# Experiment 7: Forces and Torques on Magnetic Dipoles

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1 MASSACHUSETTS INSTITUTE OF TECHNOLOY Department of Physics 8. Spring 5 OBJECTIVES Experiment 7: Forces and Torques on Magnetic Dipoles 1. To measure the magnetic fields due to a pair of current-carrying loops in the Helmholtz configuration, both with the currents in the same direction and in the opposite direction.. To observe and measure the forces and torques acting on a magnetic dipole placed in an external magnetic field. 3. To measure quantitatively the force on a magnetic dipole on the axis of a ring of current, as a function of the distance from the center of the ring. INTRODUCTION Magnetic Field of a Helmholtz Coil Consider the Helmholtz Coil Apparatus shown in Figure 1. The Apparatus consists of two coaxial coils that are separated by a distance equal to their common radii. Figure 1 Helmholtz Coil Apparatus When the current through both coils is in the same direction, the magnetic field B H at a distance x along the axis from the midpoint between the two coils is given by the sum of two equations in the form of Equation (7.1), suitably displaced from zero: E7-1

2 Nµ I R 1 B = xˆ (7.1) (x + R 3/ ) Nµ I R B H = 1 ( x R/) + R 3/ xˆ + Nµ I R 1 xˆ. (7.) 3/ ( x + R /) + R Near the midpoint ( x << R ), the above equation can be expanded in a Taylor Series as: 4 N I R 6 x 4 µ 6 x B H ( x ) = 4) 3/ 1+ 5 R +" 4 xˆ = B c 1+ +" xˆ (7.3) (5R 5 R 4 Nµ 3/ I R 4 Nµ I where B = xˆ (7.4) 4) xˆ = c 3/ (5R 5 R is the value of the magnetic field at the center of the coil configuration ( x = ). Notice that in this limit ( x R ), the field between the coils is nearly constant. We use this configuration of coils for precisely this reason it gives us a region where the magnetic field is reasonably constant near the value given in Equation (7.4). The magnetic field along the common axis of the two coils (that is, a plot of Equation (7.)) is given on the next page in Figure 4 (the black plot labeled Helmholtz Coil ). Magnetic Field of Two Coils With Helmholtz Spacing but Opposite Currents When the currents through the coils are in the opposite directions, the magnetic field near the midpoint between the coils is nearly zero and is given by the difference of two equations in the form of Equation (7.1), appropriately displaced; that is Equation (7.) with a minus before one of the terms instead of a plus. This field is plotted in the green plot labeled Reverse in Figure 4. Figure 4 also shows the field due to a single coil in the plane x = R. As we mentioned above, the calculations given indicate that when current flows in the same direction in the two coils, the resulting magnetic field is nearly uniform in the region near the midpoint between the coils. On the other hand, when currents flow in opposite directions in the two coils, the magnetic field is close to zero in that region. Note that these results assume that the same magnitude of current is flowing in each coil. E7-

3 THEORY OF TORQUE AND FORCE ON A DIPOLE 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. In general, 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 the free magnet is near any object that is a source of a magnetic field, such as a current-carrying coil. We want to explore qualitatively the nature of this torque and force, and also measure the force quantitatively. IMPORTANT NOTE: In considering the interactions between magnetic dipoles and current-carrying wires, the greek letter mu, µ or µ, is used both for the magnetic moment and for the permeability of the appropriate material, as in µ for the permeability of free space. For the purposes of these instructions, µ, with the subscript, will be used for the permeability of free space, while the erect symbol µ will be used for the magnitude of a magnetic moment, µ= µ. Figure Apparatus Figure 3 Tower assembly Our experiment uses a magnet sitting in an external magnetic field generated by currents in two coils in the Helmholtz configuration. We have two identical 168-turn coils, each E7-3

