-1 - Introduction PHY102/112 Lab 10: Measurement of a Magnetic Field 1994-2007, James J. DeHaven, Ph.D. An Electric Current Produces a Magnetic Field Since the nineteenth century, it has been known that a current-carrying wire produces a magnetic field. For example, when a compass needle is placed near a current-carrying electric wire, the needle deflects. A Magnetic Field Exerts a Force on a Current-Carrying Wire A consequence of the above observation is that a magnet will exert a force on a wire through which current is flowing, and two current-carrying wires will exert a force on one another. Newton s third law states that whenever on object exerts a force on a second object, the second object exerts an equal and opposite force on the first. Hence, a magnetic field exerts a force on a currentcarrying wire. I B B Figure 1: Magnetic field produced by current carrying wire In today s lab you will examine the magnetic field in the vicinity of a straight wire and a length of wire wound in a loop. You will measure the strength of a magnetic field resulting from a current flowing through a wire, and then directly measure the magnetic force exerted by one wire on another. Ampere s Law One of the fundamental laws of electromagnetic theory, one which relates the magnetic field in a wire to the current flowing through it is Ampere s law. Before we can discuss the mathematical formulation of this law, however, we need to examine the directional properties of a magnetic field produced by a current flowing in a wire. In general, an electric current, will produce a magnetic field which is aligned in a cylindrical fashion around the conductor, as shown in figure 1. Note that the field lines point from North to South, in the direction that the North pole of a test magnet would point in the magnetic field. The direction of the current determines the direction of the magnetic field--if the current were flowing in the opposite direction, the magnetic field lines would circle the wire in the opposite direction to that shown in figure 1.
-2 - The method for determining the direction of this magnetic field is known as the righthand rule: If you grasp the wire with your right hand so that your thumb points in the direction of the positive current, then your fingers will encircle the wire in the direction of the magnetic field. This principle is illustrated in figure 2. Note that the symbols used for the current and the magnetic field strength are I and B respectively. I I r B Figure 2: Illustration of the use of the so-called right-hand rule in determining the direction of a magnetic field produced by a current carrying wire Figure 3: Incremental evaluation of the magnetic field in a closed circle around a current-carrying wire Now back to Ampere s law: the mathematical relationship between B and I is usually formulated by considering a small portion of the magnetic field. Imagine, for example, that the magnetic field around a wire could be chopped up into small segments, each a minute length, which we call Δl. Each tiny length has a small portion of the magnetic field associated with it, with a component, B which lies parallel to the tiny length. Amperes law says that the sum of all the segmented contributions to the magnetic field will be related to the current in the wire as follows:
-3 - [1] B Δ l = µ o I In equation [1], B is the magnetic field strength in the direction of Δl, Δl is the increment of length over which B is evaluate, I is the current, and µo is known as the magnetic permeability of free space (in analogy with the electrical permittivity we encountered in electricity). The permeability has a value of 4π x 10-7 Tesla meters/amp. The tesla is the SI unit for magnetic field strength. A good strong refrigerator magnet will have a magnetic field strength of approximately 10 milliteslas. The symbol,, means take the sum of and it indicated that we should sum up the B s all around the loop. If we do that, and make sure that we stay at some distance, r, from the wire, then the Δl s all add up to the circumference of the circle: [2] B( 2πr) = µ o I and we can solve for B as a function of the current: [3] B = µ o I 2π r Another commonly encountered configuration is a loop of wire, or a length of wire wound around a cylinder, like string wound around a spool, forming several consecutive loops. This later arrangement is called a solenoid, and there can be a substantial magnetic field generated inside of it. The field strength inside a solenoid can be derived from Ampere s law, and will depend on the number of turns of wire per unit length around the solenoid, N/l, and on the current in the wire, I, as follows: [4] B = µ N o l I Therefore, it should be possible to calculate the field inside a solenoid if you know the value of the current running through it, the number of turns of wire wrapped around it, and the length of the solenoid.
