Magnetic field of single coils/ 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 1 Cobra4 Wireless Manager 12600-00 2 Cobra4 Wireless-Link 12601-00 1 Cobra4 Sensor-Unit Tesla 12652-00 1 Cobra4 Sensor-Unit Motion 12649-00 1 Cobra4 Sensor-Unit Electricity 12644-00 1 Holder for Cobra4 with support rod 12680-00 1 Induction coil, 300 turns, d = 40 mm 11006-01 1 Induction coil, 300 turns, d = 32 mm 11006-02 1 Induction coil, 300 turns, d = 25 mm 11006-03 1 Induction coil, 200 turns, d = 40 mm 11006-04 1 Induction coil, 100 turns, d = 40 mm 11006-05 1 Induction coil, 150 turns, d = 25 mm 11006-06 1 Induction coil, 75 turns, d = 25 mm 11006-07 1 Conductors, circular, set 06404-00 1 Hall probe, axial 13610-01 1 Power supply, universal 13500-93 1 Distributor 06024-00 1 Meter scale, demo. l =1000 mm 03001-00 1 Barrel base PHYWE 02006-55 1 Support rod PHYWE, square, l = 250 mm 02025-55 1 Right angle clamp PHYWE 02040-55 1 Lab jack, 200 230 mm 02074-01 1 Reducing plug 4 mm/2 mm socket, 2 11620-27 1 Connecting cord, l = 500 mm, red 07361-01 1 Connecting cord, l = 500 mm, blue 07361-04 1 Bench clamp PHYWE 02010-00 1 Stand tube 02060-00 1 Screen, metal, 300 300 mm 08062-00 1 Software measure for Cobra4 14550-61 Additionally required PC with USB interface, Windows XP or 1 higher Fig. 1: Experimental set-up. www.phywe.com P2430260 PHYWE Systeme GmbH & Co. KG All rights reserved 1
TEP Magnetic field of single coils/ 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 µ 0. 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 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. 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) - same density of turns and length but different diameter (Fig. 4) Fig. 2: Saving measurements. - same length and diameter but different density of turns (Fig. 5) The plots may look as the following diagrams: Fig.. 3: Dependance on coil length of the magnetic field with same density for 1200 ma current and 41 mm coil diameter. 2 PHYWE Systeme GmbH & Co. KG All rights reserved P2430260
Magnetic field of single coils/ TEP Fig.. 4: Independence on coil diameter of field strength with 1200 ma current and 165 mm coil length. Fig.. 5: 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 (amount and direction) at a certain emission point when a point charge and its position is given www.phywe.com P2430260 PHYWE Systeme GmbH & Co. KG All rights reserved 3
TEP Magnetic field of single coils/ Biot-Sarvat s law (2) determines the magnetic field strengths (amount and direction) emission point when a point charge moves at point with velocity at a certain 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 can be written as I d where d denotes the infinitesimal line element along the line shaped conductor at the point. 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 lie in and d is perpendicular to the plane of drawing d must also lie in the plane of drawing perpendicular to the vector. Resolving d in axial and in radial components than yields (compare Fig. 11) and 4 PHYWE Systeme GmbH & Co. KG All rights reserved P2430260
Magnetic field of single coils/ TEP Integration of the axial components dh z over the whole current loop regarding results in and The integral over the radial components dh r 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) To verify the linear dependency of B (0) on n and from the experiment the ansatz and the ansatz is used. The regression line for the measured values in Fig. 7 gives for n the dependency the exponent E 1 = 0.96±0.04 and the regression line in Fig. 8 for the R dependency the exponent E 2 = -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 www.phywe.com P2430260 PHYWE Systeme GmbH & Co. KG All rights reserved 5
TEP Magnetic field of single coils/ μ 0 = (1.28±0.01) 10-6. This value is in good agreement with the literature value = 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 PHYWE Systeme GmbH & Co. KG All rights reserved P2430260