EXPERIMENT 16 THE MAGNETIC MOMENT OF A BAR MAGNET AND THE HORIZONTAL COMPONENT OF THE EARTH S MAGNETIC FIELD

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1 THEORY EXPERMENT 16 THE MAGNETC MOMENT OF A BAR MAGNET AND THE HORZONTAL COMPONENT OF THE EARTH S MAGNETC FELD The uose of this exeiment is to measue the magnetic moment μ of a ba magnet and the hoizontal comonent B E of the eath's magnetic field. Since thee ae two unknown quantities, μ and B E, we need two indeendent equations containing the two unknowns. We will cay out two seaate ocedues: The fist ocedue will yield the atio of the two unknowns; the second will yield the oduct. We will then solve the two equations simultaneously. The ole stength of a ba magnet may be detemined by measuing the foce F exeted on one ole of the magnet by an extenal magnetic field B 0. The ole stength is then defined by = F/B 0 Note the similaity between this equation and q = F/E fo electic chages. n Exeiment we leaned that the magnitude of the magnetic field, B, due to a single magnetic ole vaies as the invese squae of the distance fom the ole. B = 2 in which k' is defined to be 10-7 N/A 2. Conside a ba magnet with oles a distance 2x aat. Conside also a oint P, located a distance fom the cente of the magnet, along a staight line which asses fom the cente of the magnet though the Noth ole. Assume that is much lage than x. The esultant magnetic field at P due to the magnet is the vecto sum of a field B N diected away fom the Noth ole, and a field B S diected towad the South ole. The distances fom P to the Noth and South oles ae - x and + x, esectively. The magnitude of the esultant field at P is B m = B N B S = ( x) 2 ( + x) 2 Putting the two tems ove a common denominato, we obtain = 4x 2 2 ( x ) 2

2 Since, x is small comaed to, we can neglect the x 2 in the denominato. 4x 4x = = 4 This can be witten in tems of the magnetic moment of the ba magnet, μ defined by the fomula μ 2x. 2 μ =. Note the similaity between the definition of the magnetic moment and the equation fo the field stength to the definition of the electic diole moment ( qd) and the equation fo the field stength of an electic diole a distance along the axis of the diole 2k ( E = ). n these equations, d is the seaation of the chages and k is Coulomb s constant. Ou fist exeimental ocedue will yield the atio of μ to B E. We will do this indiectly by comaing to B E, using a magnetomete. The magnetomete, which was also used in Exeiment, consists of a small magnetized disk attached to a long non-magnetic ointe, ivoted on a vetical axis. The ointe is mounted at ight angles to the diection of magnetization of the disk. With the ba magnet fa fom the magnetomete, the only significant magnetic field acting on the magnetized disk will be the hoizontal comonent B E of the eath's field. (The vetical comonent of the eath's field has no effect on the disk, because the disk cannot otate about a hoizontal axis.) n this case the Noth ole of the magnetized disk will oint towad magnetic noth, and the non-magnetic ointe will oint in the magnetic east-west diection. When the housing is oely oiented, the ointe will ead zeo, and a mete stick, attached to the housing, will be oiented in the magnetic east-west diection. f the ba magnet is laced on the mete stick with its Noth ole towad the magnetic east, a field, diected towad the magnetic east, will also act on the magnetized disk. The esultant field at the cente of the magnetomete is the vecto sum of BBe towad the magnetic noth and B mb towad the magnetic east. Let θ be the angle between BBe and the esultant field B. The disk and ointe must now otate clockwise though the angle θ until the Noth ole of the disk oints in the diection of the esultant field. Since the ointe oiginally ead zeo, it will now ead θ. A simle diagam shows that BBm = B E tan θ We now have an equation that will give us the atio of μ to B E.

