mechanical Magnetic Work and Thermodynamics

7/28/2019 mechanical Magnetic Work and Thermodynamics

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C H A P T E R 4

M agnetic W ork andThermodynamics

W e re ma rk e d in th e p r e v io u s c h a p te r t h a t o n e i s a b le t o e v a lu a t e t h e wo rkn e c e s s a ry t o b u i ld u p a c e r t a in c u r r e n t d i s t r i b u t io n in t h e p r e s e n c e o fma g n e t i c me d ia i f o n e k n o ws th e c o n s t i t u t i v e l a w , B(H) o r M (H) , o f t h eme d iu m. S e v e ra l me c h a n i s ms ma y a f f e c t t h e fo rm o f t h i s l a w , b u t t h e r ea r e t h r e e ma in t y p e s o f c o n t r ib u t io n s t h a t we c a n id e n t i f y : magne tos ta t i cef fects; v a r i a t i o n s o f f re e e n e rg y d u e to v a r io u s a to mic s c a le me c h a n i s m s ,like e x c hange or an i so t ropy ; an d e ne rgy d i s s ipa t ion beca use o f hys te res i s . Inth is chap te r we d iscuss the ro le o f magne tos ta t ic e f fec ts . They a re a lwaysp r e s e n t t o s o m e e x t e n t a n d r e p r e s e n t t h e s u b s t r a t e o n w h i c h w e c a nd e v e lo p a s e n sib l e d e s c r ip t io n o f a ll o th e r r e l e v a n t me c h a n i s m s . M a g n e -tos ta tic en e rgy i s, in an idea l ized sense m ad e p rec ise in Sec t ion 4 .1 .1 , them e c h a n i c a l w o r k s p e n t i n b u i l d i n g u p t h e f in a l m a g n e t i z a t i o n c o n f ig u r a -t i o n p i e ce a f t e r p i ec e , i n a w a y s imi l a r t o w h a t we d o w h e n w e c a l c u la t et h e e le c tr o st at ic e n e r g y o f a b o d y b y s u m m i n g u p t h e w o r k s p e n t t o b r i n gcharge a f te r cha rge f rom in f in i ty to the i r f ina l pos i t ion . The magne t icw o r k p e r f o r m e d o n a m a g n e t i c s y s t e m c a n b e e x p r e ss e d i n a n a t u r a l w a yin t e rms o f t h e v a r i a t i o n o f ma g n e to s t a t i c e n e rg y a n d o f a n o th e r t e rmtha t desc r ibes hys te res i s e f fec ts an d o the r in te rn a l p rocesses charac te r i s t ico f t h e me d iu m . I t i s o n th i s b a s i s th a t a p p ro p r i a t e t h e r m o d y n a m ic r e l a t io n sfo r magne t ic bod ies can be de r ived . Th is a spec t i s d i scussed in Sec t ion4 . 2 . Be c a u s e h y s t e r e s i s i s a n i n h e re n t ly o u t -o f - e q u i l i b r iu m p h e n o me n o n ,s o m e a t t e n t i o n t o n o n e q u i l i b r i u m t h e r m o d y n a m i c r e la t io n s s h a ll b e p a id .

4 .1 M A G N E T I C W O R K A N D C O N S T I T U T I V E L A W SS e v e ral m e c h a n i s m s a r e e x p e c t e d to c o n t r ib u t e t o th e e n e rg y t r a n s fo rm a -t io n s t a k in g p l a ce i n a ma g n e t i c b o d y. I n t h is s e c tio n , w e s h o w h o w th e s e

103


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104 CHAPTER 4. Magnetic W ork and Thermod ynamicsva r i ous con t r i bu t i ons can be l og ica l ly sepa ra t ed . Three m a i n s t eps w i l lbe t aken .

( i ) The body i s t r ea t ed a s an uns t ruc t u red a s sem bl y o f e l em en t a rym agne t i c m om en t s . A t t h i s s t age , one i s no t i n t e re s t ed i n t hed e t a i l s o f t h e m o m e n t a r r a n g e m e n t i n s i d e t h e b o d y o r i n a n ykind of in t e rac t ion tha t i s no t of m agn etos ta t i c or ig in . Two asp ec t scha rac t e r i ze t he sys t em : t he m agne t i za t i on spa t i a l d i s t r i bu t i onM (r ) and t he geom et r i ca l shape o f t he body . The ene rgy i nv o l vedin th i s l eve l of desc r ip t ion i s the m a g n e t o s t a t i c e n e r g y .

( i i ) In Sec t ion 3 .2 , the magnet i za t ion M(r) was t rea ted as a g ivenquan t it y , w i t h no a t t en t ion t o t he f ac t tha t i t m i gh t o r m i gh t no trepre sen t a rea li s ti c conf igu ra t ion rea l i zab le in prac ti ce . One goesb e y o n d t hi s d es c r ip t iv e a p p r o a c h , a n d a d d r e s s e s t h e p r o b l e m o fhow a ce r t a i n m agne t i za t i on d i s t r i bu t i on can be i nduced i n t hebody by app rop r i a t e ex t e rna l ac t i ons . By t he ene rgy r e l a t i onsde r i ved i n Sec t i on 3 .3 , one ca l cu l a t e s t he w ork pe r fo rm ed on am agne t i c bod y by ex t e rna l sou rces w he n a fi el d is c r ea ted i n t hereg i on occup i ed by t he body . Th is w o rk t u rns ou t t o be eq ua l t ot h e m a g n e t o s t a ti c e n e r g y w i t h t h e a d d i t i o n o f a n o t h e r c o n t r ib u -t i on , w h i ch desc r i bes t he i n t e rna l f ea t u re s t ha t m ake t he bodyd i f f e r e n t f r o m t h e u n s t r u c t u r e d m o m e n t a s s e m b l y p r e v i o u s l ycons i de red . The e s t i m a t e o f th i s con t r i bu t ion r equ i re s t he k now l -edge o f t he M (H ) r e l a t ionsh i p .( ii i) M (H) p lay s the role of c o n s t i t u t i ve l a w for the m ed ium . A c lassi fi -ca t i on can be m ade , on t he bas i s o f t he qua l i t a t i ve l y d i f f e ren tfea tures of the M(H) l aw observed in d i f fe ren t cases .

4 . 1 . 1 A m a g n e t i c b o d y a s a n a s s e m b l y o f m a g n e t i c m o m e n t sM a g n e t o s t a t i c e n e r g y , i n t he sense d i scussed in th i s sec t ion , i s a proper tyt ha t can be a t t r i bu t ed t o a m agne t i c body , w hen t he body i s r ep re sen t eda s a n a s s e m b l y o f e l e m e n t a r y m a g n e t i c m o m e n t s , o f t h e t y p e d i s c u s s e din Sec t ion 3 .3 .2 , and i t i s assumed tha t magnetos ta t i c in t e rac t ions a ret he on l y r e l evan t m echan i sm . The m agne t os t a t i c ene rgy r ep re sen t s t hem e c h a n i c a l w o r k s p e n t t o b u i l d u p t h e b o d y b y b r i n g i n g i t s m a g n e t i cmoments , one a f t e r the o ther , f rom inf in i ty to the i r f ina l pos i t ion , asp ic tu red in F ig . 4 .1 . Of course there i s a s t ron g idea l i za t ion behin d th i sde f i n i t i on , and no one w ou l d even i m ag i ne bu i l d i ng up a p i ece o f i ron


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4.1 MAGNETIC WO RK AN D CONSTITUTWE LAWS 105

Illl0FIGURE 4.1 . Bringing elemen tary mo m en t into place in a macroscopic body.

i n t h i s w a y . Y e t , t h e i m p o r t a n c e o f t h i s a p p r o a c h w i l l b e r e c o g n i z e d i nS e c t io n 4.1 .2 , w h e r e i t w i l l b e s h o w n t h a t m a g n e t o s t a t i c e n e r g y n a t u r a l l ye m e r g e s a s o n e o f t h e t e r m s c o n t r i b u t i n g t o t h e to t a l m a g n e t i c e n e r g y o fa b o d y .

