Physics 222 - Ampere's Law Lab Report #7 (4 March 2009)

A m p e r e ’s L a w E mi l y A. G at li n Pa rt n e r: Whi t n ey H e as t o n P e rf o rm ed : 4 M a rc h 2 009 11 M a rch 200 9 Joh n Ca ru t h

OBJECTIVE T h is e xp e r i m e nt s e e k s t o s ho w t he va l i d it y o f A m p e r e ’ s L a w w h i l e le a r n i n g a bo ut a n a pp l i c a t io n fo r g ia nt ma g ne t o - r e s is t i ve s e ns o r s . T hi s e xp e r i m e nt m e a s u r e s t he ma g ne t ic f ie l d s t r e ngt h a s a fu nc t io n o f t he d is t a nc e me a s u r e d p e r p e nd ic u la r l y fr o m a lo ng c u r r e nt c ar r yi n g w ir e .

INTRODUCTION S t at ic ma g ne t ic f i e ld s c o nt a i n c u r r e nt s f lo w i ng t hr o u g h w ir e s o r s p a c e, a nd in g e o me t r ic a l l y s y m m e t r ic s it u at io ns , Am p e r e ’ s L a w e na b le s t he c a lc u la t io n o f t he m a g ne t ic f ie l d i n t er ms o f c u r r e nt . Am p e r e ’ s L a w is a ma t he ma t ic a l fo r mu la t i o n t hat r e la t e s t he ma g ne t ic f i e ld s t r e ngt h w it h a n y c u r r e nt s a s so c ia t e d w it h t hat m a g ne t ic f ie l d . I t is t he c lo s e d int e g r at io n o f t he ma g n it u d e o f t he ma g ne t ic f i e l d t hat e qu a ls t he p r o du ct bet w e e n t he p e r me a b i l it y o f fr e e s p a c e a nd t he c u r r e nt e xp r e s s e d

 Bdl  B  dl  0I ( 1 . 1 ) I nt e gr at io n a lo ng t he le ng t h o f t he c ir c l e yi e ld s t he i n f i n it e s i m a l d i f f e r e nc e s o f





 B   0i





w he r e B a nd  a r e ve c t o r qu a nt it ie s ( 1 . 2 )

   I n g e ne r a l, B a nd Δ  ha ve , a pr o d u ct t hat o nl y t he co mp o ne nt o f B is mu lt ip l i e d   t i me s t he i nc r e m e nt le ng t h. I n ve c t o r m u lt ip l i c a t io n, B is t he ―d o t pr o d u ct ‖ o r t he s c a la r mu lt ip l i c a t io n g i ve n b y

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  B  B cos   B cos 

( 1 . 3)

  w he r e  = a n g l e b e t we e n B a n d Δ    B cos  is t he co mp o ne nt o f B a lo ng Δ   B

 B

 B

r

 B  B

T he c u r r e nt o n t he w ir e  d

   B  B

 B

 

r

g e ne r a t e s a ma g ne t ic f ie l d – a i

r

m a g ne t ic f ie l d t ha t fo l lo w s t he

i

 B

r ig ht - ha nd r u le . I f t he

m a g ne t ic f ie l d mo ve s c lo c k w i s e , t he c ur r e nt f lo w s i nt o t he bo a r d. T he co nve r s e i s a ls o t r u e, w he r e i f t he c u r r e nt w er e mo v i n g i nt o t he p a p e r , t he ma g ne t ic f i e l d w o u ld be c lo c k w is e . I f t he c ir c u la r p at h w e r e d i v id e d i nt o a l a r g e nu m be r o f i nc r e me nt s  a s s ho w n a bo ve , B ha s t he s a me ma g n it u d e at e ver y i nc r e me nt a nd is i n t he s a me d ir e c t io n





 a s  . H e r e, t he a ng le r be t w e e n B a nd  is z e r o . U s i ng   0 , cos   1 t he





p r o d u ct B i s B . T hu s , t he s u m m a t io n e xp r e s s io n is no w

 

