hall effect.pdf

Hall effect Experiment

HALL EFFECT EXPERIMENT

K.SURESH SENANAYAKE B.Sc. (Hon's) Physics (Sp), Grad.IP (SL)


HALL EFFECT EXPERIMENT

• Determination of Hall constant• Determination of the concentration of charge carriers• Determination of the mobility of charge carriers for Silver and Tungsten

Apparatus


INDEX

Introduction 01

Apparatus 02

Theory 03

Experimental Procedure 07

Observation 08

Results 09

Error Calculation 17

Conclusion 20

Discussion 21

References 23

Hall Effect Experiment

INTRODUCTION Hall Effect

In any conductor carrying a current and under the influence of a magnetic field component normal to the current, a potential difference perpendicular to both directions is built up due to the Lorentz force acting on the charge carriers. This phenomenon is called as the Hall Effect and the magnitude of this potential then so called “Hall Voltage”. This effect was discovered by Edwin Hall in 1879. Hall voltage depends on the magnetic flux density B, the current passing through the conductor I and the distance between the reference points (thickness of the conductor) d. Lorentz force

The force experienced by a charge moving in space where both electric and magnetic fields exist is called the Lorentz force.

Suppose there exist an electric field E at a certain point in space. The electric force Fe experienced by a charge q placed at that point is given by

Fe=qE The magnetic force Fm experienced by a charge q moving with a velocity v in a magnetic field B is given by

Fm=q (v×B) Hence the Lorentz force (i.e. total force) on a charge moving with velocity v an electric field E and magnetic field B is

F=Fm+Fe

F= q (E + v×B) A charge q moving with a velocity v in a magnetic field B experiences a magnetic

Lorentz force Fm=q (v×B)

If the three vectors Fm, v and B are mutually perpendicular to one another, v×B = vB The magnitude of Fm is qvB and it acts in the direction perpendicular to the plane of vectors v and B, given by the right-hand rule.

Drift velocity (Vd) Drift velocity is net motion of charge carriers (electrons) or the average velocity

of electrons.

Hall constant (Hall coefficient)

The ratio of the voltage created to the product of the amount of current and the magnetic field divided by the element thickness is known as the Hall coefficient. It is a characteristic of the material from which the conductor is made, as its value depends on the type, number and properties of the charge carriers that constitute the current.

1

http://en.wikipedia.org/wiki/Coefficient

http://en.wikipedia.org/wiki/Electric_charge


APPARATUS Hall effect apparatus (Silver and Tungsten)Electromagnet (pair of pole pieces, two coils with 250 turns)AmmeterMulti meterMicro voltmeterHigh current power supply(0-12 V/0-20A)DC power supply (20V/10A)Digital gauss meterConnecting wiresMeter ruleVernier calliper

2


THEORY Consider the conductor carrying a current I and under the influence of a magnetic field component B normal to the current.

Then the magnetic Lorentz force is given by BqVF dm = Where Vd is the drift velocity

For an electron q = e

Therefore BeVF dm =After forming the hall potential,

3


Then there is the electric field E due to the hall voltage between the conductor and it experiences the electric force

EeFe =At the hall voltage

dd

me

BVEBeVeEFF

=⇒=

=

tV

E H=

Where VH = Hall voltage t = width of the conductor

According to the mechanism of the current The drift velocity is given by

AneIVd =

Where A = area of the current flowing n = concentration of charge carriers (charge density)

Then

AneBIdV

dBVV

H

dH

=

=

Since A=dt where d = thickness of the conductor

dneBIVH =

Id

BRV H

H ⎟⎠

⎞⎜⎝

⎛=⇒

4


ne Where RH = 1

Hall constant

neRH

1=

By the gradient of the graph VH vs. I, the hall constant can be calculated.

If RH< 0 then the charge carriers are electrons If RH> 0 then the charge carriers are holes

Concentration of charge carriers

eRn

H

1=

Mobility of charge carriers

The mobility of charge carriers is defined as a quantity relating the drift velocity of charge carriers (electrons) to the applied electric field across a material.

