PHYS-2212 LAB Force between Electric Currents Objectives To investigate the magnetic force between two parallel current-carrying wires and determine the magnetic permeability of free space, µ o. Introduction In this experiment, a sensitive current balance is used to measure the force of attraction or repulsion F between two parallel conducting wires of equal length l. The motion of an electric charge produces a magnetic field around that charge. Thus, several electric charges flowing in a specific direction through a straight conductor, as electric current I, constitute the most elementary source of magnetic fields. Such magnetic fields produced by a current-carrying wire exist as concentric magnetic lines of force centered on the wire, as shown in Figure 1 below. Figure 1: Magnetic field lines around a current-carrying wire The magnetic field B at radial distance, r, outside an infinitely long wire carrying current I is given by Equation 1: μ0 B = 2π Eqn. 1 The magnitude of the magnetic field, B, is expressed in units of tesla (T), with the electric current, I, in amperes (A), and the radial distance, r, in meters (m), measured from the center of the wire to the point where the magnetic field is being evaluated. The magnetic permeability of free space, µ o, has a constant value 4π x 10-7 T.m/A. I r
Additionally, an electric charge moving through a region of magnetic field experiences a force due to the field. Hence, a wire carrying electric current also experiences a force in a magnetic field. The direction of the magnetic force on moving charges (or electric current) is perpendicular to both the direction of the magnetic field lines and to the direction of motion of the charges (or current). However, the magnetic force is zero if the charges (or current) move in a direction that is parallel to the direction of the magnetic field lines and a maximum when the charges move in a direction perpendicular to the magnetic field lines. When an infinitely long wire carrying current I is placed in a magnetic field of strength B, the magnetic force on the wire is given by Equation 2: F = ILB sinθ Eqn. 2 The magnetic force on the wire is in units of newtons (N), with the length of the wire inside the magnetic field, L, in meters, current in amperes (A), magnetic field in tesla (T), and θ is the angle between the directions of the electric current and the magnetic field. When two current-carrying wires are placed close enough and parallel to each other, both wires exert a magnetic force on each other. The current in one wire is perpendicular to the magnetic field due to the other wire. Thus, the magnetic field due to the current in one wire becomes the external magnetic field that exerts a force on the other wire. Consequently, each current wire exerts a magnetic force on the other. This mutual force between parallel current-carrying wires is directed perpendicular to the directions of the currents and may be attractive (pushing the wires toward each other) or repulsive (pulling the wires away from each other), depending on the directions of currents through the wires. As shown on Figure 2 and Figure 3 below, there is a force of attraction between the parallel wires when the currents through the wires are in the same direction. The force is repulsive when the currents are in opposite directions. The magnetic force per unit length (F/L) between two parallel long wires carrying currents, I 1, I 2 and separated by a perpendicular distance, d, is given by: μ0 I1I = L 2π d F 2 Eqn. 3 2008 mokafor GEORGIA PERIMETER COLLEGE Page 2 of 12
Figure 2: Currents in same direction Figure 3: Currents in opposite direction In this experiment, the Sargent-Welch current balance is used to measure the force between two parallel wires carrying currents. One straight (fixed) wire is mounted on the base of the current balance while the second (movable) straight wire is mounted on the movable arm of the beam assembly. This movable wire carries a small pan in which calibrated weights may be placed to determine the gravitational force that is equal to the magnetic force exerted on the movable wire (due to the magnetic field of the fixed wire). The wires are separated with a spacer block of known even thickness. The spot of a laser beam reflection from the balance mirror to a distant screen is marked to correspond to the desired spacing between the wires. Thus, the wires can always be returned to the same preset spacing by adjusting the laser beam to the marked spot on the screen. The distance