Download MEASUREMENT OF MAGNETIC FIELD ALONG THE AXIS OF A

Vol-12, No-3, Sept.-2012 Stewart & Gee
Experiment-373
S
MEASUREMENT OF MAGNETIC FIELD
ALONG THE AXIS OF A CIRCULAR CURRENT
CARRYING COIL USING DIGITAL STEEWART
AND GEE APPARATUS
Jeethendra Kumar P K, Santhosh K and Ajeya Padmajeeth
KamalJeeth Instrumentation & Service Unit, No-610, Tata Nagar, Bengaluru-560092. INDIA.
Email: [email protected]
Abstract
Using a search coil, microcontroller and an LCD display, the magnetic field (flux) and induced emf
produced at the center of a circular copper coil carrying current are recorded and verified vis a vis
the values obtained from the relevant theoretical equations. The permeability of free space is
calculated and the variation of magnetic field along the axis of the coil is also studied.
Introduction
Balfour Stewart and William Winston Haldane Gee designed an instrument to study the variation of magnetic
field along the axis of a circular coil. It consists of a coil with specified number turns of copper wire and a
compass that can be moved along the axis of the coil. This simple apparatus is still in use in most physics labs.
The problem with this apparatus is, however, poor quality of the magnetic compass used in it. Hence, the
magnetic field calculated using the deflection of the compass needle is not accurate.
In this experiment we have used a search coil which can be moved along the axis of the coil. The voltage
induced in the coil is noted, from which the magnetic field produced by the circular coil is calculated. A
microcontroller is used to record the induced voltage and a program is written to calculate the magnetic flux (B)
value. The display shows both the induced voltage and the magnetic field values. One can verify the
experimental results using
the values computed from the relevant theoretical equation [1].
Magnetic field along the axis of a current carrying coil
A stationary electric charge produces an electric field whereas a moving electric charge produces magnetic field
around it. This is the basic principle of electro-magnetic induction. The magnetic field at the center of a coil
carrying current I is given by
B=
where I the current flowing through the coil,
R is the radius of the coil, and
…1
Lab Experiments 1
Kamalj
Kamaljeeth Instrumentation and Service Unit
Vol-12, No-3, Sept.-2012 Stewart & Gee
µ o is permeability of free space.
If the number of turns in the of coil is ‘N’, the magnetic field, B, at the center of the coil is given by
B=N
…2
As one moves away from the center of the coil, the magnetic field decreases. Hence the magnetic field varies
along the axis of the coil. A The magnetic field at the point A at a distance Z from the center of the coil is given
by
B=
where
…3
/
I the current flowing through the coil,
R is the radius of the coil,
µ o is permeability of free space,
N is number of turns in the coil,
Z is the radial distance from the center of the coil to the point of observation.
Figure-1 shows the geometry of the field. Using Equation-3 one can determine the field along the axis of the
coil which decreases inversely with the square of the distance.
By
R
Axis
I
B
O
Z
A
BZ
Figure-1: Directions of magnetic field along the axis of the coil
Search coil detector
The above equations (Equation-3) hold good when a battery or DC source is used to energize the coil. In the
present experiment we have used an AC source and a search coil. The alternating current flowing in the field
coil induces an electromotive force (emf) in the search coil. The voltage generated in the search coil due to
electromagnetic induction is governed by the Faraday’s law of induction.
Lab Experiments 2
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Kamaljeeth Instrumentation and Service Unit
Vol-12, No-3, Sept.-2012 Stewart & Gee
The search coil used in this experiment is smaller (with 2.2cm square bobin size) compared to the filed coil
(with radius R = 9.75cm), hence the magnetic field at a given instant of time is nearly uniform over the area of
the search coil.
The magnetic flux Φ corresponding to the magnetic field vector B generated by the search coil is defined as
the product of the area A of the coil and the magnetic vector B normal to the plane of the search coil
Φ = AB cos α.
…4
where α is the angle between the planes of B and Ф. Since the search coil is also placed with its plane parallel to
the field coil
α = 0, cos 0 =1, so that Equation-4 becomes
Φ = AB
…5
The search coil is fitted to a half meter long rectangular iron rod, which acts like a core and is placed at the axis
of the field coil. The coil can be moved along the axis. The iron core increases the emf induced in the search
coil due to its high relative permittivity [2]. Hence the total flux is given by
Φ = A B µr
…6
where
A is cross sectional area of the core of the search coil,
B is the magnetic field intensity, and
µ r is the relative permittivity of the material of the core of the search coil.