-4 - The direction of the magnetic field produced by a current-carrying loop is determined again by using the righthand rule. Point the thumb in the direction of I, and curl the fingers through the center of the loop. The direction in which the fingers point will be the the direction of B. I B Figure 4: Illustration of the use of the so-called right-hand rule in determining the direction of a magnetic field produced by a current-carrying wire loop As noted above, Newton s third law requires that a wire carrying a current will experience a force from other sources of magnetic fields. Consider, for the sake of argument, a wire, carrying current, in a uniform magnetic field produced by some magnet external to the wire. The wire makes an angle, θ, with the field lines as shown in the following diagram (Fig. 9): l θ Figure 5: Wire in a magnetic field
-5 - [5] The force on the wire will be given by: where l is the length of the wire that lies within the field, I is the current, B is the field strength, and θ is the angle between the wire and the field as illustrated in figure 5. If the wire is perpendicular to the field, then the equation [5] takes on a particularly simple form: [6] If, on the other hand, the wire is parallel to the field lines then the force on the wire is zero. In part one of this lab, you will connect a wire to a battery so that current flows through it. Then you will put a compass near the current-carrying wire and and observe the deflection of the compass needle. In other words, you will observe that a current-carrying wire exerts a force on a magnetic field. You will then use the compass to map the field lines in the vicinity of a solenoid. In part two of this lab, you will predict the value of the magnetic field inside a solenoid and then measure this value. In part three of this lab, you will measure the force a magnetic field exerts on a currentcarrying wire. Experimental F = l I Bsinθ F = l I B max IMPORTANT: DO NOT LOCATE THE SOLENOID TOO CLOSE TO THE COMPUTER. YOU WILL BE GENERATING A FAIRLY LARGE MAGNETIC FIELD WITH IT. Part I: A Current-Carrying Wire Exerts a Force on a Compass Using alligator clips, hook up a wire to a DC power source. Bring a compass near the wire and observe the effect of the current-carrying wire on the compass needle. Is the deflection the same amount in all directions? Part II: Mapping the magnetic field near a solenoid ( l B ) Hook up a solenoid to a DC power supply, and an ammeter in series. Put a piece of business sized paper under the solenoid and trace out approximately the outline (or shadow) of the solenoid. Now move a compass in approximately 1 cm increments around the solenoid. Start at one end and always move in the direction in which the north pole of the compass points. Use a pencil to mark x s or dots at the center of where the compass has been. If you do this carefully, you will map the magnetic field of the solenoid. Do at least three field lines roughly 10, 15 and 20 cm from the side of the solenoid (You can start halfway along the side of the tube, moving the solenoid forwards or backwards to map out the field.) A current of 3 Amps should be about right. It is vital that you keep all electrical equipment, to the extent possible, away from your field mapping region--map the field on the side of the solenoid opposite the power supply.
-6 - Part III: A Magnetic Field Exerts a Force on a Current-Carrying Metal Loop Measurement of the Magnetic Field According to manufacturer s specifications, the solenoid contains 570 windings of wire. Calculate the magnetic field inside the solenoid using equation [4]. You will use a Hall effect transistor to measure the field inside the solenoid. The Hall effect describes a phenomenon in which a magnetic field, because it exerts a force on moving charges, can induce an EMF in materials. When this occurs in a transistor, the operating conditions of the semiconductor are substantially altered, and even relatively weak magnetic fields can be detected. A schematic diagram of the Hall probe you will use is shown in figure 6. You will use one of two different models. In the older model, a metallic probe is encased in a length of plastic tubing. The probe appears to be flattened into a paddle at the business end, and a white circular dot is painted on the tip of the probe. For the probe to be effective the flat surface of the paddle must be perpendicular to the magnetic field lines, i.e. the white dot must face so that the field lines go through its surface, not across its surface. This is illustrated in figure 7 on the next page. NEW Vernier MAGNETIC FIELD OLD Figure 6: Hall Effect Probe. The newer model has a hinged end, and the older model is enclosed in clear plastic tubing.
