Electricity Magnetostatics The effects of force in a magnetic field LEYBOLD Physics Leaflets Measuring the force on current-carrying conductors in a homogeneous magnetic field Recording with CASSY Objects of the experiment To measure the force on a current-carrying conductor in a homogeneous magnetic field as a function of the current for various conductor lengths and conductor shapes. To determine the proportionality factor between force and current as a function of the conductor length. To measure the magnetic field. 0808-Wit Principles Magnetic induction, or more simply the magnetic field B, is a vectorial quantity. A force F acts on a charge q passing through a magnetic field B with a velocity v; the size of the force depends on the strength and direction of the magnetic field. We can say: F = q (v B) (I) The Lorentz force F is also a vectorial quantity, and is perpendicular to the plane defined by v and B. We can understand the force acting on a current-carrying conductor in a magnetic field as the sum of the individual forces acting on the moving charge carriers which make up the current. In accordance with (I), the Lorentz force F acts on every single charge carrier q moving with the drift velocity v. For a straight conductor, this gives us the total force F = q nas (v B) (II) as the number of charge carriers in the conductor is the product of the density n of the charge carriers, the conductor cross-section A and the length s of the section of the conductor within the magnetic field. It is common to introduce the vector s, which points along the direction of the conductor segment. Also, the product qnav is equivalent to the current I. Thus, the force of a magnetic field on a straight, current-carrying conductor section is defined by F = I (v B) (III) and the absolute value of the force by F = I s B (IV) when s and B are perpendicular to each other. The force F and the current I are thus proportional to each other, and the proportionality factor is a s B (V). In this experiment, rectangular conductor loops are placed in a horizontal magnetic field. The aim is to measure the force acting on the horizontal segment of each conductor loop. The forces acting on the vertical segments cancel each other out. The magnetic field is generated using two coils and a U-core with pole-shoe yoke. The pole-shoe yoke consists of two soft iron blocks whose spacing can be adjusted by inserting aluminum spacers. The conductor loops are placed in the air gap of the pole-shoe yoke. A force sensor is attached to the conductor loops to measure the force. This contains a bending element with a strain gauge in which the electrical resistance varies in response to mechanical stress. The change in resistance is proportional to the force acting on the sensor. The force sensor can thus be connect to the CASSY computer-assisted measuring system via a bridge box. As the currents through the conductor loops can reach values of up to 20 A, a 30-ampere box is connected to CASSY to permit current measurement. 1
LEYBOLD Physics Leaflets Apparatus 1 Pole-shoe yoke............... 562 25 1 U-core with yoke.............. 562 11 2 Coils with 500 turns............. 562 14 1 Force sensor................ 314 261 1 Conductor loops for force measurement, set 516 34 1 Conductor loop holder........... 314 265 1 CASSYpack-E................ 524 007 1 Disk: Measuring and Evaluating...... 524 112 1 Bridge box................. 524 041 1 Multicore cable, 6-pole, 1.5 m long.... 501 16 1 30-ampere box............... 524 043 1 DC power supply, I 20 A, e.g. high current power supply........ 521 55 1 DC power supply, I 5 A, e.g. AC/DC power supply 0 15 V..... 521 50 1 Stand base, V-shape, 20 cm........ 300 02 1 Stand rod, 47 cm.............. 300 42 1 Leybold multiclamp............. 301 01 Connecting leads with conductor cross-section 2.5 mm 2 Additionally required: PC with MS-DOS 3.0 or higher Setup Notes: As the measurement quantity is very small, the measurement is easily affected by ambient disturbances: Avoid all shocks, drafts and temperature variations in the vicinity of the experiment. Subject the conductor loop holder and the conductor loops only briefly (no more than a few minutes) to loads of 20 A. Set up the experiment as shown in Fig. 1. Electromagnet: The coils must be connected properly in order for the electromagnet to function. Slide the coils onto the U-core as shown in Fig. 1 (the sockets are then located on the bottom edge of the coils.). Position the pole-shoe yoke on the U-core and set the air gap by placing two 3 mm thick spacers (b) on each side. Loosen the knurled screws (a) of the supports on all sides of the pole-shoe yoke, so that the yoke rests evenly on the U-core. The retighten the knurled screws. Connect socket A of both coils together and connect socket E of both coils to the DC voltage output of the AC/DC power supply as shown in Fig. 1. Fig. 1 Experiment setup for measuring the force on current-carrying conductors in a homogeneous magnetic field 2
