Pre-lab Quiz/PHYS 4 Eart s Magnetic Field Your name Lab section 1. Wat do you investigate in tis lab?. For a pair of Helmoltz coils described in tis manual and sown in Figure, r=.15 m, N=13, I =.4 A, wat is te magnitude of te produced magnet field at te midway of te Helmoltz coils? (Answer: elm =4.461 4 Tesla=4.46 gauss)
Lab Report/PHYS 4 Eart s Magnetic Field Name Lab section Objective In tis lab, you measure Eart s orizontal magnetic field. ackground (A) Magnetic pendulum : a bar magnet swings in magnetic Field As sown in Figure 1, wen a bar magnet is placed in a magnetic field, te net magnetic force on te bar magnet is zero. However, te net torque (τ) on te bar magnet is zero only wen te bar magnet is aligned wit (Figure 1(b). For quantitative description, one visualize te bar magnet as a magnetic dipole of magnetic moment M. Note tat M,, and τ are vector quantities and tus ave directions. Te direction of M points from te sout pole to te nort pole. Te magnitude of τ is given by sin, (1) were θ is te angle between M and. Wen te bar magnet is not aligned wit (namely, M and are not parallel wit eac oter), τ becomes non-zero and tends to rotate te bar magnet to be aligned wit (Figure 1(a) and Figure 1(c)). Denote I inertial as te moment of inertia for te bar magnet and α as te angular acceleration of te bar magnet. Ten, I inertial α equals te torque: M I inertial M sin or sin. () I inertial For small angle, sin, and Equation () becomes M. (3) I inertial Equations () and (3) resemble tose describing te motion of pendulum. For te pendulum, te torque is induced by gravity, making te pendulum swing back-and fort about te equilibrium position. At small angle, its motion is simple armonic motion. Following te discussion of pendulum in PHYS 1, te small-angle back-and-fort swing of te bar magnet is also simple armonic oscillation wit period T given by, I inertial T. (4) M Rewrite Equation (4) as
4 Iinertial 1. (5) M T I Using Equation (5), we can first determine te parameter inertial of te bar magnet by M measuring 1/T for te bar magnet in a known magnetic field, suc as elm produced by Helmoltz coils. Afterwards, we can determine te magnitude of an unknown magnetic field, suc as te Eart s magnetic field, by measuring 1/T for te bar magnet in tis field. () Eart s magnetic field Eart s magnetic field can be viewed rougly as induced by a uge bar magnet at Eart s center pointing along Eart s magnetic axis wic canges very slowly. At present, te geomagnetic sout pole is near te geograpic nort pole and te geomagnetic nort pole is near te geograpic sout pole. Like te magnetic field produced by a bar magnetic, te magnetic field on te eart surface varies from one location to anoter. At eac location, te geomagnetic field as a orizontal component ( eart, ) tangent to te eart surface and point to te geomagnetic nort pole, and a vertical component. At a location, te magnitude of eart, depends on suc as its geomagnetic latitude (not identical to geograpic latitude), tus varying from one location to anoter. At any location, te magnitude and te orientation of eart, may be affected also by te magnetic materials in te surrounding area. Terefore, it is important to map out te magnitude and te orientation of eart, on eart surface. () Te magnetic field of Helmoltz oils Figure displays a pair of Helmoltz coils consisting of two identical circular coils placed directly facing eac oter, suc tat te axial axis, wic connect teir centers, is perpendicular to bot coils. Te two coils ave te same radius r and are separated by distance r. Eac coil is wound wit N turns of wire. Wen bot coils carry current in te same direction wit te same magnitude I, very uniform magnetic field is produced between te two coils. Specifically, at te midway (Point ) between te coils, te magnetic field elm is along te axial axis wit te magnitude given by 4 NI 3/ 7 elm ( ), were 4 1 T m / A. (6) 5 r (D) Measure te unknown Eart s magnetic field using magnetic Pendulum and te known magnetic field of Helmoltz oils Align a pair of Helmoltz coils suc tat at its midway elm is parallel wit eart,, and te resultant orizontal component is 4 3/ NI elm eart, elm eart, ( ). (7) 5 r A bar magnet is placed at te mid-way of te Helmoltz coils and is restricted only to orizontal rotation. If te magnet torque dominates all oter torques, te bar magnet undergoes simple armonic orizontal small-angle swing. Following Equation (5), its time period is 1 M 4 3/ NI eart, ( ) T I r. (8) 4 inertial 5 Rewrite Equation (8) as 1 1I, wit (9) T M eart,, (1) 4 Iinertial M 4 3/ N 1 ( ). (1*) 4 I 5 r inertial
