The Hall Effect

We have seen effects of a magnetic field on free-moving charges. The magnetic field also affects charges moving in a conductor. One result is the Hall effect, which has important implications and applications.

[link] shows what happens to charges moving through a conductor in a magnetic field. The field is perpendicular to the electron drift velocity and to the width of the conductor. Note that conventional current is to the right in both parts of the figure. In part (a), electrons carry the current and move to the left. In part (b), positive charges carry the current and move to the right. Moving electrons feel a magnetic force toward one side of the conductor, leaving a net positive charge on the other side. This separation of charge *creates a voltage ε size 12{ε} {}

, known as the Hall emf, *across the conductor. The creation of a voltage across a current-carrying conductor by a magnetic field is known as the Hall effect, after Edwin Hall, the American physicist who discovered it in 1879.

Figure a shows an electron with velocity v moving toward the left. The magnetic field B is oriented out of the page. The current I is running toward the right. The force vector on the electron points downward. An illustration of the right hand rule shows the right thumb pointing left with the v vector, the fingers pointing toward 7 o’clock with the B vector, the force vector on a positive charge pointing up and the force vector on a negative charge pointing down. Figure b shows a positive charge moving toward the right. The magnetic field lines are coming out of the page. The current I is running toward the right. The force on the positive charge is down. An illustration of the right hand rule shows the thumb pointing in the direction of the charge’s velocity, the fingers pointing in the direction of B, and F pointing down away from the palm.

One very important use of the Hall effect is to determine whether positive or negative charges carries the current. Note that in [link](b), where positive charges carry the current, the Hall emf has the sign opposite to when negative charges carry the current. Historically, the Hall effect was used to show that electrons carry current in metals and it also shows that positive charges carry current in some semiconductors. The Hall effect is used today as a research tool to probe the movement of charges, their drift velocities and densities, and so on, in materials. In 1980, it was discovered that the Hall effect is quantized, an example of quantum behavior in a macroscopic object.

The Hall effect has other uses that range from the determination of blood flow rate to precision measurement of magnetic field strength. To examine these quantitatively, we need an expression for the Hall emf, ε size 12{ε} {}

, across a conductor. Consider the balance of forces on a moving charge in a situation where B size 12{B} {}

, v size 12{v} {}

, and l size 12{l} {}

are mutually perpendicular, such as shown in [link]. Although the magnetic force moves negative charges to one side, they cannot build up without limit. The electric field caused by their separation opposes the magnetic force, F=qvB size 12{F= ital "qvB"} {}

, and the electric force, Fe=qE size 12{F rSub { size 8{e} } = ital "qE"} {}

, eventually grows to equal it. That is,

qE = qvB size 12{ ital "qE"= ital "qvB"} {}

or

E=vB. size 12{E= ital "vB"} {}

Note that the electric field E size 12{E} {}

is uniform across the conductor because the magnetic field B size 12{B} {}

is uniform, as is the conductor. For a uniform electric field, the relationship between electric field and voltage is E=ε/l size 12{E=ε/l} {}

, where l size 12{l} {}

is the width of the conductor and ε size 12{ε} {}

is the Hall emf. Entering this into the last expression gives

ε l = vB . size 12{ { {ε} over {l} } = ital "vB" "." } {}

Solving this for the Hall emf yields

ε = Blv ( B , v , and l , mutually perpendicular ) , size 12{ε= ital "Blv"``` \( B,`v,`"and"`l,`"mutually perpendicular" \) ,} {}

where ε size 12{ε} {}

is the Hall effect voltage across a conductor of width l size 12{l} {}

through which charges move at a speed v size 12{v} {}

.

Diagram showing an electron moving to the left in a three-dimensional rectangular space with velocity v. The magnetic field is oriented out of the page. The electric field is down. The electric force on the charge is up while the magnetic force on the charge is down. An illustration of the right hand rule shows the thumb pointing to the left with v, the fingers out of the page with B, and the force on a positive charge up and away from the palm.

One of the most common uses of the Hall effect is in the measurement of magnetic field strength B size 12{B} {}

. Such devices, called Hall probes, can be made very small, allowing fine position mapping. Hall probes can also be made very accurate, usually accomplished by careful calibration. Another application of the Hall effect is to measure fluid flow in any fluid that has free charges (most do). (See [link].) A magnetic field applied perpendicular to the flow direction produces a Hall emf ε size 12{ε} {}

as shown. Note that the sign of ε size 12{ε} {}

depends not on the sign of the charges, but only on the directions of B size 12{B} {}

and v size 12{v} {}

. The magnitude of the Hall emf is ε=Blv size 12{ε= ital "Blv"} {}

, where l size 12{l} {}

is the pipe diameter, so that the average velocity v size 12{v} {}

can be determined from ε size 12{ε} {}

providing the other factors are known.