LEYBOLD Physics Leaflets Conductor loop: Assemble the force sensor (e), the conductor loop holder (d) and the 8 cm wide conductor loop (c) as shown in Fig. 1 and mount this assembly using stand material. Mount the conductor loop in the air gap of the pole shoe yoke and align it parallel to the soft iron blocks. To prevent short-circuiting, make sure that the uninsulated cable segments of the conductor loop holder do not touch either the conductor loop or each other. CASSY: Plug in the bridge box (g) at CASSY input A and the 30-ampere box (f) at CASSY input D (see Fig. 1). Connect the force sensor to the bridge box using the 6-conductor cable. Connect the 30-ampere box in series in the circuit for supplying the conductor loops. Measuring and Evaluating The CASSY software Measuring and Evaluating lets you select virtually all menus and options using the function keys and cursor keys. Start Measuring and Evaluating, activate program selection with F1, select the program module Multimeter and press F3 to open Select meas. quantities. Select each of the following menus using the cursor keys and press Enter to activate confirm each selection. Quantities: Reselect channel A Quantity A: Force Range A: 1.. 1 N Quantities: Reselect channel D Quantity D: Current Range D: 30.. 30 A Press F1 to activate the measuring screen. For the zeropoint compensation in the following steps, read off and write down the displayed current value I 0. Zero point compensation: Press ESC to return to the main menu, and there activate Select meas. quantities with F3. Select each of the following menus using the cursor keys and press Enter to activate confirm each selection. Quantity: Reselect channel D Quantity D: Calibrate In the window Calibrate D which subsequently appears, enter the following and press Enter after each entry: Name of meas. quantity: Current Physical symbol of meas. quantity: I Physical unit of meas. quantity: A Factor: 1 A/A Offset: here, enter the value for I 0 which you wrote down previously, but reverse the sign. In the subsequent submenu Range D, select the top measuring range and confirm with Enter. Switch to the measuring screen with F1 and check the zero point compensation for current. Carrying out the experiment Before switching on the power supplies, check to make sure that all current and voltage knobs are turned to their extreme left position. After switching the power supplies on, turn voltage knobs (h) and (k) to the exteme right position. The easiest way to regulate the supply currents to the conductor loops and the coils is to use only the current knobs (i) and (f). Record the respective values on the computer manually by pressing F1 each time. If you register a negative force value F after setting the current through the conductor loop, reverse the connecting leads at the force sensor. To compensate the zero point for the force F, press the A key. Switch on the AC/DC power supply and set the coil current I B = 2.5 A. The coils heat up after a few minutes, which changes their electrical resistance. Correct the coil current as necessary. Switch on the high current power supply, set the current I = 2 A and record the measured values by pressing F1 each time. Increase the current to 20 A in steps of 2 A and record the corresponding measured values by pressing F1 for each step. Turn down the current to I = 0 A and return to the main menu with ESC. Press F8 to activate the menu Disk operations, select the option Save meas. data and save your measured values under a suitable file name. Attach the 4 cm wide conductor loop to the force sensor in place of the 8 cm loop. Switch to the measuring screen with F1, read off the zero point of the current I 0 and correct any deviation as described under Zero point compensation. Take the previous compensation into consideration. To compensate the zero point for the force F, press the A key. Beginning at I = 2 A, increase the current to 20 A in steps of 2 A and record the corresponding measured values by pressing F1 for each step. Reduce the current to I = 0 A, return to the main menu with ESC and save your measurement series as described above. Repeat the measurements with the 2 cm and 1 cm conductor loops, and save each measurement series under a different name. 3