I n t h e d e s c r i p t i o n o u t l i n e d , t h e m a g n e t i c b o d y c o n s i s ts o f a g re a t n u m -b e r o f e l e m e n t a r y m o m e n t s m i. B i i s t h e f i e ld c r e a t e d b y m i a n d ~ i Bi i s thet o t a l f i e l d c r e a t e d b y a l l m o m e n t s . L e t u s c o n s i d e r a c e r t a i n e l e m e n t a r yv o l um e AV o f t h e b o d y, la r ge e n o u g h t o c o n t a i n m a n y m o m e n t s . W e ar en o t i n t e r e s t e d i n t h e e x a c t d i s p o s i t i o n o f t h e m o m e n t s i n s i d e AV. A c t u a ll y ,w e a i m a t c h a ra c t e r iz i n g t h e s y s t e m s i m p l y i n te r m s o f t h e a v e r a g e m o -m e n t d e n s i t y M . A c c o r d i n g l y , w e c o n s i d e r t h e c a s e w h e r e t h e m o m e n t so c c u p y r a n d o m p o s i t i o n s i n s id e AV. L e t u s c o n s i d e r o n e o f t h e s e m o m e n t s ,s a y m 0. I ts p o t e n t i a l e n e r g y i n t h e f ie l d c r e a t e d b y t h e o t h e r m o m e n t s is ,a c c o r d i n g t o E q . (3 .7 7), - m 0 9~ i Bi. B e c a u s e it o c c u p i e s a r a n d o m p o s i t i o ni n s i d e AV, m 0 w i l l e x p e r i e n c e o n t h e a v e r a g e t h e f i e ld o b t a i n e d b y a v e r a g -i n g ~ i Bi over AV. Th i s &V-average ju s t g ives , in the s ense o f Eq . (3 .1 ) ,the rna~ np tnqta t ic f ip lc l R . , e raa tpcl hxr fha hnrlxr


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106 CHAPTER 4. Magnetic Work and Therm odynam ics

of the o th e r m om en t s p re sen t in s ide A V. Ac cord ing to Eq . (3.25) , t h i sc o n t r i b u t i o n i s e q u a l t o 1

2/z0 1 2/z03 AV ~ m i = T M (4.2)i e a VT h e a v e ra g e p o t e n t i a l e n e rg y o f m 0 i s t h u s

U ~ 1 7 6 B M - s / z ~ (4.3)W h e n w e s u m u p t h e e n e rg i e s o f a ll m o m e n t s i n s id e A V, a n d t h e n w es u m u p t h e e n e r g i e s o f a l l e l e m e n t a r y v o l u m e s i n t h e b o d y , w e o b t a i nt h e f i n a l e n e rg y e x p re s s i o n

I f ( a/z0 M ) dSr (4.4)U = - ~ M . BM 3V

w h e r e t h e s u m o v e r t h e A V v o l u m e s h a s b e e n t r a n s f o r m e d i n to a s p a c e1i n t e g ra l o v e r t h e b o d y v o l u m e V, a n d t h e f a c to r 5 i s n e e d e d t o ta k e a c c o u n to f t h e f a c t t h a t e a c h m o m e n t c o n t r i b u t e s t w i c e t o t h e su m , o n c e a s f i e l ds o u r c e a n d o n c e a s t e s t m o m e n t .I t i s conven ien t to express U in t e rms o f H f i e lds ins t ead o f B f i e lds .The genera l re l a t ion BM = / ,60(H M if - M ) sh o w s t h a t BM - 2 / z 0 M / 3 =/z0(H M + M /3 ) . There fo re

f 11"~ M 2 d 3 r (4.5)--" [do H M 9 M d g r - - - ~2 V VI f t h e s e c o n d t e rm o f E q . (4 .5 ) w e re t h e o n l y i m p o r t a n t o n e , t h o se m a g n e t i -z a t i o n c o n f ig u r a t io n s s h o u l d b e e n e r g y - f a v o r e d , w h e r e M is m a d e a sl a rg e a s p o s s ib l e i n e a ch e l e m e n t a r y v o l u m e , b y a l ig n i n g t h e e l e m e n t a r ym o m e n t s a l o n g a c o m m o n d i r e c t i o n . I n m o s t c a s e s , h o w e v e r , t h i s t e r mp l a y s a m i n o r ro le . I n p a r t ic u l a r , i n f e r ro m a g n e t i c m a t e r i a l s m o m e n t a l i g n -m e n t i s d i c t a t e d b y e x c h a n g e fo rc e s , d i s c u s se d i n C h a p t e r 5 , r a t h e r t h a nb y m a g n e t o s t a t i c i n t e r a c t i o n s . E x c h a n g e fo rc e s g i v e , i n t h e m e a n - f i e l da p p r o x i m a t i o n , a t e r m d e p e n d e n t o n M 2 , jus t as in Eq. (4 .5) , but wi th ap ro p o r t i o n a l i t y c o e f f ic i en t l a rg e r b y m o re t h a n t h r e e o rd e r s o f m a g n i t u d e .The re l evan t t e rm i s t he f i r s t t e rm o f Eq . (4 .5 ) . We sha l l t hus de f ine theb o d y m a g n e t o s t a t i c e n e r g y a s

1The deriva tion of Eq. (4.2) is equivalent to the introduction of the so-called Lore n t z c av i tyf ield, because Eq. (3.25) requires that the field aroun d the mo me nt location be estimated byintegration over concentric spherical shells (see end of Section 3.1.3).


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4.1 MA GNE TIC WORK AND CONSTITUTWE LAWS 107

t1"0 f M 2U M = U -}- - ~ d g r ( 4 . 6 )vtha t i s ,

U M = - t1 "~ f H M 9 M d3r (4.7)2 vT h is e q u a t i o n c a n b e w r i t t e n i n t h e e q u i v a l e n t f o r m

= /~ 0 f H 2 d 3 r ( 4 . 8 )where the in tegra l i s now ca lcu la ted over a l l space . Equat ion (4 .8 ) i sob ta ined by express ing , in Eq . (4 .7) , M as M = BM / /~ 0 - H M and by ex -p lo i t ing the f ac t tha t the in tegra l over a l l space of HM 9 BM van ishe s , as acons eq uence o f t he f act t ha t V . BM = 0 an d V H M = 0 ( see A p pen d i x C ).

I n t h e c a s e o f a n e l l i p s o i d a l b o d y u n i f o r m l y m a g n e t i z e d a l o n g o n eof i t s pr incipal axes , Eq. (3 .49) and Eq. (4 .7) show that

U M - - /1"~N M 2 (4.9)V 2w h e r e V is t h e b o d y v o l u m e . F o r a g i v e n d e m a g n e t i z i n g f ac to r N , th ee n e r g y d e n s i t y U M / V i s i ndependen t o f t he body s i ze , and f o r g i ven Mi t r e a c h e s i t s m a x i m u m v a l u e , ~}1VI2/2 , when N --- 1 (i.e. , a flat diskm agn e t i ze d a l ong i t s r evo l u t i on ax is ). In t he mor e gene r a l ca s e o f a r b i t r a r ymagne t i za t i on o r i en t a t i on , acco r d i ng t o E q . ( 3 . 54 )

UM_I~Ov 2 ( N aM 2 + N bM ~ + N c M 2 ) (4.10)

4.1.2 Energy conservation and magnetic workI n t h e s i t u a t i o n j u s t a n a l y z e d , t h e m a g n e t i z a t i o n o f t h e b o d y i s k n o w ni n advance . O n t h i s ba s i s , one t hen s t ud i e s t he p r ope r t i e s o f t he f i e l dgene r a t ed by t he g i ven con f i gu r a t i on . I n r ea l i t y , t he magne t i c s t a t e o ft he body i s no t a p r i o r i know n , bu t i s t he r e s u l t o f s ome ac t i on , l i ket he app l i ca t i on o f an ex t e r na l fi el d . T he cen t r a l p r ob l e m i s t o com pr e hen dt he gene r a l r e l a t i ons ex i s t i ng be t w een t he ex t e r na l f i e l d and t he ens u i ngm a g n e t i z a t i o n .