4 with a resistance of approximately 8Ω. (Figure ). The coils are placed parallel to each other, separated by a distance equal to their mean radius R = 7. cm. 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 (the x-axis in the above equations) within a transparent tube Figure 3). Along the x-axis, the magnetic field of the coils is parallel to the x-direction and its magnitude is a function of the distance x along the axis: B ( x ) = B ( x )î. (7.5) 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 L of the loop: x µ I L A = I A nˆ (7.6) ( I is used for the loop current to distinguish from the current in the coils used to L generate the external magnetic field). 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. The torque on a small current loop or magnetic dipole placed in a magnetic field is directly related to the strength of the field: If the dipole is free to move, it will rotate until µ Notes, Section 8.4, for a detailed discussion). L τ = µ B (7.7) is parallel to B (see the 8. Course The force on the magnet, however, depends NOT on B itself but on the spatial rate of change (gradient) of the component of magnetic field parallel to the dipole moment. (See the 8. Course Notes, Solved Problem 8.9.4, for a presentation of a similar but equivalent discussion of this effect.) For this apparatus, we have since B and µ will point in the x-direction. B F x dipole =µ î, (7.8) x E7-4

5 FIELDS AND FIELD RADIENTS FOR THE CONFIURATIONS WE USE From Equations (7.7) and (7.8) above we see that the toque on a dipole depends on the local value of the magnetic field but the force on a dipole depends on the local spatial gradient of the magnetic field. We will use three different field configurations in this experiment, and here we discuss the field and the gradient of the field along the x -axis for each of these configurations, since these are the relevant quantities that determine the torque and force on the dipole. Figure 4 shows the x -component of the magnetic field for the three configurations we use in this experiment: (1 - red) current in a single (the top) coil; ( - black) currents flowing in the same direction in the two coils (the Helmholtz configuration); and (3 - green) currents flowing in opposite directions in the two coils (the Reverse configuration). Figure 4: The x-component of the magnetic field and its derivative for configurations, plus the same quantities for a single coil. ( Helmholtz is current in the both coils flowing in the same direction, Reverse is current in both coils flowing in opposite directions). See page the last page of this writeup for an iron-filings representation of these three field configurations. E7-5

6 The salient features of Figure 4 that we will use in this experiment are that: 1. The Helmholtz field at the center of the two coils has a large absolute value and zero gradient. Therefore, a magnet place there should feel a large torque but zero force.. The Reverse field at the center of the two coils is zero, but has a large gradient. Therefore, a magnet placed there should feel zero torque but a large force. 3. For the One Coil configuration, the gradient along the axis is well defined and we will in fact measure that gradient (see Figure 7), which we derive analytically below. The radient of the Field Along the Axis of One Coil We here derive explicitly the equation for the gradient of the field on the axis of a single coil, since we will be able to measure this gradient directly in this experiment, by measuring the force on the magnet. The magnetic field along the axis of a coil is N µ I R 1 B = î (7.9) (x + R ) 3/ (see the 8. Course Notes, Section 9.8, for a derivation of the above result). If we place a magnetic disk of magnetic moment µ in the field, the force on the disk would be B x ˆ 3µ NI R x i = F disk = µ µ 5/ î. (7.1) x (x + R ) The force is a maximum when the derivative of the x -component of the force is zero: F disk, x =. (7.11) x We can solve for the location of the maximum force by explicitly calculating this derivative; F x 3 µ = µ N I R R 4x = (7.1) x (x + R ) 7 / R which implies that the maximum occurs at two points along the x -axis, at x =±. E7-6

7 The radient For Two Coils with Currents in Opposite Directions We also give, but do not derive, the expression for the gradient of the magnetic field at the center of coils carrying current in the reverse direction (see Figure 4), since we need this to interpret our experimental results below. This gradient is given by db x dx 3 4 5/ N µ I = (7.13) 5 R x= Therefore the force on the dipole when the dipole is placed at the midpoint between the coils in the Reverse configuration is non-zero and given by Bx F dipole (x) = µ x= x 5/ î = µ 3 4 N µ I î = µ.859 ) µ N I ( x= 5 R R î (7.14) SETUP Measuring the Magnetic Field of the Coils Connecting the Magnetic Field Sensor to the 75 Interface The magnetic field sensor is plugged into the first analog channel A of the 75 Interface, as in previous experiments. Figure 5 The top of the magnetic field sensor, showing (from right to left) the RANE SELECT switch, the TARE button, and the RADIAL/AXIAL switch, which is set to RADIAL E7-7