Diagram showing a tube with diameter l with one end between the north and south poles of a magnet. The charges are moving with velocity v within the tube and out of the page. The magnetic field B is oriented across the tube, from the north to the south pole of the magnet. The force on the charges is up for positive charges and down for negative charges. e m f = B l v.

Calculating the Hall emf: Hall Effect for Blood Flow

A Hall effect flow probe is placed on an artery, applying a 0.100-T magnetic field across it, in a setup similar to that in [link]. What is the Hall emf, given the vessel’s inside diameter is 4.00 mm and the average blood velocity is 20.0 cm/s?

Strategy

Because B size 12{B} {}

, v size 12{v} {}

, and l size 12{l} {}

are mutually perpendicular, the equation ε=Blv size 12{ε= ital "Blv"} {}

can be used to find ε size 12{ε} {}

.

Solution

Entering the given values for B size 12{B} {}

, v size 12{v} {}

, and *l size 12{l} {}

ε = Blv = 0.100 T 4 . 00 × 10 − 3 m 0 .200 m/s = 80.0 μV alignl { stack { size 12{ε= ital "Blv"= left (0 "." "100"`T right ) left (4 "." "00" times "10" rSup { size 8{ - 3} } `m right ) left (0 "." "200"`"m/s" right )} {} # ="80" "." 0`"μV" {} } } {}

Discussion

This is the average voltage output. Instantaneous voltage varies with pulsating blood flow. The voltage is small in this type of measurement. ε size 12{ε} {}

is particularly difficult to measure, because there are voltages associated with heart action (ECG voltages) that are on the order of millivolts. In practice, this difficulty is overcome by applying an AC magnetic field, so that the Hall emf is AC with the same frequency. An amplifier can be very selective in picking out only the appropriate frequency, eliminating signals and noise at other frequencies.

Section Summary

Conceptual Questions

Discuss how the Hall effect could be used to obtain information on free charge density in a conductor. (Hint: Consider how drift velocity and current are related.)

Problems & Exercises

A large water main is 2.50 m in diameter and the average water velocity is 6.00 m/s. Find the Hall voltage produced if the pipe runs perpendicular to the Earth’s 5.00×10−5-T size 12{5 "." "00" times "10" rSup { size 8{ - 5} } "-T"} {}

field.

7 . 50 × 10 − 4 V

What Hall voltage is produced by a 0.200-T field applied across a 2.60-cm-diameter aorta when blood velocity is 60.0 cm/s?

(a) What is the speed of a supersonic aircraft with a 17.0-m wingspan, if it experiences a 1.60-V Hall voltage between its wing tips when in level flight over the north magnetic pole, where the Earth’s field strength is 8.00×10−5T? size 12{8 "." "00" times "10" rSup { size 8{ - 5} } `"T?"} {}

(b) Explain why very little current flows as a result of this Hall voltage.

(a) 1.18 × 10 3 m/s

(b) Once established, the Hall emf pushes charges one direction and the magnetic force acts in the opposite direction resulting in no net force on the charges. Therefore, no current flows in the direction of the Hall emf. This is the same as in a current-carrying conductor—current does not flow in the direction of the Hall emf.

A nonmechanical water meter could utilize the Hall effect by applying a magnetic field across a metal pipe and measuring the Hall voltage produced. What is the average fluid velocity in a 3.00-cm-diameter pipe, if a 0.500-T field across it creates a 60.0-mV Hall voltage?

Calculate the Hall voltage induced on a patient’s heart while being scanned by an MRI unit. Approximate the conducting path on the heart wall by a wire 7.50 cm long that moves at 10.0 cm/s perpendicular to a 1.50-T magnetic field.

11.3 mV

A Hall probe calibrated to read 1.00 μV size 12{1 "." "00"`"μV"} {}

when placed in a 2.00-T field is placed in a 0.150-T field. What is its output voltage?

Using information in [link], what would the Hall voltage be if a 2.00-T field is applied across a 10-gauge copper wire (2.588 mm in diameter) carrying a 20.0-A current?

1. 16 μV size 12{1 "." "47"`"μV"} {}

Show that the Hall voltage across wires made of the same material, carrying identical currents, and subjected to the same magnetic field is inversely proportional to their diameters. (Hint: Consider how drift velocity depends on wire diameter.)

A patient with a pacemaker is mistakenly being scanned for an MRI image. A 10.0-cm-long section of pacemaker wire moves at a speed of 10.0 cm/s perpendicular to the MRI unit’s magnetic field and a 20.0-mV Hall voltage is induced. What is the magnetic field strength?

2.00 T

Glossary

Hall effect
the creation of voltage across a current-carrying conductor by a magnetic field
Hall emf
the electromotive force created by a current-carrying conductor by a magnetic field, ε=Blv size 12{ε= ital "Blv"} {}
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