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108 CHAPTER 4 . Magnet ic Work and Therm odynam ics

I n o r d e r t o d i s c u s s t h is p o i n t , w e s h a l l c o n s i d e r t h e s i t u a t i o n s k e t c h e di n F i g . 4 . 2 . W e h a v e s o m e s e t u p ( e . g . , a s o l e n o i d ) b y w h i c h w e p r o d u c em a g n e t i c f ie ld s . F i rs t, w e c o n s i d e r t h e c a s e w h e r e n o m a g n e t i c b o d i e s a r ep r e s e n t i n t h e r e g i o n a r o u n d t h e s e t u p . C e r t a i n c u r r e n t s j a f l o w i n t h es e t u p a n d c r e a t e i n e m p t y s p a c e t h e m a g n e t i c f i e l d B a = / z 0 H a, w h i c h w ecal l the a p p l i e d fi e l d . H a s a t i s f i e s t h e e q u a t i o n s

V - H a - 0 (4 .11)V x H a = j a

N o w w e i n s e r t s o m e w h e r e i n t h e s e t u p a m a g n e t i c b o d y a n d t h e n w es w i t c h o n t h e s a m e c u r r e n t s j a a s b e f o r e . T h e f i e ld c r e a t e d b y t h e c u r r e n t sw i l l i n d u c e a c e r ta i n m a g n e t i z a t i o n M i n t h e b o d y , n o t n e c e s s a r i ly u n i f o r m .L e t u s c o n s i d e r t h e m a g n e t o s t a t i c f ie l d H M c r e a t e d b y M , i n t h e s e n s ed i s c u s s e d i n S e c t i o n 3 .2 . H M o b e y s t h e s e t o f e q u a t i o n s

V - H M = - V . M (4 .1 2)V X H M = 0

D u e t o t h e l i n e a r it y o f M a x w e l l ' s e q u a t i o n s , t h e t o t a l H f ie l d e x i s ti n gw h e n t h e c u r r e n t s ja a r e f l o w i n g a n d t h e m a g n e t i c b o d y i s p r e s e n t i s j u s tg iven by the sum o f the so lu t ions o f Eq . (4 .11 ) and Eq . (4 .12 ) :

H = H a + H M (4.13)a n d t h e c o r r e s p o n d i n g m a g n e t i c i n d u c t i o n i s

B =/~0( H + M ) = ~0(Ha + HM + M) (4 .14)L e t u s s e e w h a t w e c a n s a y a b o u t t h e e n e r g y o f t h e b od y . W e w o u l d l ik et o d e r i v e s o m e e x p r e s s i o n e x p l ic i tl y d e p e n d i n g o n H a , a s th i s is th e f i e l dt h a t w e k n o w i n a d v a n c e a n d w e c a n c o nt ro l b y a d j u s t i n g th e m a g n e t i z i n g

FIGURE 4.2 . Le ft : Setup creating ap plied f ield H a in em pty space. Right: Sam esetup in the presence of a magnetic body.


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4.1 MA GN ETIC W OR K AN D CONSTITUTIVE LAWS 109cur ren ts ja . On the o the r hand , Ha + H M , not H a, i s the f ie ld invo lved inth e p ro b le m , a n d th e ro le p o s s ib ly p l a y e d b y H a i s n o t o b v io u s .

Th e l a w o f c o n s e rv a t io n o f e n e rg y is e x p re s se d b y th e P o y n t in g th e o -r e m (Eq. 3.85). Le t u s c o n s id e r t h e fo rm t a k e n b y th i s e q u a t io n w h e n th ein t e g ra t io n v o lu m e f~ e x t e n d s t o a ll sp a c e a n d th e P o y n t in g v e c to r f l o wc o n s e q u e n t ly g o e s t o z ero . Th e wo rk 8 L p e r fo rm e d b y th e e x t e rn a l e. m .f .sE ' in the shor t t ime in te rva l 8 t is

8 L - 8 t f j a 9 E ' d gr = ~ H . 6 B d S r + 3 t ~ j 2 d S r (4.15)t rvw h e r e V i s t h e b o d y v o l u m e . L e t u s n o w s u p p o s e w e s w i t c h o n t h ema g n e t i z in g c u r r e n t s j a s o s lo wly th a t t h e e d d y -c u r r e n t d i s s ip a t io n j 2 / o ri n s id e t h e b o d y c a n b e n e g le c te d . 2 Th e w o rk e x p re s s io n th e n r e d u c e s t o

3 L = f H 9 o13 d3 r (4.16)Eq u a t io n (4.1 6 ) d o e s n o t r e d u c e to z e ro wh e n n o m a g n e t i c b o d y i s p r e s e n t .In fac t , ene rgy i s spen t to c rea te the app l ied f ie ld even in empty space .Th i s e mp ty s p a c e e n e rg y i s a lwa y s t h e s a me fo r a g iv e n s e tu p a n d h a sno re la t ion to the spec i f ic p roper t ie s o f the magne t ic body . There fo re , i ti s a p p ro p r i a t e t o s u b t r a c t i t f r o m 8L, i n o rd e r t o h a v e a wo rk e x p re s s io nd e s c r ib in g th e e n e rg y m o d i f i c a ti o n s b ro u g h t a b o u t b y th e p r e s en c e o f th eb o d y . W e s h a l l t h u s d e f in e t h e r e l e v a n t ma g n e t i c wo rk a s

8 L = ~ (H . o13 - / _ / , o H a 9 O C ~ a ) d 3 r (4.17)By ma kin g use o f Eq. (4 .13) and Eq . (4.14), th i s in teg ra l c an be expresseda s

8L = / .to f r a 9 H M ) d 3 F + f l~ 3 f H M " b 'l- IM d 3 r 4 - ,U,of H - b ' l~ d 3 r ( 4 . 1 8 )The in tegra l over a l l space o f H a 9 H M is ze ro , beca use V 9 H a -- 0 an dV x H M = 0 (s ee Ap p e n d ix C) . On th e o th e r h a n d ,

[.1, f H M 9 o I " IM d 3 r = r M (4.19)w he re UM is the m agn e tos ta t ic ene rgy , de f ined b y Eq . (4 .8) . There fo re wec o n c lu d e th a t

t~L = t~UM 4 - / , t o ~ H . b 'M d 3 r (4.20)./v

2The precise condit ions under which this approximation is acceptable wil l be discussedin Chapter 12. In any case, this assumption implies that any eddy-current dissipat ion thatkeeps on giving an appreciable contr ibut ion under arbi t rar i ly slow exci tat ion rate shouldnecessarily be included in the B(H) constitutive law (see also Section 3.3.4).


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110 CHAPTER 4. M agne tic W ork and Thermodynamicswh ere n ow the in tegra l is ca lcula ted over the bod y vo lum e V only , beca useM = 0 in outer space . Equ at ion (4.20) can be expressed in an a l te rna t iveinterest ing form by consider ing that , according to Eq. (4.7) ,

r = - ~ f H M" 6 M d g r - ~ f M 9 O~ = - r o o f HM " o~(4.21)

where the las t equal i ty der ives f rom the appl ica t ion of the rec iproc i tytheorem s of App end ix C. By inser t ing Eq. (4 .21) in to Eq. (4 .20) and bytakin g into a ccou nt Eq. (4.13) , w e obtain8L = 1~o [ H a " b'M dSr (4.22)

dvBo th Eq. (4.20) an d Eq. (4.22) are re m arka ble. Eq. (4.22) satisfies our initialrequi rem ent , to f ind a wo rk express ion co nta in ing the app l ied f ie ld H a,even i f th i s is not the t rue f ie ld involved in the problem. Equ at ion (4.20) ,on the o the r hand , g ives t he decompos i t i on o f t he magne t i c work in tomagn e tos t a ti c ene rgy and an add i t i ona l te rm, dep end en t on the ma te r ia lconst i tut ive law, M(H).

