LEP
4.3.02
-15
Magnetic field of single coils / Biot-Savart’s law
Related topics
Wire loop, Biot-Savart’s law, Hall effect, magnetic field, induction, magnetic flux density.
Principle
The magnetic field along the axis of wire loops and coils of different dimensions is measured with a Cobra3 tesla measuring
module and a Hall probe. The relationship between the maximum field strength and the dimensions is investigated and a
comparison is made between the measured and the theoretical effects of position.
Equipment
Induction coil, 300 turns, d = 40 mm
Induction coil, 300 turns, d = 32 mm
Induction coil, 300 turns, d = 25 mm
Induction coil, 200 turns, d = 40 mm
Induction coil, 100 turns, d = 40 mm
Induction coil, 150 turns, d = 25 mm
Induction coil, 75 turns, d = 25 mm
Conductors, circular, set
Hall probe, axial
Power supply, universal
Distributor
Meter scale, demo, l = 1000 mm
Barrel base -PASSSupport rod -PASS-, square, l = 250 mm
Right angle clamp -PASS-
11006.01
11006.02
11006.03
11006.04
11006.05
11006.06
11006.07
06404.00
13610.01
13500.93
06024.00
03001.00
02006.55
02025.55
02040.55
1
1
1
1
1
1
1
1
1
1
1
1
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1
2
G-clamp
Lab jack, 200230 mm
Reducing plug 4 mm/2 mm socket, 2
Connecting cord, l = 500 mm, blue
Connecting cord, l = 500 mm, red
Bench clamp -PASSStand tube
Plate holder
Silk thread, l = 200 m
Weight holder 1 g
Cobra3 Basic Unit
Cobra3 Force/Tesla Software
Tesla measuring module
Cobra3 sensor, 6 A
Movement sensor with cable
Adapter, BNC-socket/4 mm plug pair
Adapter, BNC-socket - 4 mm plug
Power supply, 12 VRS232 cable
PC, Windows® 95 or higher
02014.00
02074.01
11620.27
07361.04
07361.01
02010.00
02060.00
02062.00
02412.00
02407.00
12150.00
14515.61
12109.00
12126.00
12004.10
07542.27
07542.20
12151.99
14602.00
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Tasks
1. Measure the magnetic flux density in the middle of various
wire loops with the Hall probe and investigate its dependence on the radius and number of turns.
2. Determine the magnetic field constant m0.
3. Measure the magnetic flux density along the axis of long
coils and compare it with theoretical values.
Fig. 1: Experimental set-up for measuring a magnetic field.
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
P2430215
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LEP
4.3.02
-15
Magnetic field of single coils / Biot-Savart’s law
Set-up and procedure
Connect the “Tesla” module to the “Modul” port of the Cobra3
unit and to the Hall probe. Connect the “6 A-Sensor” to the
“Analog In 2/S2” port of the Cobra3 unit. Connect the
“Movementrecorder” to the Cobra3 unit according to Fig. 2.
Click the “Options…” button and run the calibrations on the
“Angle / Distance” and “Calibration” chart. For good distance
accuracy wind the silk thread one time around the axis of the
movement recorder. You may use a slightly heavier weight for
the thread’s tension in this experiment if the provided weight
is insufficient to make the thread drive the recorder correctly.
red
black
yellow
BNC1
BNC2
Fig. 2. Connection of the movement sensor to the Cobra3
Basic Unit
Connect the Cobra3 unit to your computer to port COM1,
COM2 or to USB port (for USB computer port use USB to
RS232 Converter 14602.10). Set up the experiment according
to Fig. 1 and start the “measure” program on your computer.
Select the “Gauge” “Cobra3 Force / Tesla”. Set the parameters according to Fig. 3.
Fig. 3: Parameter setting for the Force / Tesla gauge.
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Fig. 4: The “Calibration” chart
Fig. 5: The “Angle / Distance” chart
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
LEP
4.3.02
-15
Magnetic field of single coils / Biot-Savart’s law
The settings on the “Flux density” chart should be as seen on
Fig. 6.
With the “Continue” button the measurement can then be
started. Select an appropriate current e.g. the maximum current indicated on the coils using the power supply as a constant current supply. The power supply is in the constant current mode when the red LED above the current control is on.
Set the voltage control sufficiently high as to achieve this. Else
the power supply is in the constant voltage mode and the current will decrease with the warming of the coils and this may
disturb your measurement. 1200 mA may be chosen for all the
solenoid coils. Once you have adjusted the current, you may
leave the current control untouched so as to measure all the
coils with the same current. But do turn down the voltage
before you break the circuit unplugging the coils to avoid
spikes (!).
Data recording is then started with either the “return” or
“space” key or clicking the “start measurement” button.
Measure the magnetic field strength in the centre of the circular conductors e.g. with 5 A currrent strength. Asymmetry in
the set-up and interference fields may be eliminated by measuring the changes in field strength when turning on the power
with both polarizations of current and taking the average value
of the change for each polarization.
Fig. 6: “Flux density” chart settings
And the “Voltage / Current” settings should look like Fig. 7.
Measure the magnetic field strength along the z-axis of the
solenoid coils sliding the Hall probe mounted to a barrel base
along the meter and recording the position with the movement
sensor. If you keep the barrel base sliding on just one edge of
the meter, you can achieve a fairly straight movement through
the centre of the coils.