Therefore it is given by the equation

EVd=µ

Since Ane

IVd =

AneEI

=µ

Since A=dt

dtneEI

=µ

Also E can be written as

hVE =

Where V= the applied voltage across the plate (conductor) h= the length of the plate

Therefore

Idtne

hVdtneV

hIµ

µ =⇒=

5


Since ne

RH1

=

IhR

V H⎟⎟⎠

⎞⎜⎜⎝

⎛=

dtµ

By the gradient of the graph of V vs. I, the mobility of charge electrons can be calculated.

6


EXPERIMENTAL PROCEDURE

Ammeter

High current power supply

Micro voltmeter

DC power supply

U-core

Hall apparatus

1. The Hall Effect apparatus was fit according to the above figure into electromagnetwhose pole pieces are placed near the plate so as to keep the air gap where theSilver or Tungsten is placed as narrow as possible.

2. The first, the Silver apparatus was connected to the U-core as shown in the figureand the distance between two pole pieces was adjusted to 5 mm.

3. The current which passing through the electromagnet was adjusted to about 3.7Aby using high current power supply and digital multi-meter.

4. Then the micro voltmeter was calibrated to zero using the corresponding knob andits scale was adjusted.

5. The hall voltage and the voltage across the strip were measured by increasing thecurrent which passing through the Silver strip as 2, 4,... 20 A.

6. The magnetic flux density was measured by using digital gauss meter.7. The width, length and thickness of the Silver strip were measured by using meter

rule and vernier calliper respectively.8. The similar procedure was repeated for the Tungsten strip.

Important

Before recording the any measurements the electromagnet should be demagnetized.

7


OBSERVATION For the Silver apparatus

I (A) VH1 (without B) ×10-4 (V) VH2 (with B) ×10-4 (V) V (V) 2 -0.15 -0.14 0.14 -0.33 -0.31 0.26 -0.50 -0.47 0.38 -0.67 -0.63 0.5

10 -0.85 -0.80 0.612 -1.03 -0.97 0.714 -1.21 -1.13 0.816 -1.38 -1.30 0.818 -1.56 -1.47 0.820 -1.75 -1.65 0.9

For the Tungsten apparatus

I (A) VH1 (without B) × 10-4 (V) VH2 (with B) ×10-4 (V) V (V) 2 0.54 0.52 0.24 1.06 1.04 0.36 1.59 1.56 0.48 2.11 2.06 0.5

10 2.61 2.56 0.712 3.14 3.07 0.914 3.68 3.60 1.016 4.22 4.13 1.118 4.83 4.72 1.220 5.47 5.35 1.2

Thickness of the conductor (hall plate), d =5.00 x10-5 m Length of the hall plate, h =7.50 x10-2 m Width of the hall plate, t = 2.00x10-2 m Distance between two pole pieces, b = 5.00 x10-3 m Magnetic flux density, B = 4.30 x103 GCharge of the electron, e =1.602 x10-19C

Least Count of the Gauss Meter = 1x10-6 T Least Count of the Meter Rule = 1x10-3 m Least Count of the venire calliper = 1 x10-4 m

8


RESULTS Graph 01

9


RESULTS CALCULATION

For the Silver apparatus

I (A) VH1 (without B) ×10-4 (V) VH2 (with B)×10-4 (V) V (V) (VH1-VH2)=VH ×10-4 (V) 2 -0.15 -0.14 0.1 -0.014 -0.33 -0.31 0.2 -0.026 -0.50 -0.47 0.3 -0.038 -0.67 -0.63 0.5 -0.04

10 -0.85 -0.80 0.6 -0.0512 -1.03 -0.97 0.7 -0.0614 -1.21 -1.13 0.8 -0.0816 -1.38 -1.30 0.8 -0.0818 -1.56 -1.47 0.8 -0.0920 -1.75 -1.65 0.9 -0.10

I (A) VH ×10-4 (V) 2 -0.014 -0.026 -0.038 -0.0410 -0.0512 -0.0614 -0.0816 -0.0818 -0.0920 -0.10

From statistical method (using Origin 6.0) The gradient of the graph 01 = m1 = -5.09 x10-7 Ω

But

⎟⎠

⎞⎜⎝

⎛=

dBR

m H1

Where B =4.30 x103 G

d=5.00 x10-5 m

Since 1G = 10-4 T ⇒ B=0.43 T

Therefore

⎟⎠

⎞⎜⎝

⎛=

Bdm

RH1

10


Graph 02

11


⎟⎟⎠

⎞⎜⎜⎝

⎛ ××Ω×−=

−−

TmRH 43.0

1000.51009.5 57

( )131110918.5 −−×−= CmR AgH ⇒Hall constant for Silver

Since the hall constant is less than zero the charge carriers are the electrons.