from the mirror to the screen should be fairly large for higher sensitivity. The reflected laser beam spot on the screen is displaced through an angle twice the angle of rotation of the mirror. Therefore, a long mirror-to-screen distance significantly increases the effects of small rotations of the mirror. The swinging motion of the beam assembly is damped by a conducting blade that oscillates within the magnetic field of a permanent magnet. Apparatus Current balance, calibrated weights, (spirit) level, (0-15A) dc power supply, (0 15 A dc) digital ammeter (DMM), 0.5-Ω resistor 100 W, ⅛-inch spacer block, He-Ne laser, Vernier caliper, micrometer caliper, masking tape, and two double-pole, double-throw (DPDT) knife switches. 2008 mokafor GEORGIA PERIMETER COLLEGE Page 3 of 12
Laser Power supply DMM Calibrated weights 0.5-ohm resistor DPDT switch Current balance Figure 4: Current balance setup 2008 mokafor GEORGIA PERIMETER COLLEGE Page 4 of 12
Experimental Procedure 1. Review the overview of the current balance presented in the introduction. Identify the components of the experimental setup shown in Figure 4 above and the essential parts of the current balance labeled in Figure 5. 2. Use a micrometer calipers to determine the average thickness of the ⅛-inch spacer block. Record your measurements in Data Table 1. 3. Use Vernier calipers or metric ruler to measure and record the length of the movable wire (A) used in this apparatus. With the micrometer calipers, measure and record the diameter of the wires. You may have to lift the beam balance in order to measure the length of the movable wire. Record your measurements in Data Table 2. 4. Carefully lift and remove the balance beam assembly (B). Place a level (provided) on the base of the current balance and use the two thumbscrews (S) near the front edge to level the balance. 5. Using the compass provided, align the wires parallel to the earth s magnetic field (approximately north-south) in order to minimize its effect on the current wires. The earth s magnetic field is about 5 x 10-5 T (tesla) or 0.5 G (gauss) and this earth s field strength is significant in comparison to the magnetic field at the movable wire due to reasonable currents in the fixed wire. 6. Gently replace the beam assembly onto the knife-edge bearing surfaces, using the tapered screw points on the rotating lifting shaft (J). 7. Make sure that the movable wire (that is attached to the beam balance assembly) and the fixed wire (that is mounted on the base support posts (C)) are exactly parallel. Note that there are four sets of adjusting screws for positioning the wires. 8. Use the beam lift (J) to adjust the beam assembly and hence correctly position the upper movable wire so that it is straight and exactly parallel to the lower fixed wire, with the edges matching along their common length. Tighten the screws after adjusting the plates. 9. After positioning the beam assembly correctly, check and adjust the vertical and horizontal counterweights. The vertical counterweights (G) usually need no further adjustments and are used to set the center of gravity of the balance beam below the knife edge pivots. The resulting fairly low center of gravity provides the mechanical restoring force necessary to overcome the destabilizing effect caused by the magnetic force between the wires at the limiting conditions of the experiment. The horizontal counterweights (D) are adjusted until the beam assembly is balanced and pivots freely on its knife edges. While pivoting freely on the knife edges, the damping blade (H) should also swing freely inside the groove, without touching any damping magnets. Otherwise gently press the blade sideways, as needed, to ensure free motion of the damping blade. 2008 mokafor GEORGIA PERIMETER COLLEGE Page 5 of 12
D M B J G H A S C Figure 5: Current balance to measure magnetic force between currents 10. Connect the apparatus as shown in the schematic diagram below. 15-A dc Power Supply + 0-15 + dc A Fixed Wire DPDT1 Resistor 0.5 Ω 100 W Movable Wire DPDT2 Schematic diagram for the current balance circuit 2008 mokafor GEORGIA PERIMETER COLLEGE Page 6 of 12