4 . 1 . 3 C o n s t i t u t i v e l a w sThe v alue of Eq. (4.22) l ies in the fact that i t expresses the w or k 8Lin terms of quant i t ies of direct physical interest : the appl ied f ie ld Ha,which descr ibes the ac t ion of ex terna l sources independent ly of thep rope r t i e s o f t he body , and the magne t i za t ion M, wh ich cha rac t e r i ze sthe m agnet ic s ta te of the body. N ote tha t , i f H a i s uni form in theregion occupied by the body, then i t can be taken out of the in tegra land Eq. (4 .22) becom es

~ L = ] z 0 H a 9 8In ( 4 . 2 3 )w h e r e

m = f M(r) d3r (4.24)v

i s the to ta l magnet ic moment of the body.O n the oth er ha nd , the interest of Eq. (4.20) l ies in the fact that i t show sthe va r ious fo rms in wh ich ene rgy m ay be s to red in the sys tem. The mag ne -tos ta t ic energ y U M ac ts as a sor t of bac kgr ou nd energy , de te rm ined by thespa t ia l d i s tr ibut ion of m agne t iza t ion an d by the geom etr ica l shape of the


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4.1 MA GN ETIC W OR K AN D CONST1TUTWE LAWS 111b o dy . A d d i t i o n a l p r o c e ss e s o r d e v ia t i o n s f ro m t h e a s s u m p t i o n s m a d e i nthe ca lcu la t ion o f U M c o n t r ib u t e t o t h e e n e rg y b a l a n c e t h ro u g h th e t e rm

8 L - r M -- [-~0~H " 3 M d 3r (4.25)v

Ac c o rd in g to Eq. ( 4 .2 5 ), t h e e n e rg y s to r e d th ro u g h th e v a r io u s i n t e rn a lme c h a n i s ms a c t in g in t h e me d iu m i s r e f l e c t e d in t h e c o n s t i t u t i v e l a wM (H) . C o n s e q u e n t ly , q u a l i t a t i v e ly d i f f e re n t M (H ) c u rv e s c a n b e e x p e c t e d ,and the d i f fe rences obse rved may se rve as a bas i s fo r the c lass i f ica t iono f ma g n e t i c me d ia . Th e k e y wo rd s a r e , i n t h i s r e s p e c t , d i a m a g n e t i s m ,p a r a m a g n e t i s m , a n d f e r r o m a g n e t i s m . T h e ma in f e a tu r e s o f t h e s e k in d s o fb e h a v io r a r e s u mma r i z e d in F ig . 4 . 3 .

FIGURE 4.3 . Co nstitutive law s for different kin ds of m agne tic behavior.


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1 1 2 CHAPTER 4. Magnetic Work and Therm odyna micsD i a m a g n e t i s m . I n a d i amagne t i c s ubs t ance , t he l i nea r l aw M = xH ho l ds ,w i t h a nega t i ve s u s cep t ib i li ty . T he e f f ec t is ex t r eme l y s m a l l and l eads t ovalues of I ;~ of the order of 10 -6 to 10-5. S t r ic t ly s p eak i ng , d i a m ag ne t i s md o e s n o t f al l w i t h i n t h e f r a m e w o r k o f o u r d e s c r i p t io n o f m a g n e t i c m e d i aa s a s s e m b l ie s o f p e r m a n e n t m a g n e t i c m o m e n t s . D i a m a g n e t i s m a r i se s fr o mthe fac t tha t , in the presen ce of an ex te rn a l f i eld , o rb i t a l e lec t rons in a to m smod i f y t he i r mo t i on i n a w ay t o s h i e l d t he ex t e r na l f i e l d . I n a s ens e ,d i a m a g n e t i s m i s a m i c r o s c o p i c m a n i f e s t a t i o n o f L e n z ' s l a w , a l t h o u g h aq u a n t u m m e c h a n i c a l t r e a t m e n t i s n e c e s s a r y i n o r d e r t o c a r ry o u t a s a t is fa c -t o r y a na l y s i s o f the p r o b l em. T he f ac t t ha t X i s nega t i ve i m p l i e s t ha t t heco r r e s pon d i ng ene r g y i n t eg r a l i n E q . (4 .25) is a ls o nega t i ve . H ow eve r , t h i sd o e s n o t m e a n t h a t e n e r g y i s r e l e a s e d b y t h e s y s t e m . A c c o r d i n g t o E q .(4.17), 8 L r e p r e s e n t s t h e w o r k d o n e i n a d d i t i o n t o t h e w o r k n e c e s s a r y t oc r ea t e t he app l i ed f i e l d i n emp t y s pace . I n a d i amagne t i c s ubs t ance , t het o t a l w o r k p e r f o r m e d i s l e s s t h a n t h e w o r k p e r t a i n i n g t o e m p t y - s p a c e ,bu t i t i s s ti ll pos i ti ve . S t a r t i ng f r om t he f ac t t ha t m agne t i c w o r k i s g i ve ni n gene r a l by H 9 313 (Eq . (4 .16)) , th er m od yn am ic s tab i l i ty on ly r equ i restha t , in a l inear m e d iu m w he re B = ~ H an d M = ,u ]z >-- 0 , tha t i s,,~' >-- - 1 (se e E q. (4.37)).P a r a m a g n e t i s m . P a r a m a g n e t i c s u b s t a n c e s a r e c h a r a c t e r i z e d b y a l a w t h a ti s l inear a t low f ie lds, M = xH , wi th p os i t ive susce pt ib i l i ti es o f the o rde rof 10 -3 to 10 -5 De via t ion s f rom the l inear l aw take p lace a t ver y h igh f i e ldsw h er e t he e f fec t s a t u r a t e s. A t yp i ca l o r de r o f m ag n i t u de m ay be 10 s A m -1a t r o o m t e m p e r a t u r e . P a r a m a g n e t i s m is t h e m a n i f e s t a t io n o f t h e e x is te n c eo f p e r m a n e n t m a g n e t i c m o m e n t s i n m a t t e r . P a r a m a g n e t i c s u s c e p t i b i l i t yi s t he r e s u l t o f t he compe t i t i on be t w een t he ac t i on o f t he ex t e r na l f i e l d ,w h i ch t r i e s t o a l i gn al l m om en t s a l on g i ts d ir ec t ion , and t he r m a l ag i t a t i on ,w h i c h t e n d s t o d e s t r o y a n y a l i g m n e n t p o s s i b l y p re s e n t. S o m e c o n s i d e ra -t i o n s o n p a r a m a g n e t i s m a r e m a d e a t t h e b e g i n n i n g o f C h a p t e r 5 , w h e r eW ei s s mean f i e l d t heo r y o f f e r r omagne t i s m i s d i s cus s ed .F e r r o m a g n e t i sm . T h e bas i c fi nge r p r i n t s o f f e r r om agn e t i s m a r e t he ex i s tenceo f a s p o n t a n e o u s m a g n e t i z a t i o n a n d h y s te r e si s. A f e r r o m a g n e t i c s u b s t a n c ecanno t be cha r ac t e r i zed by any s i mp l e , s i ng l e - va l ued cons t i t u t i ve l aw ,and an i n f i n i t e s e t o f d i f f e r en t magne t i za t i on cu r ves can be obs e r ved ,d e p e n d i n g o n p a s t f i e l d h i s t o r y . T h e s e a s p e c t s h a v e a l r e a d y b e e n a d -d r e s s ed i n chap t e r s I an d 2 , and w i l l be t he s ub j ec t o f de t a i l ed d i s cus s i oni n s ubs equen t chap t e r s . I n f e r r omagne t i c ma t e r i a l s , t he p r i nc i pa l mecha -n i s m s d e t e r m i n i n g t h e m a t e r ia l c o n s t it u t iv e l a w a r e e x c h a n g e a n d a n i so t -r opy , t o be d i s cu s s ed i n C h ap t e r 5 . T hes e m ech an i s m s a r e r e s pons i b l e f o rt h e f o r m a t i o n o f m a g n e t i c d o m a i n s . T h e c o n s t it u t iv e l a w M ( H ) d e s c r ib e st he ma t e r i a l on a g r os s s ca l e , l a r ge r t han t ha t o f doma i ns . O n t h i s s ca l e ,t h e d o m a i n s t r u c t u r e i s n o l o n g e r a p p r e c i a t e d a n d t h e m a t e r i a l a p p e a r s


