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 2 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 2 1 1 1 2 1 1 1 1 1 1 1 1 1 1 2 1 1 1 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 1 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. 2 P2430215 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 3 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. 4 P2430215 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 5 LEP 4.3.02 -15 6 Magnetic field of single coils / Biot-Savart’s law P2430215 PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH • 37070 Göttingen, Germany