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Lab 6: the Earth's Magnetic Field

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Lab 6: the Earth's Magnetic Field Lab 6: The Earth's Magnetic Field N Direction of the 1 Introduction S magnetic dipole moment m The earth just like other planetary bodies has a mag- netic ¯eld. The purpose of this experiment is to mea- Figure 1: Magnetic ¯eld lines of a bar magnet sure the horizontal component of the earth's magnetic Magnetic Axis ¯eld BH using a very simple apparatus. The measure- ment involves combining the results of two separate o Axis of Rotation 23.5 experiments to obtain BH . The ¯rst experiment will o involve studies of how a freely suspended bar magnet 11.5 interacts with the earth's magnetic ¯eld. The second experiment will involve an investigation of the com- bined e®ect of the magnetic ¯eld of a bar magnet and Plane of the S the earth's magnetic ¯eld on a compass needle. Earths orbit A magnetic dipole is the fundamental entity in mag- netostatics just like a point charge is the simplest con- N ¯guration in electrostatics. A bar magnet is an exam- ple of a magnetic dipole. The magnetic ¯eld lines of the magnet are closed loops that are directed from Geographic south north to south outside the magnet, and from south pole to north inside the magnet, as illustrated in ¯gure 1. The magnetic dipole moment (by convention) is a vector drawn pointing from the south pole toward Figure 2: Earth's magnetic ¯eld the north pole. At any point in space, the direction of the magnetic ¯eld is along the tangent to the ¯eld THE BACKGROUND CONCEPTS AND EX- line at that point. The earth is another example of ERCISES 3-4, 6-8 AND 10-13 PERTAIN TO a magnetic dipole, with the north magnetic pole lo- THE EXPERIMENTAL SECTIONS. cated at an inclination of » 11:5± with respect to the rotational axis passing through the north and south geographic poles. In fact, the north magnetic pole should be labeled as the south pole of the earth's mag- 2 Background netic dipole, and the south magnetic pole should be labeled as the north pole of the dipole (¯gure 2). This can be inferred by considering the behavior of a freely Gauss's Method: The procedure to be used in this ex- suspended bar magnet or compass needle. The north periment was developed by Gauss more than a century pole of the magnetic needle is attracted toward the ago. At that time it was known that the earth had a south pole of the earth's magnetic dipole. magnetic ¯eld, but the ¯eld had not been measured. Gauss used principles of classical mechanics and very Studies of the evolution of the earth's magnetic ¯eld simple tools. In fact, with the exception of a digital have revealed that magnetic poles migrate relative to stop watch (completely unnecessary, but nevertheless the geographic pole, and that the direction and the compact) the apparatus you will use is perhaps no magnitude of the ¯eld changes. Most interestingly, the di®erent from what Gauss may have used! magnetic poles reverse periodically, once every several hundred thousand years! The ¯rst part of the experiment involves measuring the period of oscillation of a bar magnet suspended in EXERCISES 1, 2, 5 AND 9 PERTAIN TO the earth's magnetic ¯eld, in order to obtain a value 6.1 for the product of m and BH . Here, m is the mag- F N netic dipole moment of the bar magnet and BH is the m horizontal component of the Earth's magnetic ¯eld. The second part of the experiment involves a deter- BH mination of the ratio m by measuring the deflection BH F of a compass needle due to the combined e®ects of the S magnetic ¯elds of the bar magnet and the horizontal component of the earth's magnetic ¯eld. Figure 3: Torque on a bar magnet BH can be measured by combining the results of both experiments. The result can be compared to current ² Two aluminum bases attached to a wooden board to support post holders (accepted) values of BH . To appreciate that some things have indeed become easier, you will be allowed ² Magnetic ¯eld sensor, controller and shield to measure the earth's magnetic ¯eld using a conven- tional method involving a Hall probe. ² 5V DC power supply for sensor ² Digital voltmeter 3 Suggested Reading ² BNC to banana cable to connect sensor to volt- meter Refer to the chapters on the Magnetic Field, 5 Experiment I: Oscillation period of R. Wolfson and J. Pasacho®, Physics with Mod- ern Physics (3rd Edition, Addison-Wesley Longman, a bar magnet suspended in the Don Mills ON, 