The ratio of inductance of the search coil with air as the core and inductance of the search coil with iron as the
core (in the form of a rod on which the coil is fixed for movement) gives the relative permittivity of air with
respect to the material of the core
µr =
…7
`
Faraday’s law of induction gives the voltage induced per turn of the search coil as
V = -n
Ф
…8
where
n is the number of turns in the search coil,
V is the induced emf in the search coil, and
Ф is the flux.
Substituting for flux (Ф) from Equation-6, one can rewrite Equation- 8 as
V = -n
!"
V = -n A µ r
#
"#
…9
…10
Lab Experiments 3
Kamalj
Kamaljeeth Instrumentation and Service Unit
Vol-12, No-3, Sept.-2012 Stewart & Gee
Due to the current induced in the search coil it produces a magnetic field in the opposite direction as indicated
by the minus sign in Equation-10. The magnetic field produced by the search coil opposes any change in the
magnetic field produced by the field coil, as described by the Lenz’s law.
The current I flowing through the field coil vary sinusoidally with time so that the magnetic field B also varies
in the same manner. The magnitude of the magnetic field B produced by the field coil can be written as
B=
"$$
cos ωt
where
…11
Bpp is the peak to peak value of the magnetic field, and
ω is the angular frequency of AC signal in the field coil.
This is the general equation representing sinusoidal voltage (or current) produced by the field coil.
Substituting for B in Equation-10, one obtains
"$$
%
V = -n A µ r
V = -n A µ r
",,
-
cos ωt+
.#
...12
By differentiating cos (ωt ) and substituting it in Equation-12 we get
V = -n A µ r
"//
(-ω sinωt)
"//
V = nAω µ r
sinωt
...13
…14
This is the instantaneous (time varying) voltage induced in the search coil. Hence the peak to peak voltage is
given by
VPP = nAω µ r
"//
…15
The RMS value can be written as
Vrms =
!ωµ01
√
//
=
!ω3µ01
4√
//
…16
Hence the magnetic field can be determined by measuring the induced emf in the search coil. The magnetic
field is given by
BPP =
4√ 5067
.!
0
Substituting for ω =2πf, we get
Lab Experiments 4
Kamalj
Kamaljeeth Instrumentation and Service Unit
Vol-12, No-3, Sept.-2012 Stewart & Gee
BPP =
BPP =
4√ 5067
8!
0
, or
√ 5067
8! 0
…17
Using this equation one can calculate the magnetic field at the center of the coil as well as along its axis on
either side.
For a give set of search coils, the parameters n, A, f, µ r are constants, hence the magnetic field is proportional to
the induced emf, as given by Equation-17.
In the present work, instead of using a magnetic compass we have used a search coil. Hence AC power supply
is used to energize the coil. A rectangular iron rod, with dimensions 1.65 cm x 1.65 cm, which acts as the axis
of the coil is provided with the set-up. The search coil is fixed in this rectangular bar which acts like a core for
the search coil. A pointer fitted on the search coil is used to read the distance from the center of the coil, as
shown in Figure-2.
Figure-2: Search coil fitted along the axis of the coil and the pointer for reading its position
Determination of permeability of free space
Since a core is placed along the field axis, by changing the material of the core the relative permittivity of the
core material and permeability of free space can be determined using this instrument. The magnetic field at
the center of the coil is given by Equation-2. In the presence of the core, the magnetic field is multiplied by µ
B=N
μ
…18
A plot of B versus I provides the average value of B/I from which knowing the values of N, R and µ
permeability of free space can be determined from the equation
µo =
"
0
=
0
X Slope
,
the
…19
Lab Experiments 5
Kamalj
Kamaljeeth Instrumentation and Service Unit
Vol-12, No-3, Sept.-2012 Stewart & Gee
Apparatus used
Digital Stewart-Gee apparatus, AC power supply 2-12V, AC ammeter 0-2A, Rheostat 100Ω, digital vernier,
and LCR meter. The complete experimental set-up is shown in Figures-3 and 4.
