that is equal
and opposite to the magnetic force (magnitude 兩q兩vd B). After that, there is no
longer 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 the Hall 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 are positive, 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 in
1879, 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 positively 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. A missing negative 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 field Ee for the
positive-q case is in the −z-direction; its z-component Ez is negative. The magnetic
z
Jx
b
– –
– –
By
– –
– –
–
–
Fz
vd
–
Ee
q
+ +
+ +
+ +
+ +
By
+ +
a
(a)
T10.1 Forces on charge carriers in a conductor in a magnetic field. (a) Negative
charge carriers (electrons) are pushed
toward the top of the strip, leading to a
charge distribution as shown. Point a is at
a higher potential than point b. (b) With
positive charge carriers, the polarity of the
potential difference is opposite to that of
part (a).
z
y
Jx
Jx
x
+ +
+ +
+
Ee
q
+
– –
– –
–
By
y
b
+ +
+ +
+
Fz
vd
– –
– –
–
a
(b)
By
Jx
x
field is in the +y-direction, and we write it as By . The magnetic force (in the +zdirection) is qvd By. The current density Jx is in the +x-direction. In the steady
state, when the forces qEz and qvd By are equal in magnitude and opposite in direction,
qEz 1 qvdBy 5 0, or Ez 5 2vdBy
This confirms that when q is positive, Ez is negative. The current density Jx is
Jx 5 nqvd
Eliminating vd between these equations, we find
nq 5
2JxBy
Ez
(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 measure Jx , By , and Ez , so we can compute the product nq. In both metals and semiconductors, q is equal in magnitude to the electron charge, so the Hall
effect permits a direct measurement of n, 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.
Example
T10.1
Using the Hall effect
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 m V higher than at the top. From this measurement, determine the concentration of mobile electrons in copper.
SOLUTION
We’ll use Eq. (T10.1) to find n. First we find the current density Jx
and the electric field Ez:
75 A
I
5
5 2.5 3 106 A/m2
A
1 2.0 3 1023 m 2 1 1.50 3 1022 m 2
V
0.81 3 1026 V
Ez 5 5
5 5.4 3 1025 V/m
d
1.5 3 1022 m
Jx 5
Then, from Eq. (T10.1),
2Jx By
2 1 2.5 3 106 A/m2 2 1 0.40 T 2
n5
5
qEz
1 21.60 3 10219 C 2 1 5.4 3 1025 V/m 2
5 11.6 3 1028 m 23
The actual value of n for copper is 8.5 × 1028 m−3, which shows that
the simple model of the Hall effect in this section, ignoring quantum 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. Halleffect devices for magnetic-field measurements and other purposes
use semiconductor materials, for which moderate current densities
give much larger Hall emfs.