The Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets


 Valerie Wilkerson
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1 . The Magnetc Feld Concepts and Prncples Movng Charges All charged partcles create electrc felds, and these felds can be detected by other charged partcles resultng n electrc force. However, a completely dfferent feld, both qualtatvely and quanttatvely, s created when charged partcles move. Ths s the magnetc feld. All movng charged partcles create magnetc felds, and all movng charged partcles can detect magnetc felds resultng n magnetc force. Ths s n addton to the electrc feld that s always present surroundng charged partcles. Ths should strke you as rather strange. Whenever a charged partcle begns to move a completely new feld sprngs nto exstence (dstrbutng busness cards throughout the unverse). Other charged partcles, f they are at rest relatve to ths new feld, do not notce ths new feld and do not feel a magnetc force. Only f they move relatve to ths new feld can they sense ts exstence and feel a magnetc force. It s as f only whle n moton can they read the busness cards dstrbuted by the orgnal movng charge, and only whle n moton does the orgnal charge dstrbute these busness cards n the frst place! Does t sound strange yet? Why the magnetc feld exsts, and ts relatonshp to the electrc feld and relatve moton wll be explored later n the course. For now, we wll concentrate on learnng how to calculate the value of the magnetc feld at varous ponts surroundng movng charges. Next chapter, we wll learn how to calculate the value of the magnetc force actng on other charges movng relatve to a magnetc feld. Permanent Magnets I clamed above that the magnetc feld only exsts when the source charges that create t are movng. ut what about permanent magnets, lke the ones holdng your favorte physcs assgnments to your refrgerator? Where are the movng charges n those magnets? The smplest answer s that the electrons n orbt n each of the atoms of the materal create magnetc felds. In most materals, these mcroscopc magnetc felds are orented n random drectons and therefore cancel out when summed over all of the atoms n the materal. In some materals, however, these mcroscopc magnetc felds are correlated n ther orentaton and add together to yeld a measurable macroscopc feld (large enough to nteract wth the mcroscopc magnetc felds present n your refrgerator door). Although ths s a gross smplfcaton of what actually takes place, t s good enough for now. 1
2 The magnetc propertes of real materals are extremely complcated. In addton to the orbtal contrbuton to magnetc feld, ndvdual electrons and protons have an ntrnsc magnetc feld assocated wth them due to a property called spn. Moreover, even neutrons, wth no net electrc charge, have an ntrnsc magnetc feld surroundng them. To learn more about the mcroscopc bass of magnetsm, consder becomng a physcs major Electrc Current Movng electrc charges form an electrc current. We wll consder the source of all magnetc felds to be electrc current, whether that current s macroscopc and flows through a wre or whether t s mcroscopc and flows n orbt around an atomc nucleus. The smplest source of magnetc feld s electrc current flowng through a long, straght wre. In ths case, the magntude of the magnetc feld at a partcular pont n space s gven by the relaton, where 2 r 0 s the permeablty of free space, a constant equal to 1.26 x 106 Tm/A, s the source current, the electrc current that creates the magnetc feld, measured n amperes 1 (A), r s the dstance between the source current and the pont of nterest, and the drecton of the magnetc feld s tangent to a crcle centered on the source current, and located at the pont of nterest. (To determne ths tangent drecton, place your thumb n the drecton of current flow. The sense n whch the fngers of your rght hand curl s the drecton of the magnetc feld. For example, for current flowng out of the page, the magnetc fled s counterclockwse.) 1 One ampere s equal to one coulomb of charge flowng through the wre per second. 2