With the two double-pole, double-throw (DPDT) knife switches, the directions of the currents through both wires can be changed. Switch DPDT2 reverses the current in one wire with respect to the other in order to change from attractive to repulsive force measurements. Switch DPDT1 reverses the directions through both wires in order to eliminate the effect of the magnetic force on the movable wire due to the earth s magnetic field. Since both wires carry the same current in series, the neutral position of either DPDT switch opens the entire circuit. 11. The 0.5-ohm resistor is shown as Figure 6 below. Note that this resistor can become very hot during the experiment. Therefore, it is important to reduce the current to zero immediately after taking a reading and switch off the power supply. Ensure free air circulation around the resistor. Figure 6: 0.5-Ω, 100-W Resistor 12. STOP!!! Ask your lab instructor to inspect your setup. After your instructor has inspected the wiring and approved your setup, you may continue with the procedure. 13. With the ⅛-inch spacer inserted between both wires, place another circular mass on top of the movable wire to hold the spacer in position. Check the beam assembly and ensure that the wires are exactly parallel and evenly separated by the thickness of the spacer. 14. Mount the laser in front of the electrostatic balance so that the laser beam is reflected from the mirror (M) on the balance to a screen approximately 2 m away from the mirror. With the spacer block held in the middle, between the wires, mark the position of the laser light beam on the screen as the reference mark for all the experimental measurements. At this reference mark, the spacing between the wires should be exactly equal to the thickness of the spacer. 15. Remove the circular mass that holds the top (movable) wire down and also remove the spacer from between the wires. 2008 mokafor GEORGIA PERIMETER COLLEGE Page 7 of 12
Acquisition of Data 16. Set switch DPDT2 for currents in the same direction in both wires so that the force between the wires will be attractive. Place a 20-mg mass in the pan on the movable wire. Adjust the horizontal counterweights to tilt the top wire upward until the laser beam returns to the reference mark on the screen. 17. Carefully remove the 20-mg mass from the pan on the movable wire. This top wire should then tilt upward such that the laser beam shifts above the reference mark on the screen. Turn on the power supply and close switch DPDT1 in the forward position. Gradually increase the current until the beam assembly is balanced with the laser beam at the reference mark on the screen. Record this forward current reading (in Data Table 3) at which the laser beam just returns to the reference mark. 18. Reduce the current through the wires to zero and throw switch DPDT1 in the reverse position. Again, gradually increase the current until the laser beam returns to the reference mark. Record this reverse current reading in Data Table 3. 19. Repeat procedure steps 15-17 above for each of the following masses: 50 mg and 70 mg. Record the forward and reverse currents obtained for each mass. 20. Set switch DPDT2 for currents in the opposite directions in both wires so that the force between the wires will be repulsive. Remove all masses from the pan on the movable wire. Adjust the horizontal counterweights until the beam assembly balances with the laser beam on the reference mark. 21. Turn on the power supply and close switch DPDT1 in the forward direction. Gradually increase the output current to about 5 A, noting that the repulsive force between the wires causes the laser beam to shift upwards above the reference mark. Now carefully add small masses (about 20 mg) to the pan on the movable wire to bring the laser beam close to the reference mark. Make very small adjustments to the output current necessary to return the laser beam to the reference mark. Record the mass in the pan and the final (forward) current reading on the meter in Data Table 4. 22. Reverse the current by throwing the switch DPDT1 in the reverse direction. Readjust the output current as necessary to return the laser beam to the reference mark. Record this new reverse current in Data Table 4. 23. Repeat procedure steps 20-21 above with additional masses (e.g. 50 mg and 70 mg) and currents of about 8 A and 10 A respectively. Record the final forward and reverse currents obtained with the corresponding masses placed on the movable wire. 24. Reduce the current through the apparatus to zero and switch off the power supply. 2008 mokafor GEORGIA PERIMETER COLLEGE Page 8 of 12
