tructors X New Podcax ODE Lab 6 Table X DDE Lab 06 Eart X PDE Lab 06 Eart X m/OneDrive/Documents/Phys%20224%20LAB%20SP%2022/Lab%206%20report/Lab%2006%20 Instructor's Lab Manual / PHYS 224 Earth's Magnetic Field Name Lab Section_ Objective In this lab, you will measure the Earth's horizontal magnetic field. Background IN T=0 S (a) T points out of paper (b) T=0 (c) T points into paper Figure 1 (A) "Magnetic pendulum": When a bar magnet is placed in a magnetic field B, the net magnetic force on the bar magnet is zero. However, the net torque (7) on the bar magnet is zero only when the bar magnet is aligned exactly along B (Figure 1(b) ). When the bar magnet is not collinear to B (Figures 1(a) and 1(c)), there is a net torque acting on the bar. For a quantitative description, one can visualize the bar magnet as a magnetic dipole moment H. Note that #, B, and t are vector quantities and thus have directions. The direction of # points from the south pole to the north pole. The magnitude of r is given by: LOU 9 w 57%4 50 F Mostly cloudynstructors | X New Podcast PDE Lab 6 Table: X PDE Lab 06 Earti X PDE Lab 06 Earti X sam/OneDrive/Documents/Phys%20224%20LAB%20SP%2022/Lab%206%20report/Lab%2006%20% south pole to the north pole. The magnitude of r is given by: T = HB sin(0), (1) where 0 is the angle between u and B. When the bar magnet is not aligned with B (namely, u and B are not parallel with each other), t becomes non-zero (Figures 1(a) and 1(c)) and tends to rotate the bar magnet to be aligned with B (Figures 1(b)). Denote linertial as the moment of inertia for the bar magnet and a as the angular acceleration of the bar magnet. Then, the torque t is given by: t = linertial a = -H B sin(0), or a = - UB sin(0) (2) linertial For small angle, sin(0) ~ 0, and Equation (2) becomes a =_HBO Tinertial ( 3 ) Equations (2) and (3) resemble those describing the motion of a pendulum. For the pendulum, the torque is induced by gravity, making the pendulum swing back-and forth about the equilibrium position. At small angle, its motion is a simple harmonic motion. Following the discussion of pendulum in PHYS 221, the small-angle back-and-forth swing of the bar magnet is also simple harmonic oscillation with period T given by, T = 27 Inertial (4) Rewrite Equation (4) as L O U Q 9 w 57% 50.F Mostly cloudyDE Instructors | X New Podca: X DDE Lab 6 Table X (DDE Lab 06 Eart X PDE Lab 06 Earth X rs/ejsam/OneDrive/Documents/Phys%20224%20LAB%20SP%2022/Lab%206%20report/Lab%2006%20 B = 4n linertial 1 ( 5 ) Using Equation (5), we can first determine the parameter inertial of the bar magnet by measuring 72 for the bar magnet in a known magnetic field, such as the magnetic field BHelm created by Helmholtz coils. Afterwards, we can determine the magnitude of an unknown magnetic field, such as the Earth's magnetic field BEarth, by measuring 72 for the bar magnet in this field. (B) Earth's magnetic field: Earth's magnetic field can be viewed roughly as induced by a huge bar magnet at Earth's center pointing along Earth's magnetic axis which changes very slowly. At present, the geomagnetic South Pole is near the geographic North Pole and the geomagnetic North Pole is near the geographic South Pole. Like the magnetic field produced by a bar magnetic, the magnetic field on the earth surface varies from one location to another. At each location, the geomagnetic field has a horizontal component (BEarth,h) tangent to the earth surface and point to the geomagnetic north pole, and a vertical component. At a particular location, the magnitude of BEarth,h depends on its geomagnetic latitude (not identical to geographic latitude), thus varying from one location to another. At any location, the magnitude and the orientation of BEarth,h may be affected also by the magnetic materials in the surrounding area. Therefore, it is important to map out the magnitude and the orientation of BEarth,h on the Earth's surface. (C) The magnetic field of the Helmholtz Coils: Figure 2 displays a pair of Helmholtz coils consisting of two identical circular coils placed directly facing each other, such that the axial axis, which connect their centers, is perpendicular to both coils. The two coils have the same L O JO 9 W 57%4 50F Mostly cloudyPDE Instructors | X New PodcastDE Lab 6 Table X PDE Lab 06 Earth X (DDE Lab 06 Earti X sers/ejsam/OneDrive/Documents/Phys%20224%20LAB%20SP%2022/Lab%206%20report/Lab%2006%20% radius