Hall effect Experiment
HALL EFFECT EXPERIMENT
K.SURESH SENANAYAKE B.Sc. (Hon's) Physics (Sp), Grad.IP (SL)
Hall effect Experiment
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
Hall effect Experiment
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.
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Hall Effect Experiment
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
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Hall Effect Experiment
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,
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Hall Effect Experiment
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 ⎟⎠
⎞⎜⎝
⎛=⇒
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Hall Effect Experiment
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 =⇒=
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Hall Effect Experiment
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.
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Hall Effect Experiment
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.
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Hall Effect Experiment
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
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Hall Effect Experiment
RESULTS Graph 01
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Hall Effect Experiment
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
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Hall Effect Experiment
Graph 02
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Hall Effect Experiment
⎟⎟⎠
⎞⎜⎜⎝
⎛ ××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
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Hall Effect Experiment
Graph 03
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Hall Effect Experiment
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
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Hall Effect Experiment
Graph 04
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Hall Effect Experiment
⎟⎟⎠
⎞⎜⎜⎝
⎛ ××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
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Hall Effect Experiment
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
Mobility of charge carriers
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
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Hall Effect Experiment
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
Mobility of charge carriers
( ) ( )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
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Hall Effect Experiment
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
Mobility of charge carriers
( )( )
( )( )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
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Hall Effect Experiment
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 -
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Hall Effect Experiment
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.
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Hall Effect Experiment
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.
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Hall Effect Experiment
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
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