-7 - Figure 7: Illustration of the proper and improper alignment of the older Hall effect probe. The flat surface (with the white dot) should not point towards the side of the solenoid, but should be aligned parallel to the long axis of the solenoid Right Wrong A The newer probe consists of a long black tube, with the Hall transducer mounted in a black cylinder at the end, which can rotate on a hinge (see figure 6). The older transducer is connected to a box whose settings are labeled High and Low. The newer deice has a switch labeled Range, located on the side, which can be set to 6.4 mt or 0.3 mt. Figure 8: Solenoid, DC power supply, and ammeter in circuit used for measurement of magnetic field Plug the Hall probe into port 1 of the interface box connected to your computer. Start the computer and load the experiment file entitled Ex210Field2007.cmbl. Answer connect if the computer complains about needing sensor confirmation. Set up the solenoid, the power supply and the large orange ammeter in a circuit as shown on the left in figure 8. The older Hall effect probe has a small amplifier ( a black box) with a switch. The switch should always be set to low (x10) amplification. The
-8 - newer probe has a Range switch with settings labeled 0.3 mt and 6.4 mt. Set it to 6.4 mt. In order to accurately measure the magnetic field inside the solenoid, the older Hall probe must be inserted at a precise and known angle. This is not a problem with the newer probe because of the hinged measuring head. You simply need to make sure that the end of the probe sits flush against the bottom of the solenoid, and that it is lined up with the longitudinal axis of the solenoid, and not be tilted in any way (see figure 9). You should find that, no matter how far into the solenoid you place it, you will get the same magnetic field reading. End of Solenoid Vernier MAGNETIC FIELD Newer Hall Probe Figure 9: Solenoid, DC power supply, and ammeter in circuit used for measurement of magnetic field For the older probes, we will work with a 45 degree angle as shown in figures10 and 11. The older hall probe is contained in a 14.7 cm long plastic tube. The tube is flattened by 0.3 cm at one end. In addition, inner surface of the solenoid is roughly 1.5 cm above the table top. Therefore the Hall probe will be at an angle of 45 degrees with the magnetic field when the lower edge of the plastic tubing is 0.3 + 14.7(sin45) + 1.5 = 12.2 cm above the lab table Your probe may be a little different in length, and your solenoid may not be at exactly this height so you need to measure these things yourself. You should always add in the 0.3 cm for tube distortion however.
-9 - End of Solenoid approximately 12.2 cm above table top Hall Probe Figure 10: Illustration of the proper geometry for taking a measurement with the Hall Probe B meas Figure 11 : Geometry of the Hall probe and the magnetic field, B, which it is used to measure. 45 o 45 o B actual B meas = B cos θ actual
-10 - Measure the magnetic field for 6 or more values of the current. The sensors saturate at about 4.5 mt (old) and 5.0 mt (new), so your magnetic field data should fall short of these values. You probably will not be able to go above 1.5 amps, and you may need to stay quite a bit lower. Compare what you get to the results you obtain from using equation [4]. In making your measurement, it is fair to rotate the older probe about its axis and look for the maximum reading for the magnetic field (when the surface of the paddle is not tilted with respect to the field lines). You can t judge this by eye, since the probe is inside the solenoid, so you must use the largest field measurement you can obtain as you rotate the probe around the long axis of the glass tube (being careful to maintain your 45 degree angle with the horizontal). Again, the newer probe should not require this kind of manipulation. Measurement of the force on a wire You will measure the force on a wire consisting of a conductive film of metal painted on to a circuit board. You will do this by balancing the board on a copper strip and adding small weights to balance the force on the wire from the magnetic field. Your weights will actually consist of staples, whose mass you can determine on a milligram balance. Set up the circuit as shown in figures 12 and 13. Note that the copper strips which run down the side of the solenoid conduct the current to the printed circuit board which comprises the magnetic balance, and theses strips also serve as pivots. Copper Strip End of Solenoid mg F Figure 12: Geometry of the current balance
-11 - A Solenoid Figure 13: Circuit diagram for the current balance Wire on Circuit Board Ammeter Figure 14: Schematic diagram for the current balance and its supporting components Common +15 Solenoid staples + Power Supply