Plot the results for
– same diameter and denstiy of turns but different length of
coil (Fig. 8)
- same density of turns and length but different diameter
(Fig. 9)
- same length and diameter but different density of turns
(Fig. 10)
The plots may look as the following diagrams:
Fig. 7: “Voltage / Current” settings
Fig. 8: Dependance on coil length of the magnetic field with
same density of turns for 1200 mA current and 41 mm
coil diameter
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
P2430215
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LEP
4.3.02
-15
Magnetic field of single coils / Biot-Savart’s law
For application of (2) to the present experiment the following
experimental constraints must be considered:
1. The geometry of the experiment as shown in Fig. 11.S
2. For a current I through
a line shaped
conductor Q · n can
S
S
be written as I · d rQ where d rQ denotes the infinitesimal
line element along the line shaped conductor at the point
S
rQ.
3. In the experiment only the magnetic field along the z-axis is
of interest.
Formula (2) can those expressed in the form
S
r
m0 · I d l x S
S
dB .
3
4p
r
(3)
S
Fig. 9: Independence on coil diameter of field strength with
1200 mA current and 165 mm coil length
Due toS the properties of the cross product and since
r lie in
S
dH must also
and d l is perpendicular to the plane of drawing S
lie in the plane
of drawing perpendicular to the r vector.
S
Resolving dH in axial and in radial components than yields
(compare Fig. 11)
dBz m0I dl
·
· sin 1g2
4p r2
(4)
dBr m0I dl
·
· cos 1g2 .
4p r2
(5)
and
Fig. 10: Linear dependance on number of turns of field
strength for 1200 mA current and 26 mm coil diameter
Theory and evaluation
Part I: Magnetic field of wire loops
Biot-Sarvat’s law is the magnetostatic analogue to Coulomb’s
law in electrostatic.
Coulomb’s law (1) determines the electric field strengths
S
S
E 1S
r 2 (amount and direction) at a certain emission point r
S
when a point charge and its position rQ is given
S
r rS
1
Q
S
E 1S
r 2
Q S
3
4pe0
0 r rS
Q 0
(1)
Integration of the axial components dHz over the whole curR
rent loop regarding r 2R2 z2 and sin (g) 2R2 z2
results in
Bz 1z2 m0 I
R2
· 2
.
2
1R z2 2 3>2
(6)
The integral over the radial components dHr vanish since the
components cancel each other due to symmetry reasons.
If n identical loops are close together the magnetic flux density is obtained by multiplying (6) with the number of turns n.
At the centre of the loop (z = 0)
B102 m0 · n · I
2R
(7)
is obtained.
Biot-Sarvat’s law (2) determines the magnetic field strengths
S
S
(amount and direction) B 1 S
r 2 at a certain emission point r
S
S
when a point charge moves at point rQ with velocity n
S
r rS
n x1S
m0
Q 2
S
B 1S
r 2 Q
.
3
S
S
4p
0 r rQ 0
(2)
For several point charges the field strengths (electric and magnetic) at the emission point is the superposition of the contributions of the different point charges.
(1) and (2) can be derived directly from Maxwell’s equations
and can be extended to charge density or current density distributions, respectively.
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Fig. 11: Drawing for the calculation of the magnetic field along
the axis of a wire loop.
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
LEP
4.3.02
-15
Magnetic field of single coils / Biot-Savart’s law
To verify the linear dependency of B (0) on n and
experiment the ansatz
B = A1 · nE1
1
R
from the
(8)
and the ansatz
Part II: Magnetic field along the axis of a (long) coil
The calculation of the magnetic flux density on the axis of a
uniformly wound coil of length l and with n turns yields the
result
B1z2 E2
B = A2 · R
(9)
is used.
The regression line for the measured values in Fig. 12 gives for
n the dependency the exponent
E1 = 0.96±0.04.
and the regression line in Fig. 13 for the R dependency the
exponent
E2 = -0.97±0.04.
Those the experimental data confirm the theoretical expected
form of a linear dependency.
The slope of the linear dependency can be used to determine
the magnetic field constant.
From the experimental data follows the value
m0 = (1.28±0.01) · 10-6.
zl> 2
zl> 2
m0·I· n
b
· a
2l
2R21zl> 2 2 2 2R21zl> 2 2 2
(10)
For the middle of the coil, z = 0 follows
B 102 m0 · I · n
l
·
.
2
2l
2R l2> 4
For a long coil (l >> R), a solenoid, the upper equation finally
reduce to
B 102 m0 · I · n
.
l
Therefore the magnetic field strength is for solenoids independent from the coil diameter.
The independence on the coil diameter can directly be seen in
Fig. 9 whereas the dependence on number of turns is shown
in Fig. 10.
This value is in good agreement with the literature value mLit.
0
= 1.257 · 10-6.
Plot B (z) of equation (10) with data of the used solenoid coil
with 41 mm and compare with the measured results.
Fig. 12: Magnetic flux density at the centre of a coil with n
turns, as a function of the number of turns (radius
6 cm, current 5 A).
Fig. 13: Magnetic flux density at the centre of a single turn, as
a function of the radius (current 5 A).
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH • 37070 Göttingen, Germany
P2430215
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LEP
4.3.02
-15
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Magnetic field of single coils / Biot-Savart’s law
P2430215
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH • 37070 Göttingen, Germany