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4.1 MA GNE TIC WORK AND CONSTITUTWE LAWS 113t o b e h o m o g e n e o u s . 3 W e s t re s s o n c e m o r e t h a t t h e e n e r g y c a l c u la t e df r om E q . ( 4 . 25 ) i s no t neces s a r i l y s t o r ed i n t he med i um by r eve r s i b l em echa n i s m s . I n gene r a l, pa r t o f i t w i l l be t r ans f o r m ed i n t o t he r m a l ene r gybecause of hys te res i s e f fec t s , a f ac t tha t i s r e f l ec ted in the h i s to ry-depen-d e n t , m u l t i b r a n c h n a t u r e o f M ( H ) .

I t i s w or t h po i n t i ng ou t t ha t i n th i s boo k w e a r e u s i ng t he exp r e s s i onf e r r o m a g n e t i c m a t e r i a l or m a g n e t ic m a t e r ia l t o r ef e r gene r i ca l l y to a ny m a t e -r i a l w h e r e s p o n t a n e o u s m a g n e t i z a t i o n a n d h y s t e r e s i s a r e o b s e r v e d . W ei gno r e t he f ac t t ha t t he s pon t aneous magne t i za t i on a r i s e s f r om l o n g - r a n g eo r d e r i n g o f t h e m a g n e t i c m o m e n t s a n d t h a t t h i s l o n g - r a n g e o r d e r c a n b echa r ac t e r i zed by d i f f e r en t s pa t i a l s ynm~ e t r i e s . T hus w e w i l l encompas su n d e r t h e s a m e t e r m o r d i n a r y f e r r o m a g n e t s , w h e r e i d e n t i c a l m o m e n t soccupy a s i ng l e l a t t i c e and a r e a l l a l i gned a l ong t he s ame d i r ec t i on , a sw e l l a s f e r r i m a g n e t s , w h e r e m o m e n t s p o i n t a l o n g d i f fe r en t d i re c ti o n s, d e -p e n d i n g o n w h i c h s u b la t ti c e t h e y b e l o n g t o , a n d t h e t ot a l s p o n t a n e o u sm a g n e t i z a t i o n i s t h e r e s u lt o f c o m p e t i n g c o n t r ib u t i o n s c o m i n g f r o m v a r i -ous s ub l a t t i c e s . I n add i t i on , no t h i ng w i l l be s a i d abou t a n t i f e r r o m a g n e t s ,w h er e t he s ub l a t ti c e con t r i bu t i ons exac t l y cance l ou t , s o tha t t he me d i u m ,i n s p i t e o f l ong - r ange o r de r i ng , exh i b i t s no s pon t aneous magne t i za t i onat al l .A f t e r t he s e cons i de r a t i ons , i t i s app r op r i a t e t o recons i de r t he exam pl e ,p r e s e n t ed i n Fig . 1.5 , o f t he dep end enc e o f m agn e t i za t i on cu r ves on geom -e tr y. F i gu r e 4.4 r ep r e s en t s t he ex pec t ed o u t com e o f an exp e r i m en t w h e r e ,s t a r t i ng f r om t he demagne t i zed s t a t e , w e app l y a ce r t a i n f i e l d H a t o t her i n g s p e c i m e n a n d t o t h e h o r s e s h o e s p e c i m e n , m a d e u p o f t h e s a m emagne t i c ma t e r i a l . T he f i e l d and t he magne t i za t i on bo t h l i e a l ong t hes pec i m en ax i s i n bo t h ca se s . T hus , t he mag ne t i z a t i on cu r ve a nd a ll ene r g yr e l a t i ons can be exp r e s s ed i n t e r ms o f t he app l i ed f i e l d i n t ens i t y H a a n dt h e a v e r a g e m a g n e t i z a t i o n i n t e n s i t y i n t h e s a m p l e v o l u m e ,

1(M) = ~ f ]M(r)l d 3r (4.26)v

T he t w o magne t i za t i on cu r ves exh i b i t ev i den t d i f f e r ences . T he r ea s onbecomes c l ea r w hen w e cons i de r t he s haded a r ea s o f F i g . 4 . 4 , w h i ch ,acco r d i ng t o E q . ( 4 . 22 ) , a r e p r opo r t i ona l t o t he magne t i c w or k pe r -f o r med on t he s y s t em. I n t he ca s e o f t he r i ng s pec i men , M f l ow s a l ongt he r i ng by a p r ac t i ca l l y d i ve r gence - f r ee pa t h , s o t ha t , w i t h V - M ~ 0 ,H M ~ 0 , an d H a ~ H . Th ro ug h Eq. (4 .25), the w or k p er f or m ed i s d i r ec t lyr e l a t ed t o t he p r ope r t i e s o f t he M ( H ) cons t i t u t i ve l aw , becaus e U M ~ 0 .

3This poin t was already considered in Section 1.1.3 and Section 3.3.4.


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114 CHAPTER 4. Magnetic Work and Therm odynam ics

FIGURE 4.4. Influence of geom etry on magn etization curves and on wo rk per-formed on the system (shaded areas) .I n t he ca s e o f t he ho r s e s hoe s pec i men , on t he con t r a r y , magne t i c cha r gesa r e fo r m e d a t th e s p e c i m e n e d g e s , w h i c h e n t a il s ig n i fi c a n t d e m a g n e t i z i n ge ff ec ts a n d t h e p r e s e n c e o f a n i m p o r t a n t m a g n e t o s t a t i c e n e r g y c o n t r ib u -t i on . A cco r d i ng t o E q . ( 4 . 20 ) , t he g r ea t e r w or k needed t o magne t i ze t hes p e c i m e n is a d ir e c t m e a s u r e o f t h e e n e r g y s t o r e d a s m a g n e t o s t a t ic e n e r g y .T h i s a n a ly s i s s u g g e s t s th a t o n e m i g h t t r y s o m e c o r r e c ti o n to t h e m a g n e -t i za t i on cu r ve o f t he ho r s e s ho e s pe c i men , i n o r de r t o ex tr ac t t he M ( H ) cons t i -t u t iv e la w . T o t h is e n d , a n i n d e p e n d e n t e s t i m a t e o f d e m a g n e t i z i n g e ff ec tsi s necessary . This i s in p r inc ip le poss ib le i f the spec imen i s o f e l l ipso ida ls hap e a nd i f i t is un i f o r m l y m agn e t i zed . I n t h i s ca s e, in fac t, w e kn ow f r omS ec t ion 3 .2 .3 t ha t t he d em ag ne t i z i ng f ie l d is a ls o un i f o r m i n s i de t he s pec i -m e n a n d is e q u a l t o - N M , w h e r e N i s a k n o w n f u n c t io n o f t he s p e c i m e ng e o m e t r y . T he r e fo r e, g i v e n t h e m e a s u r e d c u r v e M (H ~ ) , w e o b t a i n t h e m a t e -r ia l c o n st i tu t i v e l a w s i m p l y b y p l o t t in g M ( H a - N M ) . T h e s h o r t c o m i n g o ft h i s a p p r o a c h i s t h a t t h e t w o m e n t i o n e d r e q u i r e m e n t s - - e l l i p s o i d a l s h a p ea n d u n i f o r m m a g n e t i z a t i o n ~ a r e q u i t e i d e a l i z e d a n d u s u a l l y d i f f i c u l t t om ee t i n p r ac ti ce . A s a cons equenc e , co r r ec t ions f o r dem agn e t i z i n g e f fec tsm ay l ead t o un r e l i ab l e r e s u l ts , if no t t r ea t ed w i t h a t t en t ion .