-12 - With no current in the apparatus, balance the metal loop in the coil (solenoid). This setup is known as a magnetic balance. If the balance is not quite level without current in it, attach a small piece of tape at the appropriate location to make it level. It is essential that you develop a systematic method so that the balance point is defined in some easily measurable way. For example, you could mount a ruler at the end of the balance and be sure that it was in balance at the same place on the ruler for each measurement. IMPORTANT: IF YOU ARE USING A FISHER POWER SUPPLY (THE ONE WITH TWO SEPARATE SETS OF OUTPUTS, PLEASE READ APPENDIX A BEFORE PROCEEDING. Now turn on the power supply, and advance the voltage control to the center of its scale with the current control set to zero. Note: In this part of the experiment, do not let the current get higher than 5A. Now apply an amp or so to the balance, just to see how things work. If the end of the loop in the coil moves upward instead of downward, the current flow is in the wrong direction, and you should reverse the connection to either the coil or the loop, but not to both. You will employ staples or other small, light objects to serve as weights. You must place the weights so that their distance the pivot point is the same as the corresponding distance for the metal strip at the other end of the balance. Start by placing one staple on the balance. Adjust the current until pivoting board is brought precisely into balance. Record the mass of the staple and the current. You should be able to calculate the magnitude of the B field by using your own, earlier measurements with the Hall transistor, and also the predicted value of the field from equation [4]. Continue to find the current needed to level the balance with 2, 3, 4, 5, and 6 staples on the apparatus. However, as you do this do not let the current get higher than 5 A or the loop contacts will corrode and the coil will overheat. Make a table of different masses in kg, their weights Newtons, two values for the magnetic field (calculate a theoretical value using equation [4] and the actual value by extrapolating your experimental results ) and, finally, the current in Amperes needed to balance them. Since the balance is aligned so the end of the U-shaped metal loop is perpendicular to the magnetic field while the sides are parallel to it, only the end will be subject to the force from the magnetic field. This is illustrated in figure 15 on the next page. Results How should B be related to I? What is the physical significance of the slope of a line resulting from a plot of F versus B? Which value for the magnetic field yields a better value for the force? If one value is better than the other, suggest reasons why this might be so. Does one of the methods overestimate or underestimate the field strength? If so, suggest reasons why this might be the case. Do your diagrams of the field around the solenoid remind you of anything you have done in class? Make sure you hand in your field diagram with your report. Put the name of both lab partners on the diagram.
-13 - l Figure 15: Schematic diagram showing the length l used in calculations of the force Appendix A: The Fisher DC Power Supply The Fisher power supply is a dual power supply that supplies 3A maximum from each output. To extract 5 Amps from this supply, it is necessary to run the two supplies in parallel. This is accomplished by connecting the two positive terminals together. When using the Fisher supplies, set BOTH current controls to zero (fully ccw), and set both voltage controls to the same value (roughly half way cw). Use the left hand supply for voltage 3A or under. Control the output by rotating the current control, but do not exceed 3 amps. When you need more than three amps, you can get additional current by adjusting the current control for the right hand supply. In other words, if you needed 5 Amps, you would use the left hand supply to get the first 3, and then get the extra 2 from the right hand supply. In each case, always carefully monitor your output using the large orange external ammeter. To Apparatu s + + Wire Connecting Positive Posts of Dual Supplies Figure A1: Configuration of jumper wire on Fisher power supply when more than 3 Amps is required
-14 - Report: Introduction: Write a brief introduction stating the objectives of the experiment, and a concise summary of the methods that will be used. Experimental: Describe the experimental apparatus and precisely what variables will be measured and how they will be measured. Results: Summarize the results of the experiment. Show sample calculations. If you are attaching computer generated tables or graphs, briefly explain them here. Discussion: Explain the significance of your results and their connection with more general physical principles. Where it is possible, compare your numbers with accepted values. Explain any sources of error.