The concentration of the charge carriers for the Silver

eRn

H

1=

1911 10602.110918.51

−− ×××=n

( )32910055.1 −×= mn Ag ⇒ Concentration of the charge carriers for the Silver

For the mobility of the charge carriers

I (A) V (V) 2 0.14 0.26 0.38 0.510 0.612 0.714 0.816 0.818 0.820 0.9

From statistical method (using Origin 6.0) The gradient of the graph 02 = m2 = 0.04515 Ω

22 dtm

hRdthR

m HH =⇒⎟⎟⎠

⎞⎜⎜⎝

⎛= µ

µ

Ω×××××××

= −−

−−−

04515.01000.21000.510918.51050.725

13112

mmCmmµ

( )11251083.9 −−− Ω×= CmAgµ ⇒Mobility of the electrons for Silver

12


Graph 03

13


For the Tungsten apparatus

I (A) VH1 (without B) ´10-5 (V) VH2 (with B) ´10-5 (V) V (V) (VH1-VH2)=VH ×10-5 (V) 2 0.54 0.52 0.2 0.024 1.06 1.04 0.3 0.026 1.59 1.56 0.4 0.038 2.11 2.06 0.5 0.05

10 2.61 2.56 0.7 0.0512 3.14 3.07 0.9 0.0714 3.68 3.60 1.0 0.0816 4.22 4.13 1.1 0.0918 4.83 4.72 1.2 0.1120 5.47 5.35 1.2 0.12

I (A) VH ×10-4 (V) 2 0.024 0.026 0.038 0.0510 0.0512 0.0714 0.0816 0.0918 0.1120 0.12

From statistical method (using Origin 6.0) The gradient of the graph 03 = m1 = 5.88 x10-7 Ω

But

⎟⎠

⎞⎜⎝

⎛=

dBR

m H1

Where B = 4.30 x103 G

d=5.00 x10-5 m

Since 1G = 10-4 T ⇒ B=0.43 T

Therefore

⎟⎠

⎞⎜⎝

⎛=

Bdm

RH1

14


Graph 04

15


⎟⎟⎠

⎞⎜⎜⎝

⎛ ××Ω×=

−−

TmRH 43.0

1000.51088.5 57

( )131110837.6 −−×= CmR WH ⇒Hall constant for Tungsten

Since the hall constant is greater than zero the charge carriers are the holes.

The concentration of the charge carriers for the Tungsten

eRn

H

1=

1911 10602.110837.61

−− ×××=n

( )32810130.9 −×= mn W ⇒ Concentration of the charge carriers for the Tungsten

For the mobility of the charge carriers

I (A) V (V) 2 0.24 0.36 0.48 0.510 0.712 0.914 1.016 1.118 1.220 1.2

From statistical method (using Origin 6.0) The gradient of the graph 02 = m2 = 0.06212 Ω