T h e s i m p l e s t c o n s t i t u t i v e l a w t h a t o n e m a y i m a g i n e i s p r o b a b l y t h eo n e w h e r e H = 0 e v e r y w h e r e i n s i d e t h e b o d y . T h e m a t e r i a l i s n o t a b l e


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4.2 THERM ODY NAM IC RELATIONS 115to sus ta in any in te rna l p ressure exer ted by H and acqu i res a ma gne t iza t ionun de r w ha tev er smal l value of H (Fig . 4 .5) . This ideal ized case is them agn etosta t ic equ ivalen t of a perfect conductor , wh ere the externa l e lec t ricf ie ld is fu l ly shie lded by an appropr ia te d is t r ibut ion of e lec t r ic chargesa t the body sur face . Under these c i rcumstances , a l l the work pe r fo rmedon the system is s tored as magnetosta t ic energy. Descr ipt ions of th is k indm ay be use fu l a s a s impl i f ied appro ach to so f t mag ne t ic m ate r ia l s , wh ichare eas i ly magnet ized under smal l f ie lds . In par t icular , they wil l be em-p loyed in Chap te r 12 , to s tudy ce r ta in aspec t s o f magne t ic losses . Note ,how ever , tha t the cond i t ion H = 0 m us t necessa r i ly fa il w he n the m ate r ia lapproaches sa tu ra t ion .

4 .2 T H E R M O D Y N A M I C R E L A T IO N SThe der iva t ion o f a su i tab le express ion fo r the ma gne t ic wo rk , d i scussed inthe prev ious sect ion, i s the s tar t ing poin t for the analysis of the ther m od y-namic p roper t i e s o f magne t ic media . Th is ana lys i s i s compl ica ted by thepresence o f hys te resi s , which can on ly be t r ea ted in the f rame of nonequi l ib -r ium the rmodynamics . We sha l l f i r s t a ssume tha t hys te res i s phenomena

FIGURE 4.5 . Ideal step like con stitutive law a nd corresponding magn etizationcurve in the presence of demagnetizing effects.


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116 CHAPTER 4. Magnetic Work and Thermodyn amicsa re absen t and w e sha ll de r ive t he rm odyn am i c r e la t ions t ha t w o u l d app l yi f the sy s tem were a lway s in equi l ib r ium, tha t i s, i f i t had as cons t i tu t ive l awthe anhys te re t i c magnet i za t ion curve . Then we wi l l comple te the p ic tureby ad d i ng f ea tu re s typ ica l of noneq u i l i b r i um t he rm odyn am i cs . A c ruc ia laspec t in th i s respec t is , how ever , t ha t non equ i l ibr ium therm ody nam ics i scom m only base d on the so-ca ll ed pr inc ip le of loca l equi l ib r ium, s t a t ing tha tt he rm odynam i c equ i l i b r i um re l a t i ons can be app l i ed t o each e l em en t a ryvolum e, ev en tho ug h the sys tem i s no t g loba l ly in equi l ib r ium. Thi s hyp oth-es is has to be a ban do ned i f hys te res i s i s a l ready rooted in the cons t i tu t ivel aw d esc r i b ing t he behav i o r o f each e l em en t a ry vo l um e .

4.2.1 Thermodynamic potentialsW e beg i n w i t h a b r ie f sum m ary o f t he p r inc i pa l t he rm odyn am i c r e la t ionsva l id for a gener i c sys tem wi th homogenous proper t i es . La te r , we wi l lspecial ize this descript ion to the magnet ic case.F i r s t la w o f t h e r m o d y n a m i c s . This l aw expresses the conserva t ion of energyand s t a t e s t ha t , unde r a gene r i c t r ans fo rm a t i on w here w ork 8 L i s per-fo rm ed on t he sys t em and hea t 8 Q i s abso rbed by i t ,

d U = 8 L + 8 Q (4.27)where U, the i n t e r n a l e n e r g y , i s a s t a t e func t ion . We assume tha t the work8 L can be exp res sed i n t e rm s o f t w o approp r i a t e c o n j u g a t e w o r k v a r i a b l e s ,H and X, accord ing to the express ion

8 L = H d X (4.28)X i s a s t a t e var i ab le , descr ib ing some proper ty of the sys tem, and Hcharac te r i zes the ex te rna l ac t ion exer t ed on the sys tem.S e c o n d la w o f t h e r m o d y n a m i c s . This l aw in t rod uces a second s t a t e func tion ,the e n t r o p y S , which , g iven any revers ib le or i r revers ib le t ransformat ion ,always sat i sf ies the inequal i ty

>_ - ~ (4.29)S 1The equa l s ign appl i es to revers ib le t ransform at ions . In tha t case one has

8 Q = T d S (4.30)There i s a nice symmetry between Eq. (4.28) and Eq. (4.30) . T and Sare conjuga te var i ab les wi th respec t to thermal energy in the same way


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4.2 THE RM OD YNA MIC RELATIONS 117a s H a n d X a r e c o n ju g a t e v a r i a b l e s w i th r e s p e c t t o t h e e n e rg y c o min gf ro m e x te rn a l wo rk . T a n d H a r e i n t e n s i v e v a r i ab l e s , w h e re a s S a n d X ar ee x t e n s i v e ones .T h e r m o d y n a m i c p o t e n t i a l s . T h e r m o d y n a m i c t r a n s f o r m a t i o n s c a n t a k e p l ac eu n d e r v a r io u s c o n s t r a in t s , l i k e c o n s t a n t t e mp e ra tu r e , c o n s t a n t e n t ro p y ,a n d s o o n . Th e c o n s t r a in t w i l l i n g e n e ra l c o n c e rn b o th e x t e rn a l wo rka s p e c t s a n d th e rma l e n e rg y a s p e c t s , a n d w i l l c o r r e s p o n d to f i x in g o n evar iab le in each o f the (T ,S) an d (H,X) se ts . Give n a con s t ra in t , one cani n t r o d u c e a c o r r e s p o n d i n g t h e r m o d y n a m i c p o t e n t i a l , tha t i s, a s ta te fun c t ionc o n t ro l l i n g h o w th e t r a n s fo rma t io n w i l l e v o lv e . Th e re a r e f o u r p o s s ib l ec h o ic e s for t h e c o n s t r a in t v a ri a b le s , s o fo u r t h e rm o d y n a m ic p o te n t i a ls c a nb e d e f in e d . Th e y a r e t h e i n t e r n a l e n e r g y U ( X , S ) , th e e n t h a l p y E ( H , S ) , t h e

f r e e e n e r g y F ( X ,T ), a n d t h e G ib b s f u n c t i o n G ( H ,T ). Ea c h o f t h e m h a s t h ep ro p e r ty t h a t it n e v e r i n c re a s e s i n a n y t r a n s fo rm a t io n wh e re i t s a rg u m e n t sa re k e p t f ix ed . T h e r m o d y n a m i c e q u i l ib r i u m i s r e a c h ed w h e n t h e a p p ro -p r i a t e t h e r m o d y n a m ic p o te n t i a l a t ta in s it s g lo b a l m in im u m . W e a r e i n t er -es ted , in pa r t icu la r , in the p rope r t ie s o f the po ten t ia l s fo r t rans f o rm at ion su n d e r c o n s t a n t t e m p e r a t u r e .F r e e e n e r g y F ( X , T ) . This func t ion i s a l so ca l led the H e lm h o l t z f u n c t i o n orH e l m h o l t z f r e e e n e r g y . I t i s de f ined as

F = U - TS (4.31)Eq u a t io n (4. 27 ) t h ro u g h Eq . ( 4.2 9 ) s h o w th a t

d F


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118 CHAPTER 4. Magnetic W ork and ThermodynamicsThus , d G < - 0 under cons tant H and T. At equi l ibr ium

