THE CURRENT BALANCE Physics 258/259

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1 DSH 1988, 005 THE CURRENT BALANCE Physcs 58/59 The tme average force between two parallel conductors carryng an alternatng current s measured by balancng ths force aganst the gravtatonal force on a set of known masses. The relatonshp between the tme-average force and the root-mean-square current s nvestgated usng several methods of analyss. I. INTRODUCTION The MKS system of unts defnes the ampere n terms of the force between two parallel conductors both carryng a current: One ampere s that unvaryng current whch, f present n each of two parallel conductors of nfnte length and one meter apart n empty space, causes each conductor to experence of force of exactly 10 7 newton per meter of length. The current to be measured n ths experment s passed n opposte drectons through two parallel horzontal bars that are connected n seres. The lower bar s fxed and the upper bar s balanced a few mllmeters above t on a par of knfe-edges. The upper bar has a small pan nto whch analytcal weghts are placed, thereby causng the upper bar to drop down toward the lower one. When the current s turned on and ncreased suffcently, the repulsve force between the two bars causes the upper bar to rse back up to ts equlbrum poston. At ths pont, the repulsve force s equal and opposte to the gravtatonal force on the analytcal mass. The poston of the bar s observed by means of a mrror mounted on the balance beam and a He-Ne laser and meter stck placed a few meters from the mrror. II. THEORY Consder two parallel straght rods, each of length L and separated by a dstance d as shown n Fg. 1. The current I along the z axs n the lower rod produces a magnetc feld B at the upper rod of B = ki d ˆx, (1) 1

2 FIG. 1: The current n the lower rod s n the +ẑ drecton, whereas t s along ẑ n the upper rod. where k = µ o /π = 10 7 N/A. carryng a current I n the ẑ drecton. Ths force s Ths feld creates a force on the upper rod that s F = IL( ẑ B) = kil d ŷ. () For a snusodal tme dependence of the currents,.e. I(t) = I 0 sn(ωt), the tme average force s F = kl d 1 τ τ 0 ( ) πt I0 sn dt, (3) τ where we have ntegrated over one perod τ = π/ω. The term n the brackets s I rms and thus Eq. (3) can be wrtten as F I rms = kl d. (4) The total magnetc feld at the upper rod s comprsed of the harmonc feld as descrbed above plus any statc stray felds, such as the earth s magnetc feld. Expand the analyss to nclude a statc feld B s and show that Eq. (4) s also vald n ths case. Ths s why we use an AC current nstead of a DC current for ths experment. III. PROCEDURE The current balance s a delcate nstrument and should be handled carefully. Partcularly senstve to damage are the two knfe edge suspenson ponts and the surfaces upon whch they rest. The upper balance assembly can be rased off of the knfe edges by a lft mechansm. The electrcal wrng schematc s dagramed below. The transformers T 1 and T have an adjustable output from 0 to 135 volts AC for a 115-volt AC nput and T 3 s a fxed stepdown transformer wth a 6.3/115 rato. The ammeter s a calbrated for a root-mean-square readng. Also note that t has a reflectve strp that s used to prevent parallax errors n

3 FIG. : The wrng dagram for the current balance. readng the needle poston. Before you take any measurements, be sure to level the base by usng the two thumb screws at front of the unt. Use the lft arms to reposton the balance assembly on the posts. Practce ths a few tmes so that you can do ths smoothly. Wth no mass n the pan and no current flowng, the gap between the two rods should be a few mllmeters. If t s not, carefully adjust the counter-weght behnd the mrror. Then use the lft arms to reposton the balance assembly. Be careful to never touch the unt or make any mechancal adjustments to the apparatus whle the current s flowng. Place the 500 mg mass n the pan, whch should brng the rods nto contact. Record the poston on the He-Ne laser beam on the meter stck. Carefully remove the mass and record the new poston of the laser beam when the balance assembly s n equlbrum. Practce placng and removng the mass untl you can do t smoothly, wthout jarrng the balance. If you do bump t, use the lft arms to reposton the knfe-edges. Also note that the balance s susceptble to small ar currents. Wth the power to the transformers turned off, set T to ts mnmum value and T 1 to about full scale. Turn the power on and slowly ncrease T untl the ammeter reaches 0 A. (If the fluctuatons n the current readng are more than 0.1 A out of 0 A, notfy your nstructor. These fluctuatons can be caused by Ohmc heatng coupled to a temperature dependent electrcal resstance.) You should observe the result of the repulsve force between the parallel rods as ndcated by the movement of the He-Ne laser spot. Use T 1 to adjust the current (never exceed 0 A ) leavng T fxed. Set the current to zero and carefully place the 50 mg mass n the pan. Slowly ncrease the current untl the laser beam ponts to the equlbrum spot. Record the value of the current. Remove the mass and turn the current down to zero. Verfy that the laser spot returns to the same equlbrum poston. If not, use the lft assembly to reposton the knfe-edges 3

4 and record the new postons for when the rods are touchng and when the balance s n equlbrum and repeat ths step. You should measure the current requred to brng the balance to ts equlbrum poston for masses of 50, 100, 150, 00 and 50 mg. Verfy that the zero-current and zero-mass equlbrum poston of the laser spot has not shfted durng the placement or removal of the masses. It s good expermental technque to calculate mg/irms (whch should stay nearly constant) as you go along. When you have a good set of data ponts, adjust the counterweght slghtly to gve you a dfferent equlbrum poston and then take a second set of data. Record the followng dstances: () L, the length of the upper rod as measured from center to center of ts two supportng rods. () a, the lever arm, from the center of the front bar to the knfe edge. (Do ths for both sdes and use the average.) () b, the dstance from the mrror to the meter stck scale. (Note that the mrror s slvered on ts back surface.) (v) d u, the dameter of the upper rod. (v) d l, the dameter of the lower rod. IV. DATA ANALYSIS From your data calculate the mean value of F /Irms and ts uncertanty. Plot F as a functon of Irms. Draw a straght lne on your graph that passes through the orgn and has a slope m equal to the mean value calculated above. Use dashed lnes to draw addtonal lnes whose slopes are m + m and m m. A smple geometrc argument shows that the dstance d can be calculated from d Da b + d u + d l, (5) where D s the dfference between the just touchng and the equlbrum readngs for the poston of the He-Ne laser beam spot on the meter stck. Use ths value of d and the measured value of L to calculate the rght hand sde of Eq. (4). Compare wth your mean value of F /Irms from above. Another way to analyze your measurements of F as a functon of Irms s to do a least squares ft of the data to a straght lne of the form y = a + bx. The value of a that you calculate should be small, but may not be exactly zero. Snce Eq.(4) predcts a lnear relatonshp between F and Irms wth an ntercept of zero, you should try a least squares 4

5 ft to an equaton of the form y = cx. Frst you must mnmze the varance, σ = 1 N (y cx ) (6) wth respect to c and show that 1 the best ft value of c s c = x y x, (7) and that the uncertanty n c s 1 σ c = σ. (8) x Use these results to ft F aganst I rms. Compare to your prevous values from above. 1 These results are excercse problems 8.5 and 8.18 n: John R. Taylor, An Introducton to Error Analyss, nd Ed. (Unversty Scence Books, 1997). 5