 B     B cos0   B  B   0i ( 1 . 4 ) w he r e    2r , r  radi us of ci rcul ar path T hu s , B 2  r  0i W he n r e a r r a ng e d, t his e q u a t io n ma k e s

 B  0I ( 1 . 5 ) 2r

B 1 a l i n e a r fu nc t io n o f t o g ive a s t r a ig ht i r

l i n e w he n g r a p he d :

 1 B  0 i 2 r

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T he ve r y lo ng s t r a ig ht le ng t h o f w ir e c a r r yi ng a c u r r e nt ve r i f i e s A m p e r e ’ s la w u s i ng t his r e la t io ns h i p . T he ma g ne t ic f i e ld d e c r e a s e s in ve r s e l y w it h d is t a nc e a nd m e a s u r e d r a d ia l l y o u t w ar d fr o m t he w ir e .

PROCEDURE Apparatus T he a p p a r at u s u s e d t o ve r i f y t he a fo r e m e nt io ne d Voltage

r e la t io ns h i p is r o u g hl y d e p i c t e d o n t he l e ft . T he d is t a nc e is d e t er m i ne d w it h a d ig it a l s c a le t o a p r e c is io n o f 0 . 0 1 - m m a nd t he r a d iu s is t he d is t a nc e

be t w e e n t he c e nt e r o f t he w ir e a nd t he p o int t ha t t he s e ns o r is lo c at e d be ne a t h t he s ur fa c e o f it s mo u nt i ng p a c k a g e . A G ia nt M a g ne t o r e s is t iv e ( G M R ) s e ns o r m e a s u r e s t he ma g ne t ic f ie l d w ho s e c ha ng e i n vo lt s i s d ir e c t l y p r o po r t io na l t o t he c ha ng e in ma g ne t ic f i e l d s t r e ngt h. T he d ig it a l a na m e t e r d et e r m i ne s t he me a s u r e o f t he c u r r e nt p r e s e nt in t he w ir e . We g a t he r e d d at a at a p p r o xi m a t e l y 0 . 1 - m m i nt e r va ls i n t he r a ng e 0. 2 - m m t o 1 . 0 0 - m m, 0 . 5 - m m i nt e r va ls i n t he r a ng e 1 - m m t o 5 - m m, 1 . 0 - m m i nt e r va l s in t he r a ng e 5 - m m t o 1 0 - m m, a nd 5 . 0 - m m i nt e r va ls i n t he r a ng e 1 0 - m m t o 50 - m m. A ft e r t he co l le c t io n o f d a t a, t he g r a p h o f

B 1 versus  a l lo w s t he a c qu is it io n o f i r

t he l i ne a r r e g r e s s io n e q u a t io n i n o r d e r t o g a in t he me a s u r e d va lu e 0 . T he s lo p e



m  o f t hi s l i n e g i ve s t he va lu e 20 . ~ Page 3 ~

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T he r e fo r e , u s i ng t he s lo p e in t he l i n e a r r e gr e s s io n e q u a t io n, t he me a s u r e d va lu e o f 0 w a s c a lc u l a t e d a nd co mp a r e d t o t he t he o r et ic a l va l u e i n o r d e r t o ve r i f y A mp e r e ’ L a w .

DATA/C ALCULATIONS T he t a b le be lo w s u m ma r i z e s t he me a s u r e d d at a: radius of wire

sensor depth

r’

s

0.00118 0.00118 0.00118 0.00118 0.00118 0.00118 0.00118 0.00118 0.00118 0.00118 0.00118 0.00118 0.00118 0.00118 0.00118 0.00118 0.00118 0.00118 0.00118 0.00118 0.00118 0.00118 0.00118 0.00118 0.00118 0.00118 0.00118 0.00118 0.00118 0.00118

