Magnetic field of single coils/
Biot-Savart’s law with Cobra4
TEP
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 Cobra4 Sensor-Unit Tesla 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
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Cobra4 Wireless Manager
Cobra4 Wireless-Link
Cobra4 Sensor-Unit Tesla
Cobra4 Sensor-Unit Motion
Cobra4 Sensor-Unit Electricity
Holder for Cobra4 with support rod
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
Hall probe, axial
Power supply, universal
Distributor
Fig. 1:
12600-00
12601-00
12652-00
12649-00
12644-00
12680-00
11006-01
11006-02
11006-03
11006-04
11006-05
11006-06
11006-07
13610-01
13500-93
06024-00
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Meter scale, demo. l =1000 mm
03001-00
Barrel base PHYWE
02006-55
Support rod PHYWE, square, l = 250 mm 02025-55
Right angle clamp PHYWE
02040-55
Lab jack, 200×230 mm
02074-01
Reducing plug 4 mm/2 mm socket, 2
11620-27
Connecting cord, l = 500 mm, red
07361-01
Connecting cord, l = 500 mm, blue
07361-04
Bench clamp PHYWE
02010-00
Stand tube
02060-00
Screen, metal, 300×300 mm
08062-00
Software measure for Cobra4
14550-61
Additionally required
PC with USB interface, Windows XP or
1
higher
Experimental set-up.
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Magnetic field of single coils/
Biot-Savart’s law with Cobra4
TEP
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 µ0.
3. Measure the magnetic flux density along the axis of long coils and compare it with theoretical values.
Set-up and procedure
Connect the Sensor-Unit Electricity to one Wireless-Link. 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 (!).
Now connect the Sensor-Unit Tesla to this Wireless-Link and to the Hall probe. Connect the Sensor-Unit
Motion to the other Wireless-Link.
Set up the experiment according to Fig. 1, start the “measure” program on your computer and load the
“Biot-Savart’s law” experiment. (Experiment > Open experiment). All pre-settings that are necessary for
measured value recording are now carried out.
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 motion 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.
Click on
in the icon strip to start measurement and slide the Hall probe
along the meter for about 40 cm. Click on the
icon in the icon strip to
end measurement and send the data to measure (Fig. 2).
Plot the results for
- same diameter and denstiy of turns but different length of coil (Fig. 3)
Fig. 2: Saving measurements.
- same density of turns and length but different diameter (Fig. 4)
- same length and diameter but different density of turns (Fig. 5)
The plots may look as the following diagrams:
Fig.. 3:
2
Dependance on coil length of the magnetic field with same density for 1200 mA
current and 41 mm coil diameter.
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Magnetic field of single coils/
Biot-Savart’s law with Cobra4
Fig.. 4:
Independence on coil diameter of field strength with 1200 mA current and
165 mm coil length.
Fig.. 5:
Linear dependence on number of turns of field strength for 1200 mA current
and 26 mm coil diameter.
TEP
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 𝐸⃗ (𝑟) (amount and direction) at a certain emission point 𝑟 when a point charge and its position 𝑟𝑄 is given
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Magnetic field of single coils/
Biot-Savart’s law with Cobra4
TEP
⃗ (𝑟)at a certain
Biot-Sarvat’s law (2) determines the magnetic field strengths (amount and direction) 𝐵
emission point 𝑟 when a point charge moves at point 𝑟𝑄 with velocity 𝜈
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.
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. 6.
2. For a current I through a line shaped conductor Q · 𝜈 can be written as I · d 𝑟𝑄 where d 𝑟𝑄 denotes the
infinitesimal line element along the line shaped conductor at the point 𝑟𝑄 .
3. In the experiment only the magnetic field along the z-axis is of interest.
Fig. 6:
Drawing for the calculation of the magnetic field along the axis of
a wire loop.
Formula (2) can those expressed in the form
Due to the properties of the cross product and since 𝜌 lie in and d𝑙 is perpendicular to the plane of draw⃗ must also lie in the plane of drawing perpendicular to the 𝜌 vector.
ing d𝐻
⃗ in axial and in radial components than yields (compare Fig. 11)
Resolving d𝐻
and
4
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Magnetic field of single coils/
Biot-Savart’s law with Cobra4
TEP
Integration of the axial components dHz over the whole current loop regarding 𝜌 = √𝑅 2 + 𝑧 2 and sin(𝛾) =
𝑅
results in
2
2
√𝑅 +𝑧
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)
1
To verify the linear dependency of B (0) on n and 𝑅 the ansatz
and the ansatz
is used.
The regression line in Fig. 7 gives for n the dependency the exponent
E1 = 0.96±0.04
and the regression line in Fig. 8 for the R dependency the exponent
E2 = -0.97±0.04.
Fig. 7:
Magnetic flux density at the centre of a coil with n
turns, as a function of the number of turns (radius 6cm,
current 5 A).
Fig. 8:
Magnetic flux density at the centre of a single
turn, as a function of the radius (current 5 A).
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
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TEP
Magnetic field of single coils/
Biot-Savart’s law with Cobra4
μ0 = (1.28±0.01) · 10-6.
This value is in good agreement with the literature value
𝜇0𝐿𝑖𝑡. = 1.257 · 10-6.
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
For the middle of the coil, z = 0 follows
For a long coil (l >> R), a solenoid, the upper equation finally reduce to
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. 5 whereas the dependence on number of turns is shown in Fig. 6.
Plot B (z) of equation (10) with data of the used solenoid coil with 41 mm and compare with the measured results.
6
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