11 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 three one-gram ball bearings (provided) on the magnet and measure the elongation L produced. Be careful not to over-stretch the spring, which could permanently damage the spring, which is somewhat fragile. Enter your measurements into the appropriate boxes in the spreadsheet exp7.xls, which will then determine k = F / L (7.15) in SI units, newtons per meter. Replace the cap assembly on the apparatus. Now, place the disk magnet at the center position (x = ) between the coils. The goal is to measure the force on the dipole produced by various currents (but not to exceed3a!), and then use equation (7.14) above to deduce the magnitude of the dipole moment of the magnet. Turn on the power supply and adjust the current to the desired value, to begin with around.5 A. Then, with the power on and the current flowing position the magnet at the midpoint between the two coils. Record the current I. Then turn off the power supply and record the position x of the magnet. Repeat this same procedure with currents around. A, 1.5 A and 1. A. In the spreadsheet, enter your data in the table provided. The force corresponding to the position will be calculated, using your value of the spring constant determined previously, and the force as a function of current plotted. You should see a plot that looks the plot on the next page, Figure 6. The force on the magnet is given by Equation (7.14). With N =168 and R =.7 m, we have F (x) B x T =µ î =µ.37 I î (7.16) disk x= x A m x= The force is predicted to be directly proportional to the current in the coil, which means that your graph ought to be a straight line. The slope of the best-fit straight line will be computed and the magnetic moment found from µ = slope A m. (7.17).37 T E7-11

12 Question 1 (answer on your tear-sheet at the end): What is your measured value for µ in SI units? Figure 6 Force as a function of current for the reverse current configuration for the magnet at the midpoint between the two coils. Force on a Dipole On The Axis Of A Single Coil The goal here is to measure the force on the dipole at different positions above the coil while the current remains constant. According to equation (7.8), measuring the force is the same as a measurement of the gradient of the magnetic field along the axis, and we know theoretically what that gradient should be from equation (7.9). Leave the magnet taped in place in the gimbal as above. Remove the connections to the bottom coil so that current now flows only through the upper coil. Place the disk magnet above the upper coil, a centimeter or so. Set the DC power supply set for 1A and turn on the power supply. If the magnet is not attracted to the coil, reverse the connections to the power supply. You will now leave the current at a constant value of around 1A for the all the measurements below. With the power supply turned on, position the disk magnet 1cm above the center of the top coil, and record that position as x f in your spreadsheet. You record this by using the scale on the plastic tube. Note that this scale is zero at the midpoint between the two coils, so your reading should be 4.5 cm when the magnet is 1 cm above the center of the top coil. Then turn off the power supply and record the new position of the magnet as E7-1

14 One Coil Helmholtz Reverse E7-14

15 MASSACHUSETTS INSTITUTE OF TECHNOLOY Department of Physics 8. Spring 5 Experiment Summary 7: Forces and Torques on Magnetic Dipoles roup and Section (e.g. 1A, L: Please Fill Out) Names Magnetic field of Two N-turn Coils in Helmholtz Configuration Question 1: Does your graph of the field along the axis look like the appropriate graph in Figure 4? Magnetic field of Two N-turn Coils with Currents in Opposite Directions Question : Does your graph of the field along the axis look like the appropriate graph in Figure 4? Torque and Force on A Magnetic Dipole in a Uniform Magnetic Field Question 3: Did the disk magnet rotate? (Was there a torque on the magnet?) Question 4: Did the spring stretch or compress? (Was there a force on the magnet?) Question 5: What happened to the orientation of the magnet when you changed the current direction in the coils in the Helmholtz configuration? Torque and Force On A Magnetic Dipole in a Non-Uniform Magnetic Field Question 6: Did the disk magnet rotate? (Was there a torque on the magnet?) Question 7: Did the spring stretch or compress? (Was there a force on the magnet?) E7-15

16 Question 8: Describe what happened when you moved the magnet past the midpoint of the two coils with the current in the Reverse configuration. Question 9: Why did this happen? Measuring the Dipole Moment of Your Disk Magnet Question 1: What is your measured value for µ in SI units? Force on a Dipole on the Axis of a Single Coil Question 11: From your graph, determine the distance (in cm) along the x -axis where the force is a maximum R Question 1: How does the theoretical result of x = ± for the maximum force magnitude (see Equation (7.1)) compare to your experimental result? E7-16

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