0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005

position of GMR sensor

Voltage without current

Voltage current present

Current

p

V0

V

I

0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.0010 0.00150 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005 0.006000 0.0070 0.0080 0.0090 0.0100 0.0150 0.0200 0.0250 0.0300 0.0350 0.0400 0.0450 0.0500

0.5720 0.5557 0.5630 0.5700 0.5680 0.5700 0.5770 0.5770 0.5780 0.5750 0.5750 0.5790 0.5660 0.5700 0.5670 0.5660 0.5620 0.5557 0.5560 0.5520 0.5520 0.5490 0.5430 0.5440 0.5400 0.5370 0.5330 0.5300 0.5290 0.5270

8.9300 8.3000 7.9200 7.5800 7.2000 6.9800 6.7400 6.5100 6.2800 5.3100 4.6200 4.04 3.57 3.243 2.91 2.666 2.447 2.067 1.78 1.58 1.448 1.309 0.993 0.859 0.781 0.724 0.687 0.659 0.639 0.623

7.5600 7.4800 7.4900 7.5100 7.5300 7.5000 7.5200 7.5200 7.5400 7.5300 7.5000 7.52 7.51 7.53 7.51 7.53 7.53 7.53 7.52 7.5 7.55 7.53 7.55 7.54 7.53 7.54 7.53 7.52 7.51 7.54

k= 4 T

1.11 10

1.10 x 10--4 1.10 x 10--4 1.10 x 10-4 1.10 x 10--4 1.10 x 10--4 1.10 x 10--4 1.10 x 10--4 1.10 x 10--4 1.10 x 10-4 1.10 x 10-4 1.10 x 10-4 1.10 x 10-4 1.10 x 10-4 1.10 x 10-4 1.10 x 10-4 1.10 x 10-4 1.10 x 10-4 1.10 x 10-4 1.10 x 10-4 1.10 x 10-4 1.10 x 10-4 1.10 x 10-4 1.10 x 10-4 1.10 x 10-4 1.10 x 10-4 1.10 x 10-4 1.10 x 10-4 1.10 x 10-4 1.10 x 10-4 1.10 x 10-4

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v

1 1  r r ' s  p

DV V  V0   I i

B kDV  I i

531.914894 505.050505 480.769231 458.715596 438.596491 420.168067 403.225806 387.596899 373.134328 314.465409 271.73913 239.23445 213.675214 193.050193 176.056338 161.812298 149.700599 130.208333 115.207373 103.305785 93.6329588 85.6164384 59.9520384 46.1254613 37.4812594 31.5656566 27.2628135 23.9923225 21.4224507 19.3498452

1.105555556 1.035334225 0.982242991 0.933422104 0.880743692 0.854666667 0.819547872 0.788962766 0.756233422 0.628818061 0.539333333 0.460239362 0.4 0.35498008 0.311984021 0.278884462 0.250332005 0.200703851 0.162765957 0.137066667 0.118675497 0.100929615 0.059602649 0.041777188 0.032005312 0.024801061 0.020451527 0.017154255 0.014647137 0.012732095

0.0001227 0.0001149 0.000109 0.0001036 9.776 x 10-5 9.487 x 10-5 9.097 x 10-5 8.757 x 10-5 8.394 x 10-5 6.98 x 10-5 5.987 x 10-5 5.109 x 10-5 0.0000444 3.94 x 10-5 3.463 x 10-5 3.096 x 10-5 2.779 x 10-5 2.228 x 10-5 1.807 x 10-5 1.521 x 10-5 1.317 x 10-5 1.12 x 10-5 6.616 x 10-6 4.637 x 10-6 3.553 x 10-6 2.753 x 10-6 2.27 x 10-6 1.904 x 10-6 1.626 x 10-6 1.413 x 10-6

A mp e re ’ s L aw

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B/i vs. 1/r 0.00014 y = 0.0000002405x - 0.0000066242