Measuring 1/T as a function of I and fitting te measured 1/T -versus-i curve by Equation (9), we obtain and 1. Dividing Equation (1) by Equation (1*), we can derive from eart, 5 3/ r eart, ( ). (11) 1 4 N Te SI units for te relevant pysical quantities in te above equations are: in Tesla (T); r in 7 meter (m); I in Ampere (A); T in second (s). 4 1 T m / A. Te SI units for is 1/s and tat for 1 is 1/(A s ). EXPERIMENT Apparatus Te main part of te set-up used in tis lab is te pair of Helmoltz coils sown in Figure 3. Te radius of te coils is r =.15 m. Eac coil is wound wit N=13 or 15 turns of wire. Note: te maximum current for te Helmoltz coils is 1. A!!! A small bar magnet is anged approximately at te mid-way between te two coils by te anging wire. Procedures 1. Set up te Helmoltz coils and te bar magnet Using te small bar magnet (wen it stops rotating) to determine te direction of te local eart, (te orizontal component of te Eart s magnetic field). Now, slowly reorient te Helmoltz coils to make te axial axis of te Helmoltz coils parallel to eart, (tis ave been done by your TA). Slowly sift te anging wire of te bar magnet to make te bar magnet align as orizontally as possible, to ensure tat te bar magnet is anged by te wire at its center-of-gravity. eck weter te bar magnet points along te axial axis. Read te marked turns of wire wound in eac coil and record N in Table 1. Te average radius of eac coil is also given in Table 1.. Set up te circuit (Figure 4) onnect te ground of te power supply to reference connector of te variable resistor. onnect te sliding connector to te common terminal of te ammeter. On te base of te Helmoltz coils, tere are two connectors labeled as Field. onnect one Field connector to te positive terminal of te power supply, and te oter to te positive terminal of te ammeter. At tis moment, te power supply sould remain off. Ask your TA to ceck te circuit! Note: te maximum current for te D Power Supply is. A!!! Now, turn on te power supply. Increase te current to 1. A. If te bar magnet rotates orizontally by 18º, it means tat te Helmoltz coils produces elm anti-parallel to eart,, and you need to excange te connections for te two field connectors. On te oter and, if tey are connected as desired in tis lab, te Helmoltz coils produces elm parallel to eart,.
3. Measure te time period T wen I= Turn off te power supply. So I= and elm =. Gently turn te bar magnet orizontally to a small angle (<15º) off te axial axis (wic is also te direction of eart, ). After being gently released, te bar magnet sould undergoes simple armonic motion by swinging back and fort around te axial axis under te restoring torque due to eart,. Measure te time duration for cycles of swing, t. Record te t value in Table. 4. Measure te time period T wen I Turn on te power supply. Set te current I in turn from.15 to 1. A in steps of.15 A. (Note: te maximum current for te D Power Supply is. A!!!) For eac current setting, gently turn te bar magnet orizontally to a small angle (<15º) off te axial axis. After being gently released, te bar magnet sould undergoes simple armonic motion. Measure te time duration for cycles of swing, t 1. Record te t 1 value in Table 3. TALE 1 r (m) N.15 TALE I (A) T (s) T (s) 1/T (1/s ) TALE 3 I (A) T (s) T (s) 1/T (1/s ).15.3.45.6.75.9 1.5 1.
Analysis 1. For te t data in Table and Table 3, calculate and record te corresponding time periods T=t / and also 1/T.. Using te data in Table 3 (do not use te I= data of Table ), plot 1/T -versus-i and use Equation (9) to fit te curve. Record te fitting parameters below = 1 = 3. Using Equation (11), te fitted and 1, and te r and N values in Table 1, calculate * (Denote ) eart, 5 ( 4 * 3/ eart, ) 1 r N eart, 4. Using Equation (1), te fitted 1, and te r and N value in Table 1, calculate M 1 4 3/ N ( = 4 I 5 r ) inertial 1 5. Using Equation (9), te calculated value, and te 1/T data measured at I= (Table ), ** calculate (Denote as ) eart, eart, ** eart, 1 1 ( ) T Questions 1. Wic value, * eart, or ** eart,, sould be more accurate? Wy?