3 Another common source of magnetc feld s electrc current flowng through a crcular loop of wre. In ths case, the magntude of the magnetc feld at the center of the loop s gven by the relaton, where R s the radus of the loop, 0 2 R and the drecton of the magnetc feld s perpendcular to the loop. (To determne ths drecton, agan place your thumb n the drecton of current flow. The sense n whch the fngers of your rght hand curl s the drecton of the magnetc feld. For example, for current flowng counterclockwse around the loop the magnetc feld nsde the loop ponts out of the page, and the magnetc feld outsde of the loop ponts nto the page.) 3
4 . The Magnetc Feld Analyss Tools Long, Parallel Wres Fnd the magnetc feld at the ndcated pont. The long, parallel wres are separated by a dstance 4a The magnetc feld at ths pont wll be the vector sum of the magnetc feld from the left wre and the magnetc feld from the rght wre. For the left wre, I ve ndcated the drecton of the magnetc feld. Remember, wth your thumb pontng n the drecton of the current (out of the page), the drecton n whch the fngers of your rght hand curl s the drecton of the tangent vector (counterclockwse) The magntude of the magnetc feld from the left wre s: left left left 2 r 2 5a (2) 0 (2a) 2 ( a) 2 To determne the drecton of ths feld, notce that the magnetc feld vector s at the same angle relatve to the yaxs that the lne connectng ts locaton to the source current s relatve to the xaxs. Ths lne forms a rght trangle wth gven by: a tan 2a tan a a
5 Thus, leftx leftx leftx sn a (0.45) 5a a lefty lefty lefty cos a (0.89) 5a a Repeatng the analyss for the magntude of the feld from the rght wre gves: rght rght rght 0 2 r ( ) (2a) 5a 0 2 ( a) 2 Wth your thumb pontng n the drecton of the current (nto the page), the drecton n whch the fngers of your rght hand curl s the drecton of the tangent vector (clockwse). Notce that the magnetc feld s at the same angle relatve to the yaxs as before, although ths magnetc feld has postve x and ycomponents. a  2a Thus, rghtx rghtx rghtx sn a (0.45) 2 5a a rghty rghty rghty cos a (0.89) 2 5a a 5
6 The resultant magnetc feld at ths pont s the sum of the felds from the two source currents: x x x leftx rghtx a a a y y y lefty rghty a a a Thus, the magnetc feld s predomnately n the +ydrecton wth a slght leftward (x) component. The magntude of ths feld could be determned by Pythagoras theorem, and the exact angle of the resultng feld determned by the tangent functon. Loops of Wre The nner col conssts of 100 crcular loops of wre wth radus 10 cm carryng 3.0 A counterclockwse and the outer col conssts of 500 crcular loops of wre wth radus 25 cm carryng 1.5 A clockwse. Fnd the magnetc feld at the orgn. The magnetc feld at the orgn wll be the vector sum of the magnetc felds from the two cols. Each loop of the nner col produces: 0 2R 6 (1.26x10 )(3) 2(0.10) 18.9x10 Snce there are 100 dentcal loops of wre makng up ths col, the total feld produced by the nner col s: Ths feld s drected out of the page. 1.89x10 6 T T 100(18.9x T ) 6
7 Each outer col produces: Ths feld s drected nto the page. N( ) 2R 6 (1.26x10 )(1.5) 500 2(0.25) 1.89x10 3 T Snce the two cols produce equal magntude but opposte drected magnetc felds at the orgn, the net magnetc feld at the orgn s zero. (Ths does not mean, however, that the feld s equal to zero at any other ponts n space.) 7
8 . The Magnetc Feld Actvtes 8
9 Determne the drecton of the net magnetc feld at each of the ndcated ponts. The wres are long, perpendcular to the page, and carry constant current ether out of (+) or nto () the page. a. +  b
10 Determne the drecton of the net magnetc feld at each of the ndcated ponts. The wres are long, perpendcular to the page, and carry constant current ether out of (+) or nto () the page. a.   b
11 Determne the drecton of the net magnetc feld at each of the ndcated ponts. The wres are long, perpendcular to the page, and carry constant current ether out of (+) or nto () the page. a b
12 Determne the drecton of the net magnetc feld at each of the ndcated ponts. The wres are long, perpendcular to the page, and carry constant current ether out of (+) or nto () the page. a b