Experimental procedure
The experiment consists of two parts:
Part-A: Determination of permeability of free space (µo)
Part-B: Variation of magnetic field along the axis of the coil
The number of turns and diameter of the field coil are noted as
Number of turns in the field coil, N = 50
Diameter of the coil including its thickness, d = 19.5cm
Hence radius of the coil, R = 9.75x10-2m.
Frequency of AC power supply =50Hz
The number of turns, core area and relative permittivity of the search coil are noted from the Stewart- Gee
apparatus as
Number of turns in the search coil (n) = 2751
@.4BC
Relative permeability, µ = @.DD@ =4.42
Area of the search coil core (16.5mm x 16.5mm) = 2.73x10-4 m2
These data are provided by the manufacturer of the Stewart- Gee apparatus.
Lab Experiments 6
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Kamaljeeth Instrumentation and Service Unit
Vol-12, No-3, Sept.-2012 Stewart & Gee
Figure-3: Stewart - Gee apparatus with the search coil
Figure-4: Microcontroller based magnetic field and induced voltage indicator
Part-A: Determination of the permeability of free space (µo)
1. The Stewart-Gee apparatus is connected to the AC power supply in series with a digital AC current
meter and rheostat, as shown in Figure-5. (There is no need to align the instrument along the magnetic
meridian.)
2. The search coil cable is connected to the digital field along the axis of the instrument and it is switched
on.
3. The AC voltage in the power supply is set to 10V. The current through the field coil is adjusted to the
value 0.1A using a rheostat.
Rheostat 100 Ohms
0-2A
Coil
2-12V AC
power supply
Figure-5: Circuit connections
4. The magnetic flux (Bpp) and the induced emf (Vrms) are recorded from the LCD display and are given in
Table-1.
Lab Experiments 7
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Kamaljeeth Instrumentation and Service Unit
Vol-12, No-3, Sept.-2012 Stewart & Gee
Vrms = 0.07V; Bpp = 3.02 Gauss; Brms =
"$$
√
= 1.06 Gauss
The readings obtained are compared with the value obtained from Equation-17
Bpp =
√ 5067
π ! µ0
=
√ F@.@G
π FH@ F GHDF 4.4 F .GIFD@JK
=
@.DLB
H D.4I
= 0.000379 Tesla = 3.79 Gauss
This value is slightly higher than the value (3.02Gauss) read by the meter.
5. The experiment is repeated by varying the field coil current in suitable steps up to the maximum value
of 1A. In each case the flux and Vrms values are recorded in Table-1 and verified vis a vis the value
obtained from Equation-17.
6. A graph is drawn taking field coil current along the X-axis and the magnetic flux along the Y-axis, as
shown in Figure-6. The slope of the straight line is given by
4.C S
Slope = G.C S @.D = 12.10
In the SI unit, magnetic flux is specified in Tesla, therefore converting flux into Tesla unit, the value of
slope becomes
D .D
Slope = D@@@@ = 12.1V10W4
µo =
"
µ0
=
FL.GHFD@J
[email protected]
12.1X10W4 = 1.06X10WC
This value is about 15% lower than the standard value of µo of free space which is 1.256x10-6.
Table-1
Field coil current Induced emf Vrms
Magnetic Flux (BPP) Gauss
(A)
(V)
(Bpp) Meter
Brms
From Eq.-17
0.070
3.02
1.06
3.97
0.101
0.130
7.18
2.54
7.38
0.208
0.180
10.18
3.60
10.22
0.298
0.250
14.20
5.02
14.20
0.408
0.300
16.30
5.76
17.04
0.500
0.360
20.36
7.20
20.45
0.595
0.450
25.46
9.00
25.56
0.757
0.520
29.52
10.44
29.54
0.868
0.560
31.56
11.16
31.81
0.919
0.560
31.56
11.16
31.56
0.926
0.810
38.64
13.66
39.20
1.137
Variation of magnetic field at the centre of the coil
Lab Experiments 8
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Kamaljeeth Instrumentation and Service Unit
Vol-12, No-3, Sept.-2012 Stewart & Gee
Part-B: Variation of magnetic field along the axis of the coil
In this part of the experiment the search coil is moved along the axis of the coil and magnetic flux is
determined.