1999) earth's magnetic ¯eld D. Halliday, R. Resnick and K. S. Krane, Physics (Volume 2, 5th Edition, John Wiley, 2002) Consider a bar magnet suspended in the earth's (uni- form) magnetic ¯eld. If the magnet is displaced from the position of equilibrium, a torque due to the earth's ¯eld acts on the magnet and tends to align it with the 4 Apparatus direction of the ¯eld, as shown in ¯gure 3. The torque is given by, ¡! ¡! ¡! ¿ = m £ BH (1) Refer to Appendix E for photos of the appara- Here, ¡!m is the magnetic moment of the magnet, and tus ¡! BH is the horizontal component of the earth's mag- netic ¯eld. ² Bar magnet Exercise 1: If the magnet is displaced from its posi- ² Sensitive compass needle mounted on a protractor tion of equilibrium by a small angle £, show that it will oscillate with a period T given by, ² Stopwatch for timing the oscillations s I ² Balance to measure the mass of the magnet T = 2¼ M (2) mBH ² Aluminum post and clamp for suspending the magnet Here, IM is the moment of inertia of the bar magnet. It is given by, ² Torsionless string for suspending the magnet M(L2 + a2) I = (3) ² Meter stick (without magnetized metal ends) M 12 ² Two aluminum posts and two clamps for support- where, L is the length of the magnet, a is its width, ing the meter stick on 3" post holders and M is its mass (which can be measured using a 6.2 Suspension axis Suspension axis R N Clamp Meter stick w a SN Magnet L Compass L a L (b) (a) Figure 5: Magnetometer setup, top view Figure 4: Moments of Inertia for the bar magnet, (a) suspension m axis perpendicular to the surface of area La and (b) suspension 6 Experiment II: Determination of B axis perpendicular to the surface of area Lw H This part of the experiment requires the use of a com- pass needle that is extremely sensitive to magnetic ¯elds. So make sure that apart from the bar mag- balance). This is the moment of inertia with the `sus- net, there are no other magnetized materials in the pension axis' of the bar magnet perpendicular to the vicinity of the compass needle. This is crucial for surface of area La, ¯gure 4a. obtaining accurate results In ¯gure 4b the moment of inertia with suspension The experimental setup consists of a magnetometer as axis perpendicular to the surface of area Lw is IM = 2 2 shown in ¯gure 5. To set up the magnetometer, ¯nd M(L +w ) . It is important to realize that the torque 12 the direction of the Earth's magnetic ¯eld (i.e. the on the magnet restores the magnet to its equilibrium N-S direction) using the compass needle. Then place position. Hence ¿ = ¡mBH sin £. To solve for the the axis of the bar magnet perpendicular to the N/S period of oscillation, use the rotational analog of New- direction, with the magnet's center at a distance R ton's second law. from the center of the compass needle. It is assumed that the distance R is very much larger than L, the Exercise 2: Solve equation 2 to obtain an expression length of the magnet. for mBH . What are the SI units of m and BH ? If R À L, the magnitude of the magnetic ¯eld Bb Suspend the magnet from its midpoint by a torsion- produced by the bar magnet at the compass is given less string, so that only the horizontal component of by, the Earth's magnetic ¯eld will cause a torque. Deter- 2k m B = m (4) mine the period of small amplitude oscillations. Re- b R3 peat the experiment several times and obtain the aver- ¹0 ¡7 N age value of the oscillation period. During each trial where, km = 4¼ = 10 A2 is a constant, m is the measure the period of oscillation by timing »10 os- magnetic dipole moment of the bar magnet and R cillations. Make sure that there are no other is the distance from the center of the magnet to the magnets or magnetized materials in the vicin- center of the compass. ity of the magnet. Also make sure that you shield the experiment from air currents. If a Note that equation 4 can represent the magnetic ¯eld wooden bar is used instead of a magnet there would due to a single current carrying loop of wire for R À r, be no restoring torque. Why? where r is the radius of the loop (in this case m is the magnetic moment of the loop). Exercise 3a: Organize your experimental data in a table. Next determine the moment of inertia of the Equation 4 is also analogous to the expression for the bar magnet, using equation 3. Estimate the experi- electric ¯eld perpendicular to the axis of an electric mental errors associated with each measured quantity.
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