22 dtm

hRdthR

m HH =⇒⎟⎟⎠

⎞⎜⎜⎝

⎛= µ

µ

Ω×××××××

= −−

−−−

06212.01000.21000.510837.61050.725

13112

mmCmmµ

( )11251025.8 −−− Ω×= CmAgµ ⇒Mobility of the electrons for Tungsten

16


ERROR CALCULATION Hall constant

Bdm

RH1=

BdmRH lnlnlnln 1 −+=⇒

By integrating BB

dd

mm

RR

H

H δδδδ++=⇒

1

1

Where d is given by the manufacturer so it is a constant then δd=0

Therefore BB

mm

RR

H

H δδδ+=⇒

1

1

HH RBB

mm

R ⎟⎟⎠

⎞⎜⎜⎝

⎛+=⇒δδ

δ1

1

Concentration of charge carriers

eRn

H

1= eRn H lnlnln −−=⇒

By integrating H

H

RR

nn δδ=⇒ Qfor the maximum error

Therefore

nRR

nH

H⎟⎟⎠

⎞⎜⎜⎝

⎛=

δδ


22

lnlnlnlnlnln mtdRhdtmhR

HH −−−+=⇒= µµ

By integrating

2

2

mm

tt

RR

hh

H

H δδδδµδµ

+++= For the maximum error

µδδδδδµ ⎟⎟

⎠

⎞⎜⎜⎝

⎛+++=

2

2

mm

tt

RR

hh

H

H

17


Error calculation for the Silver

( ) ( )AgHAgH RBB

mm

R ⎟⎟⎠

⎞⎜⎜⎝

⎛+=⇒δδ

δ1

1

δB=1×10-6 T by using digital gauss meter

δm1=1.818×10-8 Ω by the statistical method using Origin 6.0

( )1311

6

7

8

10918.543.0101

1009.510818.1 −−

−

−

−

××⎟⎟⎠

⎞⎜⎜⎝

⎛ ×+

××

=⇒ CmR AgHδ

( )131110211.0 −−×= CmR AgHδ ⇒Error for the hall constant for the Silver

Concentration of the charge carriers

( )( )

( )( )Ag

AgH

AgHAg n

RR

n ⎟⎟⎠

⎞⎜⎜⎝

⎛=

δδ

( )329

11

11

10055.110918.510211.0 −

−

−

××⎟⎟⎠

⎞⎜⎜⎝

⎛××

= mn Agδ

( )32910038.0 −×= mn Agδ ⇒ Error for the concentration of charge carriers

of the Silver


( ) ( )AgH

HAg m

mtt

RR

hh µ

δδδδδµ ⎟⎟⎠

⎞⎜⎜⎝

⎛+++=

2

2

δm2=0.00427 Ω by the statistical method using Origin 6.0

( )1125

2

3

11

11

2

3

1083.904515.000427.0

102105.0

10918.510211.0

105.7105.0 −−−

−

−

−

−

−

−

Ω××⎟⎟⎠

⎞⎜⎜⎝

⎛+

××

+××

+××

= CmAgδµ

( )11251018.2 −−− Ω×= CmAgδµ ⇒ Error for the mobility of charge carriers of the

Silver

18


Error calculation for the Tungsten

( ) ( )WHWH RBB

mm

R ⎟⎟⎠

⎞⎜⎜⎝

⎛+=⇒δδ

δ1

1

δB=1×10-6 T by using digital gauss meter

δm1=2.984×10-8 Ω by the statistical method using Origin 6.0

( )1311

6

7

8

10837.643.0101

1088.510984.2 −−

−

−

−

××⎟⎟⎠

⎞⎜⎜⎝

⎛ ×+

××

=⇒ CmR WHδ

( )131110347.0 −−×= CmR WHδ ⇒Error for the hall constant for the Tungsten

Concentration of the charge carriers

( )( )

( )( )W

WH

WHW n

RR

n ⎟⎟⎠

⎞⎜⎜⎝

⎛=

δδ

( )328

11

11

10130.910837.610347.0 −

−

−

××⎟⎟⎠

⎞⎜⎜⎝

⎛××

= mn Wδ

( )32810463.0 −×= mn Wδ ⇒ Error for the concentration of charge carriers

of the Tungsten


( )( )

( )( )W

WH

WHW m

mtt

RR

hh µ

δδδδδµ ⎟⎟⎠

⎞⎜⎜⎝

⎛+++=

2

2

δm2=0.00346 Ω by the statistical method using Origin 6.0

( )1125

2

3

11

11

2

3

1025.806212.000346.0

102105.0

10837.610347.0

105.7105.0 −−−

−

−

−

−

−

−

Ω××⎟⎟⎠

⎞⎜⎜⎝

⎛+

××

+××

+××

= CmWδµ

( )11251014.1 −−− Ω×= CmWδµ ⇒ Error for the mobility of charge carriers of the

Tungsten

19


CONCLUSION

Experimental values Standard values Substance RH ×10-11 (m3C-1)

n ×1029 (m-3)

µ ×10-5 (m2C-1Ω-1)

RH ×10-11 (m3C-1)

n ×1029 (m-3)

µ ×10-5 (m2C-1Ω-1)

Silver -5.9±0.2 1.05±0.04 9.8±2.2 -8.9 0.66 -

Tungsten 6.8±0.3 0.91±0.05 8.2±1.1 11.8 0.53 -

20


DISCUSSION There can be occurred some errors in this experiment.

