MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2005 Experiment 8: Magnetic Forces OBJECTIVES 1. To investigate the magnetic force between two current carrying wires. 2. To measure the permeability constant µ0. INTRODUCTION The magnetic force on a current-carrying conductor underpins every electric motor - turning the hands of electric watches and clocks, transporting tape in Walkmans, starting cars, operating refrigerator compressors, etc. In this experiment, you will investigate the magnetic force between two current carrying wires. One wire will be a coil of 10 turns and the other will be a coil of 38 turns. The 10-turn coil will be taped to one end of a pivoted balance beam. The beam pivots on two pins. It also makes electrical contact through the two pins, allowing current to flow onto the beam and through the 10-turn coil. The 38-turn coil will be positioned on the table directly below the 10-turn coil. A current traveling through both coils will produce a magnetic force between the coils, either attraction or repulsion, depending on the relative direction of the currents. You will measure the magnitude of this force by noting when the magnetic torque produced by this force between the coils is balanced by the torque due to the force of gravity on known weights. In this experiment we will determine a value for the constant µ0 . To do this we will depend on the multimeter calibration of the amount of current flowing through the coils in amperes. −7 ⋅Usually we define the ampere so that the constant µ= 4π×10 T m/A , using measured forces 0 between current-carrying wires. Here we will do the inverse—we assume our multimeter gives an accurate value for the current, and we measure forces to determine the constant µ0 . This requires a calculation of the force expected between two current-carrying coils. THEORY Consider a system of two coils having currents flowing in opposite directions, as shown in G Figure 1. What is the force between the coils? Is it attractive or repulsive? In the figure, B2 is the magnetic field at coil #2 produced by the current in coil #1. E08-1E08-2 Figure 1 Force diagram on coil # 2 for repulsive force In our experimental setup, the two coils are separated by a distance d which is much smaller than r, the radius of either coil (for purposes of clarity, the figure is not to scale). As a first approximation, we can treat the two coils as if they were parallel wires separated by a distance d. In this limit we neglect the contribution to the magnetic field, 1BG, from parts of the lower coil that are not directly below the small current element in the upper coil. This will over-estimate the force somewhat (can you see qualitatively why?) but the error is not more than about 10% with your arrangement. The expression for the magnitude of the force per unit length between two infinite wires (an approximation to very long wires or, in our case, the coils separated by a distance small compared to the common coil radius) is derived in the 8.02 Course Notes, Section 9.2; 1202IIforcelength dµπ= (8.1) where 1I and 2I are the currents and the separation is d . The force will be attractive if the currents in the coils are in the same direction, repulsive if in opposite directions. For this experiment the currents will be 11InI= and 22InI= , where 1n and 2n are the numbers of turns in the coils and Iis the common current. The length to be used is the circumference of the coils, 2 rπ, where r is the common radius. The result is that the magnitude of the force is ()212mag 02nn I rforceFrlength dπµ== (8.2) By Newton’s third law, the total force on coil #1 is equal in direction and opposite in magnitude of the force on coil #2. The magnetic force is balanced by aluminum foil weights that are placed at an equal distance from the pivot as the center of the upper coil. If the weights are all the same – 2 cm 1 cm × rectangles of aluminum foil – their weight will beF =nmg = nρAtg (8.3)g210 −4 2where g = 9.8 m/s2 , A =× m is the area of each piece of foil, t =1.8×10 −5m is the 3 3thickness of the foil, ρ= 2.7 ×10 kg/m is the density of the foil and n is the number of pieces of foil. The balance reaches equilibrium when the magnitude of the torque from the magnetic force equals the magnitude of the torque from the aluminum weight. Since the moment arms are equal, the forces must also be equal; F =Fmag , (8.4)gor 2 nAtg =µnn I r 012ρ . (8.5)d This equation shows us that the current squared depends linearly on the number of foil pieces present, specifically I2 =⎛⎜ ρAt g d ⎞ ⎟n. (8.6)µnn r ⎝ 012 ⎠ The slope of the I2 vs. n plot is given by slope =ρAt g d . (8.7)µnn r 012 From this slope, determined from your data, you can in principle calculate the magnetic permeability of space using µ= 1 ⎛⎜ ρAt g d ⎞⎟. (8.8)0 slope nn r ⎠⎝ 12 EXPERIMENTAL SETUP The setup of the experiment is depicted in Figures 2 –7 below: E08-3Figure 2 Top view of balance with 10-turn coil taped to underside of foam core Figure 3 Underneath side of balance with 10-turn coil Figure 4 Wires soldered to pins E08-4Figure 5 Spacing between the 10-turn and 38-turn coils The coil end of the beam is much heavier than the other end. The beam is approximately balanced by taping one penny, appropriately located, on the light end. Balancing the beam will determine the distance d between the coils used in Equations (8.1) - (8.8) above. In the experiment, when a current runs through the coils the beam will return to this equilibrium position when the weight of the aluminum foil squares is just equal to the magnetic force. EXPERIMENT Depending upon the winding direction of the coils, the current through them will generate magnetic forces such that the coils will either attract or repel each other, depending on the relative direction of the currents. The current direction is reversed with the switch shown in Figure 7 (but not in the circuit diagram, Figure 6). For this experiment, it is not practical to measure the current directly. Instead, a voltmeter is put in parallel with known resistance; from Ohm’s Law, the current in the circuit, and hence in the coils, will be proportional to the measured voltage. Start by downloading the Excel spreadsheet exp08.xls and saving to your desktop. Measure the following quantities; record your values both in the Excel spreadsheet and on the tear-off
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