Measurement of the Horizontal Component (H) of Earth's Magnetic Field Dr. Tim Niiler, WCU based on lab by Dr. Harold Skelton

Transcription

1 Measurement of the Horizontal Component (H) of Earth's Magnetic Field Dr. Tim Niiler, WCU based on lab by Dr. Harold Skelton Background The Earth's magnetic field is of interest to scientists due to its interaction with the Sun, its ability to protect us from harmful extraterrestrial radiation, its effect on compasses, and many other reasons. The magnitude of the field depends greatly on solar activity. When solar activity is high, the field lines are compressed and the field strength increases. The Earth's field is threedimensional: magnetometer stations about the globe typically report on the declination, the inclination (or dip angle), and horizontal component of the field. The declination, D, is the angle the field varies from true north. The inclination, I, is the angle (downwards) from the horizontal, and the horizontal component, H, is the projection of the field on the surface of the Earth. Today, we will be trying to measure the horizontal component of the Earth's magnetic field at our latitude and compare it to data from a nearby geomagnetic observatory. Theory Unlike our professional counterparts, we will not be using a magnetometer to determine H. Rather, we will be using a combination of two methods involving torque balance and harmonic motion. The strategy will be to determine H, and /H separately, and then by solving the two simultaneous equations, be able to determine H of Earth, and (by extension), the magnetic moment of the magnet. This section will present some of the background theory behind the experiment. a) Determining H, the torque on the magnet In this section of the lab, a cylindrical magnet will be allowed to dangle from a string and oscillate in the H component of Earth's field. This field creates a torque on the hanging magnet: 1) =I = Hsin Illustration 1: Earth's magnetosphere. Image under GPL license from Wikipedia.org. where I is the moment of inertia of the magnet, is the angular acceleration, is the magnetic moment of the magnet, and is the angle the magnet makes with respect to H. This is, in actuality a differential equation:

2 2) I d2 dt 2 = Hsin In the case of small oscillations, sin, so this equation simplifies (with some rearrangement to: 3) I d2 H =0 2 dt or 4) d 2 dt 2 H I =0 Presuming that = o sin t, we find that: 5) d dt = 0 cos t and 6) d 2 dt 2 = 2 0 sin t Substitution of equation 6) into equation 4) gives us: 7) 2 0 sin t H I 0 sin t=0 For this to be true: 8) 2 = H I The quantity is the angular frequency which is related to the period of oscillation by: 9) T= 2

3 By combination of equations 8) and 9), one can see how to determine the torque, H: 10) B=I 2 2 T The period of oscillation can be easily measured. The moment of inertia of the cylindrical magnet is given by: 11) I= 1 12 ml2 1 4 mr2 where m is the mass of the magnet, L is its length, and R is its radius. b) Determining /H Using torque balance, it is possible to determine the ratio /H. At a large distance from the magnet, its magnetic field may be approximated by that of the axial field of a current loop: 12) B= 0 2 z 3 where is the magnetic moment, and z is the distance (on axis) from the center of the magnet. Illustration 2: Setup for part b) of experiment. The distance, z, is from the center of the magnet to the center of the compass. The torque from H on the compass needle is: 13) 1 = compass Hsin

4 The torque from the magnet on the compass needle is: 14) 2 = compass B cos If the needle is in stasis the torques balance: 15) compass Hsin compass Bcos =0 The magnetic moment of the compass cancels, and with some rearrangement we have: 16) Hsin =Bcos Substituting equation 12) in for B in equation 16) we get: 17) Hsin = 0 2 z 3 cos or solving for /H: 18) H = 2 z 3 tan 0 Procedures For part a) suspend a magnet from a string tied at its center and record the time it takes for it to oscillate 10 times. Repeat this 10 times and calculate the average period of oscillation. Your uncertainty in period will be the standard deviation of this result. Use equation 10) to calculate H. For part b) arrange a compass on the center of a ruler such that when no magnet is near, magnetic north is at right angles to the ruler. Start with the magnet at 15 cm from the center of the compass, and pull the magnet back from the compass in 5 cm increments until your magnet is 40cm away. Repeat this on the far side of the compass. Using a plot of tangent of the angle (y axis) vs. z -3 (x axis), determine the slope of this graph [*** Important! If you plot this incorrectly, your results will be wildly inaccurate! ***]. The slope of your graph is equal to: 19) slope= 0 2 H Use this result to calculate /H. Then using your results from part a) as well, solve for and H individually. Estimate your uncertainty in these values.

5 Analysis In addition to the above calculations, you should compare your values to the current Fredricksburg value for H ( ) and save this graph. Read off the value that corresponds to the time you collected your data. Your lab should include this graph with a demarcation of what value of H you are using. This is your gold standard. Is this value of H within your experimental uncertainty? What sources of error contribute to inaccurate results. DO NOT STATE HUMAN ERROR OR CALCULATION ERROR!