8.0T Fall 001 Magnetic Dipoles: Force and Torque 1 (10/5/01) Goal: To observe and measure the forces and torques acting on a magnetic dipole placed in an external magnetic field. Introduction When a magnet that is free to move is placed near another fixed magnet, two things may happen. The free magnet may rotate as well as accelerate towards the fixed magnet case. When this happens, the free magnet experiences a torque that produces the rotation, and a force that produces the acceleration. This may happen not only when the free magnet is placed near a permanent magnet, but also when it is near any object that is a source of a magnetic field such as a coil carrying current. In this experiment, the source of the magnetic field is a pair of currentcarrying, 168-turn coils, each with resistance.8ω. The coils are placed parallel to each other, a distance apart equal to their mean radius. (Figure 1). Figure 1 A small-magnetized disk is mounted in a gimbal (so that the magnet is free to rotate about a horizontal axis) and suspended from a spring. The spring is free to move along the symmetry axis of the coils (referred to as the z-axis) z within a transparent tower. (Figure ) Along the z-axis, the magnetic field of the coils is parallel to the z direction and its magnitude is a function of the distance z along the axis: B = B z (z). Figure Tower assembly For a planar loop of current, the magnetic dipole moment vector µ is defined in terms of the current I, the area A of the loop, and the unit vector nˆ perpendicular to the plane of the loop: µ I A = IAˆ n The unit normal vector nˆ points in the direction defined by your right thumb when you curl the fingers of your right hand in the direction of the current in the loop. A permanent magnet also has a magnetic dipole moment. At first glance the magnetic fields associated with permanent magnets do not appear to be related to any flow of current. However on an atomic scale there many types of moving charges forming current loops: electrons orbiting nuclei, and electrons spinning about their axes. Even though we need quantum mechanics to properly describe this motion, the macroscopic effect is that these current loops give the permanent magnet a magnetic dipole moment µ. The behavior of a dipole in a magnetic field in these experiments may surprise you. Experiment 1 Magnetic dipole in a uniform magnetic field: Check that the disk magnet is free to rotate within the gimbal mount before beginning. If you have to remove the spring and magnet from the
8.0T Fall 001 Magnetic Dipoles: Force and Torque (10/5/01) tower, be very careful with the spring. Connect the DC power supply to the coils so that they are connected in series with the current flowing in the same direction in both coils the Helmholz configuration. As you have seen in previous experiments, this configuration of coils produces a very uniform field between the coils (see Figure 3 below, curve labeled Helmholtz Configuration ). Turn on the power supply and increase both the current and voltage knobs until the voltage is about 6 V and then reduce the current knob until the current reading reaches 1 A. (As you reduce the current, the voltage will also drop, reaching about 5.6V.) Warning: do not let the current exceed 3A or you may damage the coils. Then turn off the power supply. Figure 3: Plots of the magnet field on the axis for the currents used in this experiment. Next place the center of the disk magnet at the midpoint between the coils (marked z = 0 on the scale). Use the bar magnet on the table to randomly align the disk magnet. Now turn on the power supply. Questions: 1. Did the disk magnet rotate? (Was there a torque on the magnet?). Did the spring stretch or compress? (Was there a force on the magnet?) Note the alignment of the magnet at the end of this trial. Is its north pole up or down? Turn off the power supply, and then reverse the connections to it, which causes the current to reverse in both coils. What do you expect will happen when you turn on the power supply? Turn on the power supply and see. 3. What happened? 4. Summarize the behavior of a magnetic dipole placed in a uniform magnetic field. Experiment Magnetic dipole in a non-uniform magnetic field: Reverse the connections to one of the coils. They will still be connected in series but now the currents in the coils are flowing in opposite directions. Leave the DC power supply off for now. You should realize that the magnetic field between the coils is NOT uniform in this arrangement. At z = 0, the magnetic field must be zero since the fields of the coils are oppositely directed, and field reverses direction as you cross the z = 0 plane, that is, the sign of B z changes (see Figure 3 above, curve labeled Currents Opposite Directions ).