r and are separated by distance r. Each coil is wounded with N turns of wire. When both coils carry current in the same direction with the same magnitude I, very uniform magnetic field is produced between the two coils. Specifically, 00 at the midway between the coils, the magnetic field BHelm is along the axial axis with its magnitude given Figure 2 by : BHelm = ()ON!, where Ho = 41 X 10-7T m/A. ( 6 ) (D) Measure the unknown Earth's magnetic field using a "magnetic pendulum" and the known magnetic field of Helmholtz coils: Align a pair of Helmholtz coils such that at its mid-way BHelm is parallel with BEarth,h and the resultant horizontal component is Bres = BEarth,h + BHelm = BEarth,h + () HON ( 7 ) A bar magnet is placed at the mid-way of the Helmholtz coils and is restricted only to horizontal rotation. If the magnet torque dominates all other torques, the bar magnet undergoes simple harmonic horizontal small-angle swing. Following Equation (5), its time period is described by : T2 = An liertial -4 72 Iinertial | BE that | BEarthin + ()40 N1. ( 8 ) Rewrite Equation (8) as L O JO 9 W 57%4 50.F Mostly cloudyan Instructors X New Podca: X a Lab 6 Table X PDE Lab 06 Eart X PDE Lab 06 Eart X PRE Users/ejsam/OneDrive/Documents/Phys%20224%20LAB%20SP%2022/Lab%206%20report/Lab%2006%20%20%2 72 = Co + CI , (9) where : (10) (11) Measuring - as a function of the current I and fitting the measured - versus-/ curve by Equation (9), we obtain Co and G . Dividing Equation (10) by Equation (11), we can determine BearthA from (12 ) The SI units for the relevant physical quantities in the above equations are: B in tesla (T), r in meter (m), I in ampere (A), T in second (s) and Ho = 4x x 10-7T m/A. The SI units for Co is 1/s? and that for C is 1/ ( As ? ) . EXPERIMENT TT Apparatus The main part of the set-up used in this lab is the pair of Helmholtz coils shown in Figure 3. The radius of the coils is r = Figure 3 0.105 m. Each coil is wounded with N = 200 turns of wire. A small bar magnet is hanged 9 W 58% 50F Mostly cloudyODE Instructors X New Podcast X PDE Lab 6 Table X ODE Lab 06 Eart X DDE Lab 06 Earth X PDE ers/ejsam/OneDrive/Documents/Phys%20224%20LAB%20SP%2022/Lab%206%20report/Lab%2006%20%20 0.105 m. Each coil is wounded with N = 200 turns of wire. A small bar magnet is hanged approximately at the mid-way between the two coils by the hanging wire. Procedures 1. Set up the Helmholtz coils and the bar magnet: Use the small bar magnet located in the lab (when it has stopped rotating) to determine the direction of the local Barth,h (the horizontal component of the Earth's magnetic field). Now, slowly reorient the Helmholtz coils to make the axis of the Helmholtz coils parallel to BEarthh (this may have been done by your GTA) . Slowly shift the hanging wire of the bar magnet to make the bar magnet align as horizontally as possible; this will ensure that the bar magnet is hanged by the wire at its center-of-gravity. Check whether the bar magnet points along the coil axial axis. Read the marked number of wire turns wounded in each coil and record V in Table 1. The average radius of each coil is also given in Table 1. 2. Set up the circuit (Figure 4) Ammeter At this moment, the power supply A "field" -O- - -. should remain off. On the base of the Helmholtz there are two Helmholtz i coils connectors labeled as "field". Connect DC Power Supply one "field" connector to the positive terminal of the power supply, and the Figure 4 "field" : > L O J O 9 W 58% 4 50OF Mostly cloudy x X F3 F4 C F5 F6 FZ O-- F8 F9 F10 Pris F11 F12BDE Instructors X New Podca: X DE Lab 6 Table X DDD Lab 06 Eart X PDE Lab 06 Eart X Users/ejsam/OneDrive/Documents/Phys%20224%20LAB%20SP%2022/Lab%206%20report/Lab%2006%20%20% other "field" connector to the negative terminal power supply. Note: the maximum current for the DC Power Supply is 2.0 A!! ! Now, turn on the power supply. Increase the current to 1.0 A. If the bar magnet rotates horizontally by 1809, it means that the Helmholtz coils produces BHelm anti-parallel to BEarth, and you need to exchange the connections for the two "field" connectors. On the other hand, if they are connected as desired in this lab, the Helmholtz coils produces BHelm parallel to BEarth,h . 3. Measure the time period T when I = 0 Turn off the power supply. So 1 = 0 and BHelm = 0. Gently turn the bar magnet horizontally to a small angle (