7. The current in the field coil is now adjusted to 1A and voltage is set to 12V. The field at the centre of
the coil is recorded from the LCD display as
At the center (Z =0),
Vrms = 0.60V;
Bpp = 33.60 Gauss
The search coil is now moved to the left of the centre by 1cm and the flux and the induced emf are noted
as
Z = 1cm; Vrms = 0.60V; BPP = 33.60 Gauss
Magnetic flux Brms (Gauss)
16
14
12
10
8
6
4
2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
Field coil current (A)
Figure-6: Variation of magnetic field with current at the center of the field coil
8. The experiment is repeated by moving the search coil in steps of 1cm, recording the emf and the flux
each time. The readings obtained are tabulated in Table-2.
9. A graph is drawn with distance along the X-axis and magnetic flux along Y-axis, as shown in Figure-7.
10. A line parallel to the X-axis is drawn where the curvature of the curve changes. These two pints (P and
Q) are known as the inflexion points.
11. The distance between the two inflexion points is noted as
Distance between P and Q = 12cm, which is slightly greater than the diameter of the field coil (9.75cm)
Lab Experiments 9
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Kamaljeeth Instrumentation and Service Unit
Vol-12, No-3, Sept.-2012 Stewart & Gee
Further, the curve is not exactly symmetrical about the vertical axis. This may be due the slight
realignment of the search coil axis.
Results and conclusions
Permeability of free space (µ o) is found to be 1.06x10-6 (= 3.21π x10-7), the standard value being 4π x 10-7.
Table-2
Distance
Induced emf
Magnetic
Distance
Induced emf
Magnetic
from the
Vrms (V)
Field
from the
Vrms (V)
Field
center of the
Bpp (Gauss) center of the
Bpp(Gauss)
field coil
field coil
Z (cm)
Z (cm)
Left hand side of the coil
Right hand side of the coil
0.60
33.60
0
0.61
35.62
0
0.60
33.60
1
0.61
35.62
1
0.60
33.60
2
0.60
33.60
2
0.60
33.60
3
0.56
31.56
3
0.59
33.54
4
0.54
29.58
4
0.57
31.52
5
0.50
28.50
5
0.54
30.54
6
0.47
25.46
6
0.52
28.56
7
0.43
24.48
7
0.48
26.48
8
0.40
22.30
8
0.45
24.44
9
0.36
20.36
9
0.41
22.46
10
0.33
19.34
10
0.38
20.32
11
0.30
16.30
11
0.35
19.34
12
0.28
15.28
12
0.31
17.36
13
0.25
14.20
13
0.29
15.28
14
0.23
12.28
14
0.26
14.26
15
0.21
11.26
15
0.24
12.26
16
0.19
10.14
16
0.22
11.26
17
0.17
9.12
17
0.19
10.14
18
0.16
8.10
18
0.17
9.12
19
0.14
7.14
19
0.16
8.10
20
0.13
6.12
20
Magnetic flux variation along the axis of the coil
Lab Experiments 10
Kamalj
Kamaljeeth Instrumentation and Service Unit
Vol-12, No-3, Sept.-2012 Stewart & Gee
Magnetic Flux Bpp (Gauss)
40
35
Q
30
P
25
20
15
10
5
0
-25
-20
-15
-10
-5
0
5
10
15
20
25
Distance from the center of the field coil
Figure-7: Magnetic flux variation along the axis of the coil
Based on this work, the following conclusions may be drawn:
a. The search coil method of measuring magnetic field is found to be quite accurate, as evident from the
first part of the experiment.
b. The permeability of free space is measured to an accuracy of about 10%.
c. The digital microcontroller based measurement avoids lengthy calculations involved in determination of
the magnetic flux as per Equation-3.
d. Based on this, it is planned to design experiments in future to determine permeability of free space using
core of different materials in the search coil.
e. The microcontroller based instrument is found to be more accurate and reliable.
Reference
[1]
http://physics.nyu.edu/~physlab/GenPhysII_PhysIII/MagFieldCoil.pdf
[2]
Dr Jeethendra Kumar P K and Dr J Uchil, Magnetic Hysteresis, LE Vol-6, No-4, Page-296, 2006.
Lab Experiments 11
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Kamaljeeth Instrumentation and Service Unit