13 Determne the drecton of the net magnetc feld at each of the ndcated ponts. All fgures are planar and the wres are long and carry constant current n the drecton ndcated. a. b. 2 c. 3 d. 4 2 e
14 Determne the drecton of the net magnetc feld at each of the ndcated ponts. All fgures are planar and the wres are long, nsulated from each other, and carry constant current n the drecton ndcated. a. b. 2 c. d
15 Determne the drecton of the net magnetc feld at each of the ndcated ponts. The ponts are n the parallel or perpendcular plane passng through the center of each crcular hoop. a. b. c. d. 15 2
16 For each of the current dstrbutons below, ndcate the approxmate locaton(s), f any, where the magnetc feld s zero. The wres are long, perpendcular to the page, and carry constant current ether out of (+) or nto () the page. a. + + b. +  c d e
17 For each of the current dstrbutons below, ndcate the approxmate locaton(s), f any, where the magnetc feld s zero. The wres are long, perpendcular to the page, and carry constant current ether out of (+) or nto () the page. a b
18 For each of the current dstrbutons below, determne and clearly label the regons where the magnetc feld ponts nto and out of the plane of the page. All fgures are planar and the wres are long, nsulated from each other, and carry constant current n the drecton ndcated. a. b. 2 c. d
19 A par of long, parallel wres separated by 0.5 cm carry 1.0 A n opposte drectons. Fnd the magnetc feld drectly between the wres. +  Qualtatve Analyss On the graphc above, sketch the drecton of the magnetc feld at the requested pont. Explan why the magnetc feld ponts n ths drecton. Mathematcal Analyss 19
20 Four long, parallel wres wth spacng 1.5 cm each carry 350 ma. Fnd the magnetc feld at the center of the wre array Qualtatve Analyss On the graphc above, sketch the drecton of the magnetc feld at the requested pont. Explan why the magnetc feld ponts n ths drecton. Mathematcal Analyss 20
21 Fnd the magnetc feld at each of the ndcated ponts. The long, parallel wres are separated by a dstance 4a. a. b. c. +  d. Mathematcal Analyss 21
22 Fnd the magnetc feld at each of the ndcated ponts. The long, parallel wres are separated by a dstance 4a. a. b. c d. Mathematcal Analyss 22
23 The long, parallel wres at rght are separated by a dstance 2a. Fnd the magnetc feld at all ponts on the yaxs. + + Mathematcal Analyss Questons For all ponts on the yaxs, what should y equal? Does your functon agree wth ths observaton? At y = 0, what should x equal? Does your functon agree wth ths observaton? Sketch x below. Indcate the value of the functon at y = 0. y 23
24 The long, parallel wres at rght are separated by a dstance 2a. Fnd the magnetc feld at all ponts on the yaxs. +  Mathematcal Analyss Questons For all ponts on the yaxs, what should x equal? Does your functon agree wth ths observaton? Sketch y below. Indcate the value of the functon at y = 0. y 24
25 The long, parallel wres at rght are separated by a dstance 2a. Determne the locaton(s), f any, where the magnetc feld s zero Mathematcal Analyss 25
26 The long, parallel wres at rght are separated by a dstance 2a. Determne the locaton(s), f any, where the magnetc feld s zero Mathematcal Analyss 26
27 The long, parallel wres at rght are separated by a dstance 2a. Determne the locaton(s), f any, where the magnetc feld s zero Mathematcal Analyss 27
28 Fnd the magnetc feld at the orgn. The nner radus R col conssts of N loops of wre carryng current counterclockwse and the outer radus 3R col conssts of 2N loops of wre carryng current clockwse. Mathematcal Analyss 28
29 Fnd the magnetc feld at the orgn. The nner radus R col conssts of 3N loops of wre carryng current counterclockwse and the outer radus 3R col conssts of N loops of wre carryng current 2 counterclockwse. Mathematcal Analyss 29
30 A long wre carryng 2.0 A s bent nto a crcular loop of radus 5.0 cm as shown at rght. Fnd the magnetc feld at the center of the crcle. Mathematcal Analyss 30
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