[O~H = - X [O~T = - S (4.36)T H

A spontaneous t ransformat ion can only take p lace when the sys temis in i tia lly not in equi l ibr ium. The none qui l ibr ium condi t ion i s descr ibedby o ther in te rna l var iab les , in addi t ion to the cons t ra ined ones , whichmay be space -dependen t and which evo lve in t ime un t i l t hey r each theva lue for wh ich the therm ody nam ic potent ia l a t ta ins it s g loba lly m in im umvalue . Once the sys tem is in equi l ibr ium, any sp ontane ous f luc tua t ion inthe sys tem proper t ies necessar i ly leads to a nonnegat ive var ia t ion of thepotent ia l . For example , any spo ntaneo us f luc tua t ion 8 X or 8S un der con-stant H an d T m us t be such th at 8(3 ~ 0. This s tabi l i ty req uirem ent lea dsto useful inequal i t ies . In par t icular , i t can be shown that , a t equi l ibr ium,

[ 0 ~ ] ~ 0 (4.37)T

By derivin g Eq. (4.33) w ith respect to X an d Eq. (4.36) w ith respect to Hand by m ak ing use of Eq. (4.37) , one also f inds that , a t equi l ibr ium ,[ 0 2 G ][ 0 2 F 1 > 0 l _ - ~ _ ] < 0 (4.38)

[ c ~ X a ] T - - T - -which shows tha t F i s a lways a convex func t ion of X and G a concavefunct ion of H.

4 .2 .2 Thermodynamic potent ia l s for magnet ic mediaLet us now cons ider a mag net ic sys tem subject to transforma t ions du r ingwhich the body vo lume r ema ins unchanged . Th i s means tha t we ignorea ll t hose phen om ena where the vo lum e m ay change a s a consequence o fchanges in tempera ture ( thermal expans ion) or magnet ic f ie ld (magneto-s tr ic t ion) . Therm ody nam ic re la t ions can be wr i t ten once we have a su i tab leexpression for the magnet ic work. This is given in general by Eq. (4.16) ,wh ich involves as conjugate var iab les B and H. Yet , we prefer to baseou r ana lysis on Eq. (4.23) , wh ich is par t icu lar ly sui ted to descr ibe typicals i tuat ions en cou ntere d in m agne t ic experim ents . We recal l that this expres-s ion holds w hen the appl ied f ie ld H a i s uni form over the bod y v olum eand when la rge-sca le eddy-cur rent d i ss ipa t ion i s neglec ted . In addi t ion ,i t does not inc lude the energy tha t would be spent to c rea te the appl ied


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4.2 THERM ODY NAM IC RELATIONS 119

f ie ld i n e m p t y s p a ce . T h e m a g n e t i c m o m e n t / . t o m a n d t h e f ie ld H a ar e th ev e c t o r a n a l o g u e o f t h e q u a n t i f i e s X a n d H i n t r o d u c e d i n t h e p r e c e d i n gs e c ti o n , a n d p l a y t h e r o le o f s t a te v a r i a b l e a n d e x t e r n a l p a r a m e t e r .W e a r e m a i n l y i n t e r e s t e d i n m a g n e t i c t r a n s f o r m a t i o n s w h e r e t h e s y s -t e m i s i n c o n t a c t w i t h a t h e r m a l b a t h a t c o n s t a n t t e m p e r a t u r e , a n d w es h a l l t h e n c o n c e n t r a t e o n H e l m h o l t z a n d G i b b s f r e e e n e r g y , F ( m , T ) a n dG(H,T) . For F , we have

d E ~ - / . t 0 H 9 d m - S d T (4.39)T h e e x p e r i m e n t e r , h o w e v e r , u s u a l l y p r e f e r s to d e a l w i t h s i tu a t io n s w h e r et h e a p p l i e d f i e l d , r a t h e r t h a n t h e m a g n e t i z a t i o n , i s u n d e r c o n t r o l . T h eG i b b s f r e e e n e r g y

G = F - ~0H a . m (4.40)i s t hen t he po t en t i a l o f i n t e re s t . G sa t i s f i e s t he i nequa l i t y

d G


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120 CHAPTE R 4. Magnetic W ork and Thermody namicsTh i s s h o ws th a t we c a n e s t ima te t h e two p o te n t i a l s f r o m a p lo t o f t h ema te r i a l c o n s t i t u t iv e l a w . I n Fig . 4 .6 , w e h a v e r e p re s e n te d a ma g n e t i z a t i o nc u rv e w h e r e b o th H a a n d m p o in t a lo n g th e s a m e d i r e ct io n . Th e v a r i a t i o n so f F a n d G a r e r e p re s e n te d b y th e a r e a s d e l im i t e d b y th e ma g n e t i z a t i o nc u rv e a n d th e m o r H a a x es . Re m e m b e r t h a t t h i s c o r r e s p o n d e n c e h o ld so n l y i f t h e s y s t e m i s i n t h e r m o d y n a m i c e q u i l i b r i u m a t e a c h p o i n t o ft h e t r a n s f o r m a t i o n , w h i c h m e a n s t h a t t h e m a g n e t i z a t i o n c u r v e m u s t b er e v e rs ib le , a n d d i s s ip a t io n d u e to h y s te r e s i s mu s t b e a b s e n t o r n e g l ig ib le .

4.2.3 Nonequilibrium thermodynamicsTh e id e a o f i r re v e r s ib i l it y is r o o t e d in e q u i l i b r iu m th e rm o d y n a m ic s i t se lf .I n f a c t , t h e s e c o n d l a w o f t h e rmo d y n a mic s , s t a t i n g th a t t h e e n t ro p y o f ath e rm a l ly i s o l a te d s y s t e m n e v e r d e c r ea s e s , id e n t i fi e s t h e d i re c t io n in w h ic ha s y s t e m n o t i n e q u i l i b ri u m w i l l s p o n t a n e o u s l y e v o lv e . In n o n e q u i l i b r i u mt h e r m o d y n a m i c s , e n t r o p y i s a q u a n t i t y t h a t m a y f l o w i n s p a c e a n d m a yb e c r e a t e d , a n d th e s e c o n d l a w o f t h e rmo d y n a mic s i s r e in t e rp r e t e d a s a ne n t ro py b a l anc e e qua t ion . Eq u a t io n (4.2 9 ) b e c o m e s

- - " r S i f- r S - - - - ~ - J r - r S (4.46)Sw h e r e 8eS r e p re s e n t s t h e e n t ro p y f lo win g in to t h e s y s te m, wh i l e 8iS is thee n t ro p y p ro d u c e d in s id e th e s y s t e m b y i r re v e r s ib l e p ro ce s s es . I n th i sf r a m e w o r k t h e s e c o n d l a w o f t h e r m o d y n a m i c s i s e q u i v a l e n t to t h e s ta t e-m e n t t h a t t h e e n t r o p y p r o d u c ti o n i s a lways pos i t ive :

r S ~ 0 (4.47)

FIGURE 4.6 . Helm holtz and G ibbs free energy estimated from reversible magn eti-zation curve.


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4.2 THERMO DYNAM IC RELATIONS 121

where the equal s ign holds in the l imi t ing case of a pure ly revers ib lep rocess. The cen tr a l p rob l em o f noneq u i l i b r i um t he rm odyn am i cs is t hep red i c t i on o f t he dependence o f 8iS on the var ious i r revers ib le processesposs ib ly t ak ing p lace in the sys tem.