0.00012 0.0001

B/i

0.00008 B/i vs. 1/r

0.00006

Linear (B/i vs. 1/r) 0.00004 0.00002 0 0

100

200

-0.00002

300

400

500

600

1/r

T he s lo p e o bt a i ne d m  0.0000002405 a l lo w s t he c a lc u la t io n fo r t he p er me a b i l it y o f fr e e s p a c e u s i ng m 

0 2

 0  m  2   . T he me a s u r e d va lu e o f 0 i s

c o mp a r e d a g a in s t t he t heo r et ic a l va l u e i n o r d e r t o d et e r m i ne t he me a s u r e o f e x p e r i m e nt a l e r r o r pr e s e nt .

Theoretical 1.26 × 10-06

Measured

% Error

1.51111×10-06 % Error =

Theoretical  Measured 100  20% Theoretical

T he po t e nt ia l s o u r c e s fo r e r r o r w it h i n t he e xp e r i m e nt ar e fr o m a va r ie t y o f s o ur c e s . F ir s t , o ne po s s i b i l it y fo r e r r o r is s i m p l y t he m i s r e a d i ng o f t he r a d iu s m e a s u r e me nt s d u e t o hu ma n e r r o r . S e co nd , t he i na c c u r a c y o f me a s u r e me nt s w it h i n c lo s e p r o xi m it y t o t he w ir e i s a n o t he r po t e nt ia l s o u r c e o f e r r o r . T h is is a s ig n i f i c a nt a r e a be c a u s e t he la r g e s t c ha r g e s in t he ma g ne t ic f i e l d s t r e ngt h o c c ur s

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A mp e re ’ s L aw

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a t t he s e s ma l l d i s t a nc e s fr o m t h e w ir e — t hu s , i na c c u r a c y a t t h is s t a g e le a d s t o a h ig he r va r ia nc e fr o m t he e xp e ct e d va lu e t ha n t he r e a d i ng s o bt a ine d a t g r e at va lu e s . Ot he r po t e nt ia l s o u r c e s o f e r r o r inc l u d e a nd a r e no t l i m it e d t o t he a p p r o x i ma t io n o f t he c u r r e nt me a s u r e m e nt in t he s e t - u p o f t he e xp e r i me nt ’ s a p p a r at u s, a nd t he g a i n o f t he a mp l i f i e r t o va r y fr o m t he a p p r o xi m a t e 20 0 . H o w e ve r , t his e xp e r i m e nt o ve r a l l d e mo ns t r at e d t he co r e co nc e pt o f A m p e r e ’ s l a w a c c u r at e l y.

CONCLUSION A lt ho u g h e r r o r e x is t e d w it h t he d at a , t he e xp e r i m e nt c le a r l y d e mo n s t r at e d ho w A mp e r e ’ s L a w r e la t e s t he ma g ne t ic f i e l d t o t he c u r r e nt u s i ng t he r e la t io ns h ip s s t at e d w it h i n B 

0I 2r

. T h is r e la t io ns h ip a l lo w e d t he e xp e r i m e nt t o c a lc u l a t e t he

va lu e o f t he p e r me a b i l it y o f fr e e s p a c e ( a co ns t a n t ) in t h is r e la t io ns h i p t o ve r i f y t hi s r e la t io ns h i p a s o ut l i ne d b y A mp e r e ’ s L a w . Ad d it io na l l y, t h is la b p r o vid e d a n a p p l ic a t io n fo r G M R s e ns o r s a nd me a s u r e d t he ma g ne t ic f i e l d s t r e ng t h a s a fu nc t io n o f t he d is t a nc e me a s u r e d p e r pe nd ic u l a r l y fr o m a lo ng c u r r e nt - c ar r yi n g w ir e . C le a r l y, A mp e r e ’ s L a w d e mo ns t r a t e s a t r u e p h ys i c a l p he no me no n t ha t r e p r o d u c e s t he s a me r e s u lt s a s ve r i f i c a t io n.

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