• The electromagnet is not demagnetized completely• There is no uniform magnetic field and it can be changed.• Resistance due to connecting wires• The temperature of the room is not constant• All equipments are not ideal

To minimize these errors it was got some activities as follows. • Before recording the readings, it was demagnetized the iron of the

electromagnet by allowing a current of approximately 5A a.c. which is then slowly reduced to zero, to flow through the coils for a short time.

For demonstrating the proportionalities VH-I and VH-B and for precise determination of the Hall potential VH, the Silver Hall Effect apparatus is most suitable.

Qualitative experiments with the tungsten apparatus require special care and skill of the experimenter.

• With switched on current which passing through the conductor, air circulationmay cause considerable zero point fluctuations (thermo voltages on the measuring contacts for hall voltage).

• Due to the higher electric resistance of tungsten, the thermal effects and hencethe zero-point fluctuations are higher than with Silver.

Concluding that for Silver the concentration of charge carriers, has the same order of magnitude as the density of the atoms, a confirmation of the model of free electron gas for metals.

With the knowledge of hall constant for a constant current how the dependence of magnetic flux density and the current through the magnet’s coil can be recorded in form of a characteristic B (Icoil); meaning that the hall effect can be used to measure the strength of magnetic field.

By changing from the Silver apparatus to the Tungsten conductor and by repeating the measurements, it is shown that the hall constant for tungsten is several times greater and, ever more confusing at first sight, has the opposite sign than that of the hall constant for Silver.

The assumption of a free electron gas does not hold true for non-monovalent metals like tungsten, and both effects can be explained by the band model of conduction.

This by the experimental treatment of the Hall Effect, the theoretical basis for the understanding of the conduction mechanisms in semi conducting materials is also confidingly introduced.

21


Technological applications of Hall Effect

So-called "Hall effect sensors" are readily available from a number of different manufacturers, and may be used in various sensors such as fluid flow sensors, current sensors, and pressure sensors. Other applications may be found in some electric air soft guns and on the triggers of electropnuematic paintball guns.

Applications of Hall Effect

Hall Effect devices produce a very low signal level and thus require amplification. While suitable for laboratory instruments, the vacuum tube amplifiers available in the first half of the 20th century were too expensive, power consuming, and unreliable for everyday applications. It was only with the development of the low cost integrated circuit that the Hall Effect sensor became suitable for mass application. Many devices now sold as "Hall effect sensors" are in fact a device containing both the sensor described above and a high gain integrated circuit (IC) amplifier in a single package. Reed switch electrical motors using the Hall Effect IC is another application.

Hall probes are often used to measure magnetic fields, or inspect materials (such as tubing or pipelines) using the principles of Magnetic flux leakage.

22

http://en.wikipedia.org/wiki/Hall_effect_sensor

http://en.wikipedia.org/wiki/Flow_sensor

http://en.wikipedia.org/wiki/Current_sensor

http://en.wikipedia.org/wiki/Current_sensor

http://en.wikipedia.org/wiki/Pressure_sensor

http://en.wikipedia.org/wiki/Airsoft

http://en.wikipedia.org/wiki/Vacuum_tube

http://en.wikipedia.org/wiki/Amplifier

http://en.wikipedia.org/wiki/20th_century

http://en.wikipedia.org/wiki/Integrated_circuit



http://en.wikipedia.org/wiki/Integrated_circuit

http://en.wikipedia.org/wiki/Reed_switch

http://en.wikipedia.org/wiki/Magnetic_flux_leakage


REFERENCE Leybold catalogue: Equipment for Scientific and Technical Education. (LEYBOLD-HERAEUS GMBH) LEYBOLD DIDACTIC GMBH

Physics for class xii Bajaj, N.K. 2nd edition

TATA Mc Graw Hill.

http://www.hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

http://www.wikipedia.org/wiki/Hall_effect

http://www.svslabs.com/pro1/2k.pdf

23

http://www.hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

http://www.wikipedia.org/wiki/Hall_effect

http://www.svslabs.com/pro1/2k.pdf

hall effect.pdf

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