LEYBOLD Physics Leaflets Measuring example CASSY tables Table 1: Measurement using the first conductor loop (s = 8 cm) 1 0.028 N 2.11 A 2 0.054 N 4.06 A 3 0.081 N 6.12 A 4 0.108 N 8.10 A 5 0.136 N 10.18 A 6 0.162 N 12.16 A 7 0.192 N 14.35 A 8 0.217 N 16.23 A 9 0.245 N 18.31 A 10 0.271 N 20.28 A Table 3: Measurement using the third conductor loop (s = 2 cm) 1 0.007 N 2.12 A 2 0.014 N 4.14 A 3 0.021 N 6.24 A 4 0.028 N 8.19 A 5 0.034 N 10.19 A 6 0.042 N 12.28 A 7 0.049 N 14.19 A 8 0.056 N 16.19 A 9 0.063 N 18.28 A 10 0.070 N 20.26 A Table 2: Measurement using the second conductor loop (s = 4 cm) 1 0.014 N 2.10 A 2 0.028 N 4.10 A 3 0.041 N 6.09 A 4 0.055 N 8.13 A 5 0.069 N 10.17 A 6 0.083 N 12.20 A 7 0.096 N 14.22 A 8 0.110 N 16.21 A 9 0.124 N 18.27 A 10 0.138 N 20.28 A Table 4: Measurement using the fourth conductor loop (s = 1 cm) 1 0.003 N 2.18 A 2 0.006 N 4.07 A 3 0.010 N 6.06 A 4 0.013 N 8.22 A 5 0.017 N 10.19 A 6 0.020 N 12.22 A 7 0.023 N 14.25 A 8 0.027 N 16.23 A 9 0.031 N 18.25 A 10 0.034 N 20.26 A 4
LEYBOLD Physics Leaflets Evaluation In the main menu, open Disk operations with F8. In the submenu Disk, select the option Multigraph on and then Load meas. series. You need to press Enter to confirm each selection. In the subsequent file selection box, select the measurement series for the 8 cm, 4 cm, 2 cm and 1 cm conductor loops one after another using the cursor keys and Enter. Be sure to select them in that order. Press ESC to return to the main menu and select Evaluate in graph with F6. In the menu Select representation, select Common diagram and press Enter. In the diagram which subsequently appears, you can fit a straight line through the origin to all four measurement series by pressing Shift+F1 (see Fig. 2). Exit the multigraph with Shift+F9 and enter the following under Transfer slopes and parameters for XYZ-input (be sure to press Enter each time). Name of slope: Slope Physical symbol of slope: a Physical unit of slope: N/A No. of decimal places of slope: 4 Name of parameter: Length Physical symbol of parameter: s Physical unit of parameter: m No. of decimal places of parameter: 2 The program now displays a table with the heading Slopes and parameters for XYZ-input. Here, enter the missing parameters s: 0.08 m <ENTER>, 0.04m <ENTER>, 0.02m <ENTER> and 0.01 m <ENTER> The program automatically returns you to the multigraph screen; from there, switch to the main menu with ESC. In the main menu, press F6 Evaluate in graph, fit a straight line through the origin by pressing F1, then press the key combination Alt+F1 to display the slope of the straight line as a numerical value on the screen (see Fig. 3). Table 5: Proportionality factor a as a function of the conductor length s n s a 1 0.08 m 0.0134 N/A 2 0.04 m 0.0068 N/A 3 0.02 m 0.0034 N/A 4 0.01 m 0.0017 N/A Results The force F of a current-carrying conductor in the magnetic field is proportional to the current I for a given length s (see Fig. 2). The proportionality factors a are in turn proportional to the conductor length s (see Fig. 3). When we insert the slope into the formula a = s B(V) we obtain a value of B = 0.17 T for the magnetic field. Additional information The set of conductor loops includes two 4 cm conductor loops which form two partial rectangles or one complete rectangle. These conductor loops can be used in the experiment setup described here to show that the forces in two parallel conductor segments in a magnetic field in which the current is flowing in opposite directions cancel each other out. The force on the open conductor loop corresponds to the force on the 2 cm wide conductor loop, the force on the completely closed conductor loop is zero. Fig. 2 CASSY multigraph with the F(I) diagrams for the conductor lengths s = 8 cm, s = 4 cm, s = 2 cm und s = 1 cm. Fig. 3 CASSY graph showing the proportionality factors a as a function of the conductor length s. LEYBOLD DIDACTIC GMBH Leyboldstrasse 1 D-50354 Hürth Phone (02233) 604-0 Telefax (02233) 604-222 Telex 17 223 332 LHPCGN D by Leybold Didactic GmbH Printed in the Federal Republic of Germany Technical alterations reserved