8.0T Fall 001 Magnetic Dipoles: Force and Torque 3 (10/5/01) Place the magnetic disk at z = 0 the midpoint between the disks where B = 0. What do you expect to happen when the current is turned on? Turn on the current and check your prediction. Questions: 5. Did the disk magnet rotate? (Was there a torque on the magnet?) 6. Did the spring stretch or compress? (Was there a force on the magnet?) Lower the disk magnet to the base of the tower and very slowly raise it until it is well past the center (z = 0). Observe the behavior of the dipole carefully. 7. Describe what happened. 8. Why did this happen? Discussion. The torque on a small current loop or magnetic dipole placed in a magnetic field is directly related to the strength of the field: τ mag = µ B. If the dipole is free to move, it will rotate until µ is parallel to B. The force on the magnet, however, depends NOT on B but on the spatial rate of change of the component of magnetic field parallel to the dipole moment. In this apparatus, since B and µ point in the z-direction. F dipole Bz = µ kˆ z Experiment Three Dipole moment of a disk magnet: You will now measure the dipole moment of your magnet. Leave the coils connected as in Experiment. Remove the cap, spring and magnet to prepare them for the measurement. Be very careful with the spring. Orient the magnet so that its north pole points down inside the gimbal and tape it place so that the magnet cannot rotate. While the apparatus is disassembled, calibrate the spring so that you can calculate the force acting on the spring by measuring its elongation. To do this, hang a known weight (some 1 gram ball bearings are provided) on the magnet and measure the elongation ( L) it produces. Be careful not to over stretch the spring, which could permanently damage it. From this, compute the spring constant k = F/ L in Newtons per meter. Replace the cap assembly on the apparatus. Place the disk magnet at the center position (z = 0) between the coils. The goal is to measure the force on the dipole produced by various currents (but not to exceed 3A!). Here is a procedure to follow: Turn on the power supply and adjust the current to the desired value. Then loosen the set-screw on the cap of the apparatus and position the magnet at the center of the two loops. Record the current I. Then turn off the power supply and record the position Z of the magnet. Open a spreadsheet and enter your data in a table. Column A contains the measured values of Z. In Column B, labeled F, compute the force on the dipole by multiplying the entry in column 1 by k, the
8.0T Fall 001 Magnetic Dipoles: Force and Torque 4 (10/5/01) spring constant. Column C, labeled I, contains the corresponding current. Prepare a graph of F versus I. Bz Discussion: The force on the magnet is given by Fdisk = µ kˆ. On the symmetry axis of one coil, z the magnetic field determined from the Biot-Savart law is given by N 0I R 1 B kˆ 1 = µ, 3 / ( z + R ) where the origin for z is at the center of the coil. By direct differentiation B 3Nµ 0I R z z = 5 / z ( z +. R 1 ) R At the midpoint between two coils (where z = ± ), the total magnetic field is zero since the currents in the coils are in opposite directions, but the gradients add. You showed in a WebAssign problem that B z 48 I 0.859 I Tesla = µ 5/ on = µ on. Putting in the numbers for your apparatus, N=168 z 5 R R meter Bz Tesla and R = 0.07 m, you can compute that =.0370 I. Thus, the force on the disk is z meter.0370 Tesla F = µ I ˆ disk k, meter that is, the force is directly proportional to the current in the coil, which means that your graph ought to be a straight line. Determine the slope of the best fitting straight line and compute the magnetic slope moment of the magnet from µ =. Be careful to use the correct SI units. 0.0370 T / m Question 9. What is your measured value for µ in SI units? Get a instructor to check that your answer is in the ballpark. Then go to WebAssign and enter this value. Experiment Four Maximum force on a dipole: The goal here is to measure the force on the dipole at different positions above the coil while the current remains constant. We will then be able to show that the force is proportional to the gradient of B, since we know what that expression is (see Figure 3 above). Leave the magnet locked in place in the gimbal as in Experiment 3. Remove connections to the bottom coil so that current now flows only through the upper coil. Place the disk magnet above the coil. Set the DC power supply set for 1A. If the magnet is not attracted to the coil, reverse the connections to the power supply. Turn on the power supply, position the disk magnet 1 cm above the coil, and record the position as z f,. Then turn off the power supply and record the new position of the magnet as z o.
8.0T Fall 001 Magnetic Dipoles: Force and Torque 5 (10/5/01) Open a spreadsheet program such as Excel and enter the data in a table. In Column A, labeled ZO, enter the position z o of the disk magnet when the current is zero. In Column B, labeled ZF, enter the position z f of the magnet when the current is on. In Column C, labeled Height, compute the height of the magnet above the center of the coil, which ought to be (z f 3.5 cm). In the fourth column of your data table labeled Elongation compute z o - z f. Repeat this procedure for several values of z f from 0 to several centimeters in increments of a centimeter or less. Make a graph of the Elongation vs. Height. Since the elongation of a spring is directly proportional to the force acting on it, this graph shows the force on the magnet at various distances from the coil. Question 10. From your graph, determine the distance (in cm) along the z-axis where the force is a maximum. 11. If you have time, use the expression for B z of a single coil (given on page 4 above) to determine theoretically the distance at which db z /dz reaches a maximum, and compare this to your experimental result.