Analysis of Data 25. Table 1: Thickness of spacer block Micrometer zero reading = Thickness of the spacer block Micrometer readings Trial 1 Trial 2 Trial 3 Average Thickness of the ⅛-inch spacer block 26. Table 2: Dimensions of the wires Micrometer zero reading = Vernier zero reading = Dimensions of the current wires Micrometer readings Trial 1 Trial 2 Trial 3 Average Diameter of the movable wire Diameter of the fixed wire Length of movable wire Radius (r 1 ) of the movable wire = Radius (r 2 ) of the fixed wire = Length of movable wire, L = 2008 mokafor GEORGIA PERIMETER COLLEGE Page 9 of 12
27. Determine the average length of the movable wire and the average radius of each wire from three independent measurements of the diameters as recorded in Table 2 above. Record these results. 28. From the average thickness of the spacer block shown in Table 1 above, and the radius of each wire determined from Table 2, calculate and record the wire separation (d). The wire separation is the distance between the centers of the wires. Wire separation (d) with the ⅛-inch spacing = 29. Currents in the Same Direction (attractive force): From the measurements shown in Table 3 below, calculate the square of the currents for the forward and reverse current sets and the matching magnetic force. Enter these results in Table 3. Recall that this magnetic force (in newtons) is equal to the gravitational force (mg) on the masses removed from the pan of the movable wire. 30. Table 3: Measurements for currents in the Same Direction (attractive force) Mass on pan of movable wire Forward current Same Direction (Attractive Force) Reverse current Squared currents Magnetic Force 2008 mokafor GEORGIA PERIMETER COLLEGE Page 10 of 12
31. Currents in Opposite Direction (repulsive force): From the measurements shown in Table 4 below, calculate the square of the currents for the forward and reverse current sets and the matching magnetic force. Enter these results in Table 4. Again, this magnetic force is equal to the force of gravity (in newtons) acting on the masses on the pan of the movable wire. 32. Table 4: Measurements for currents in the Opposite Direction (repulsive force) Mass on pan of movable wire Forward current Opposite Direction (Repulsive Force) Reverse current Squared currents Magnetic Force 33. Using data from Tables #3 and #4, plot a curve of the Squared Currents (A 2 ) as ordinates against the matching values of the Magnetic Forces (F) as abscissa. Plot all the points representing Squared Currents (in same and opposite directions) as positive values. Distinguish between points representing attractive forces (currents in same direction) and those for repulsive forces (currents in opposite directions). Mark the points for attractive forces as circles around points and for repulsive forces as squares around points. The plot area of this curve should cover, at least, two-thirds of the entire sheet of graph paper, with the origin (0, 0) near the lower left corner of the sheet. The origin indicates the zero force at zero current through the apparatus. 34. Draw the best straight line through all the plotted points. The origin should be a point on this line. This line averages variations in the meter readings as well as shifts caused by reversing the current through the apparatus relative to the Earth s field direction. 2008 mokafor GEORGIA PERIMETER COLLEGE Page 11 of 12
35. Calculate the measured value of µ o, magnetic permeability of free space, using Eqn. 3 and the value of the slope. Compare this result with the accepted value of µ o by finding the percent error. Slope of graph = Value of µ o = Percent error = 36. In a typical experiment to study the force between electric currents, identical fixed and movable wires were separated using a spacer of thickness 0.6065 cm. The radius of each wire was determined to be 0.162 cm and the length of the movable wire was 26.5 cm. The apparatus was set up so as to produce a repulsive force when a current of 9.3 A passed through the wires in opposite directions. Assume the magnetic permeability of free space, µ o = 4π x 10-7 T.m/A. (a) Calculate the magnetic field strength at the center of the movable wire due to the given current through the fixed wire. (b) What mass (in milligrams) would have to be added to the weight pan on the movable wire to balance out the repulsive magnetic force when the given current passes through the fixed wire? (c) Calculate the magnitude of the total magnetic field due to the given current in the wires, along the perpendicular line between the wires at a point y = 0.404 cm, measured from the surface of the fixed wire when: (i) the current flows in the opposite direction in the two wires, and (ii) the current flows in same directions. 2008 mokafor GEORGIA PERIMETER COLLEGE Page 12 of 12