3 B E 2μ = cot θ Fo this exeiment, we will measue the angle θ fo seveal diffeent values of. The atio of μ to B E can be calculated fom the sloe of a gah of cot θ vesus. The oduct of μ and B E may be found by susending the ba magnet fom a nealy tosion-fee sting, giving it a small angula dislacement fom equilibium, and allowing it to oscillate in simle hamonic motion. The eath's hoizontal field B E ovides the estoing toque. The toque exeted by a unifom magnetic field BBE on a ba magnet of magnetic moment μ is given by τ = μ BBE o τ = μb EB sinφ in which φ is the angle between μ and B E, which is the same as the angle between the instantaneous osition of the ba magnet and its equilibium osition. (The vetical comonent of the eath's magnetic field has no effect in this at of the exeiment eithe, because the ba magnet is not fee to otate about a hoizontal axis.) f the angle φ is small, then sin φ can be aoximated as φ and the estoing toque will be ootional to φ. f the magnet has a moment of inetia,, the angula acceleation is elated to this toque by μb α = μb Eφ o α = E φ. This equation descibes simle hamonic motion with angula fequency 2π eiod T = = 2π. ω μ B E μb ω = E and. LABORATORY PROCEDURE CAUTON: f you have a mechanical watch, kee it at some distance fom the ba magnet at all times. Use cae not to do the magnet. Jolts tend to demagnetize any magnet, educing its magnetic moment, which is one of the unknown quantities we ae measuing. 1. Place the magnetomete on to of an inveted wooden box to emove it fom feous metal in the laboatoy table. Do not tilt the magnetomete excessively, o the glass late (on some models) may fall out and beak. Place the level on to of the glass late of the magnetomete and level the instument. Remove the level some distance fom the magnetomete.

4 Remove the ba magnet some distance fom the magnetomete. Also emove any feous metals, such as the level and cetain mechanical encils. Tun the knob o wheel to aise the ivot until the ointe moves feely; howeve, do not aise the ivot so high that the ointe is ushed against the glass late. Rotate the housing so that both ends of the ointe ead zeo. f the ointe is somewhat bent, both ends should be off fom zeo by equal amounts.. f the magnetomete lacks a built-in mete stick, inset a thin mete stick into the backets, and cente it. 4. Place the ba magnet on the mete stick so that its cente is 20.0 cm fom the cente of the magnetomete. This is accomlished most easily by lacing the two ends of the magnet equally distant fom the 20.0 cm mak. The magnet must be aallel to the mete stick. Read and ecod both ends of the ointe, estimating to the neaest tenth of a degee. Teat all angles as ositive. Recod the data of stes 4-6 in tabula fom. 5. Reeat ste 4 fo distances of 22, 24, 26, 28 and 0 cm. 6. Reeat stes 4 and 5 with the ba magnet evesed in diection. 7. Remove the magnetomete a consideable distance fom the box. Set u a table clam, a vetical od, a ight angle clam, and a hoizontal od, with the hoizontal od above the box. Obtain two stings, one long and one shot, with loos at each end. Pass one loo of the long sting ove the hoizontal od. Pass the shot sting though the othe loo of the long sting. Pass the ba magnet though both loos of the shot sting. Adjust the ods and clams so that the ba magnet is susended in a hoizontal lane at the same osition as was eviously occuied by the cente of the magnetomete. (The same location is used because the Eath s magnetic field is not unifom thoughout the laboatoy oom.) 8. Twist the ba magnet 10 o 20 fom its equilibium osition, and elease it. t should oscillate in angula simle hamonic motion, twisting the sting (not simle endulum motion). Use a clock o watch to time 20 cycles. (Duing one cycle the magnet moves fom one side to the othe and back again.) t is sufficient to estimate time to the neaest second. Reeat and aveage. f the two times diffe by moe than 2%, eeat the timings. 9. Use one of the stings to susend the ba magnet above the an of the balance. Measue the mass to at least thee significant figues. Do not magnetize the an by lacing the magnet diectly on it. Do not discad the stings. 10. Use a venie calie to measue the length of the magnet. 11. f a mete stick was attached to the magnetomete in ste, emove it.

5 CALCULATONS 1. Calculate the moment of inetia of the magnet. 2. List the following quantities in tabula fom: (in metes),, θ, and cot θ. θ must be the aveage of the fou values measued at each distance.. Plot a gah of cot θ vesus. nclude the oigin on the gah. Use a staightedge to daw the staight line that best fits the lotted oints, and also asses though the oigin. Does cot θ aea to be diectly ootional to? 4. Use the Modified Least Squaes fomula to calculate the sloe of the gah. The fomula can be found in the ntoduction of this Lab Manual. 5. Fom the sloe, calculate the atio of the magnetic moment of the ba magnet to the hoizontal comonent of the Eath s magnetic field. 6. Calculate the aveage exeimental eiod of the ba magnet when oscillating in simle hamonic motion. Remembe, the eiod is the time fo one cycle. Calculate the oduct of the magnetic moment of the ba magnet and the hoizontal comonent of the Eath s magnetic field. 7. Calculate the magnetic moment of the ba magnet. 8. Calculate the hoizontal comonent of the eath's magnetic field at the location of the aaatus. Convet the esult to μt.