Equat ion (4 .46) i s an in tegra l re l a t ion apply ing to the sys tem as awhole . Yet, w he n the sys tem i s no t in equi l ib r ium , one expec t s in genera li t s p roper t i es to be space- t ime dependent , and var ious re l axa t ion anddi f fus ion processes to t ake p lace whi l e equi l ib r ium i s approached . Thi si s t rea ted by in t roducing loca l dens i t i es of the var ious thermodynamicquant i t i es of in t e res t , and by wr i t ing loca l ba lance equat ions for thesedens i ti es . The descr ip t ion can become fa i rly compl ica ted , because in gen-era l there a re severa l ba lance equat ions to be s imul t aneous ly fu l f i l l ed ,re l a t ed to conse rva t ion of mass , energy , m om en tum , and so on . We arenot go ing to d i scuss the problem in such genera l i ty . We sha l l on ly g ivean exam pl e va l i d for a m agne t i c sys t em w he re no t r an spor t o f m a t t e r o rchange in mass dens i ty t akes p lace . To each e l ementary volume AV, weassociate a local value of magnet ic induct ion B, magnet ic f ie ld H, e lect r icf i e ld E , cur ren t dens i ty j , i n t e rna l energy dens i ty u , en t ropy dens i ty s ,and en t ro py pro duc t ion or. The en t ro py b a lance equat ion , Eq . (4 .46), w he nwri t t en in loca l form, becomes

c98- - = - V . J s + cr (4 .48)3 twh ere the vec tor Js i s the en t r o p y f lo w , and cr m easu res the ra t e a t wh ichent ropy i s loca l ly produced . The next re l a t ion of impor tance i s the loca lform of the Po yn t ing theor em , 4 Eq. (3.82):

0 BH - a-- t-= - V . (E H) - j . E (4 .49)A cont inui ty re l a t ion can be wr i t t en for the in t e rna l energy dens i ty u .Energy can f low in the form of hea t f low, descr ibed by an appropr i a t ecurren t Jq, o r of e l ec tromagn et i c f low, descr ibed by the Po ynt ing vec tor:

a u = - V . J q - V - ( E H ) (4.50)3 tThe f inal re lat ion to consider i s the local form of the f i rs t law of thermo-dynam i cs . The a s sum pt i on m ade i n noneq u i l i b r i um t he rm odynam i cs ,k n o w n a s t h e hypo thes i s o f loca l equi l ibr ium, s t a t es tha t , even though thesys tem as a wh ole i s no t in equi lib r ium, equi l ib r ium holds in each e lemen-

4F or t he s a ke o f s i m p l i c i t y , w e i gno r e t he t e r m E 9 aD /Ot of Eq . (3.82) .


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1 2 2 CHAPTER 4. Magnetic Work and Thermodynam icsta ry volume, in the sense tha t the en t ropy dens i ty s i s loca l ly the samefunc t ion o f t he o the r t he rm o dyna m i c va r i ab l e s a s i n t he rm ody nam i c equ i -l ibr ium. According to the resul t s of Sect ion 4.1.2 and Sect ion 4.2.1, thismeans tha t loca l ly d u = H 9 o"B + T d s , that i s ,

T 3__z_~ 0u H . ~0B (4.51)3t 3t 3tThe hy pothe s i s of loca l equi l ib r ium plays a c rucia l ro le, as it perm i t s oneto l ink the en t ropy product ion to the o ther sys tem var i ab les . In fac t , byinsert ing Eq. (4.49) and Eq. (4.50) into Eq. (4.51), we obtain

3sT = - V . J q + j . E (4.52)3 tw hi ch can be w r i t t en i n t he equ i va l en t fo rm

O s - V . ( ~ ) J q. V T j -E (4 .53)3--t = -- T 2 T

By comparing Eq. (4.53) wi th Eq. (4.48) , we see that the ent ropy f low andt he en t ropy p roduc t i on a re g i ven byJs = J_a ( 4 . 5 4 )T

To-= - Jq" ~7T + j - E (4.55)TEquat ion (4 .55) has the typ ica l s t ruc ture exhib i t ed by en t ropy product ioni n noneq u i l i b r i um t he rm ody nam i cs . I t is a sum o f t e rm s , each desc r i b inga di fferent i r reversible process, heat f low and elect r ic current f low in thispar t i cu la r case . Each t e rm i s the product of the re l evant thermodynamicf low (Jq , j ) and the cor resp ond ing ther mo dy nam ic force ( - ~7T/T ,E) dr iv in gthe process . Because the en t ropy product ion must be pos i t ive , these prod-uc t s mu st a l so be pos i tive . In addi t ion , one kn ow s tha t bo th the force andt he fl ux shou l d v an i sh w h en t he sys tem i s i n t he rm ody nam i c equ i li b r ium .The s imp les t l aw cons i s t en t wi th these req ui rem ent s i s a linear re la t ionshipbe tween the f low and the force :

Jq = - K V T (4.56)j = e E

w ith K > 0 an d cr > 0. We recognize F o u r i e r ' s l a w of hea t conduc t i on andO h m ' s l a w . T h e existence of these li n e a r p h e n o m e n o l o g i c a l l a w s is the secon dbas ic a s sum p t i on o f nonequ i l ib r i um t he rm od ynam i cs , w h i ch , t oge t he r


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4.2 THERM ODYNA MIC RELATIONS 123wi th the hyp othes i s of loca l equi l ib r ium, pe rmi t s one to deve lo p a quant i -t a t ive descr ip t ion of the processes under cons idera t ion .

O ne m i gh t w onde r w he re i n t h i s con t ex t hys t e re s i s m ay com e i n t oplay . One might have not i ced tha t the t e rm H 9 3B/Ot, ch arac ter izing them agne t i c p rope r t i e s o f t he m ed i um , is no t i nvo l ved i n en t ropy p roduc t i on(Eq. (4.55)) . This i s the di rect consequence of assuming that Eq. (4.51)descr ibes a s i tua t ion of loca l equi l ib r ium. W he n the co ns t i tu t ive l aw B(H)exhib it s hys te res is , t h is i s no long er t rue . Energy d i ss ipa t ion a nd en t rop yproduc t i on a re a l so b rough t abou t by t he m agne t i c w ork H 9 813 a n d Eq.(4 .51) no lo nger h o lds as an equali ty . Thi s compl ica tes the ap proa ch verym uch , because one needs new p r i nc i p l e s p red i c t i ng i n w h i ch p ropor t i ont he m agne t i c w ork H 9 813 i s reversibly s tored or i r revers ibly t ran sfo rm edin to hea t . Choos ing to work wi th f iner or coarser sca les a l so p lays a ro lein this respect . As we discussed in Sect ion 3.3.4, const i tut ive laws wi thhys t e re s i s m ay be j u s t t he sho r tcu t by w h i ch w e sum m a r i ze com pl i ca tedprocesses t ak ing p lace on a f iner sca le . Note tha t , i n the approximat ionwhere ra t e - independent hys te res i s ho lds , t hese compl ica t ions a re encoun-te red no mat t e r how s low i s the appl i ca t ion of the ex te rna l f i e ld .

Tem pera t u re g rad i en t s and hea t f l ow can i n p r i nc i p l e be expec t edw h en one dea ls w i t h none qu i l i b r i um s i tua t ions i nvo l v i ng spa t ia l inhom o -gen eous s t ruc tures , li ke ma gnet i c d om ain s t ruc tures . The jo in t descr ip t ionof loca l d i ss ipa t ion coup led to hea t f low leads to fa i r ly com pl ica ted t rea t -ment s and wi l l no t be pursued any fur ther . In the fo l lowing chapters , wesha l l usua l ly l imi t our cons idera t ions to the t rea tme nt of a mag net i c bod yas a who le , ass um ed to be a t a cer t a in t em pera ture T, wi th ou t g o ing in tothe de ta i ls o f the sp ace- t im e d i s t r ibu t ion of i ts therma l energy . Thi s m ayof t en be a good app rox i m a t i on , because unde r l ow m agne t i za t i on r a t e slosses t end to be homogeneous ly d i s t r ibu ted and , on a sca le l a rger thant ha t o f dom a i ns , t hey l ead t o f a i r l y un i fo rm en t ropy p roduc t i on . U nde rthese condit ions, Eq. (4.46) can be used to rewrite Eq. (4.39) and Eq. (4.41)as equal i t i es involv ing the overa l l en t ropy product ion in the body. Inpart icular, Eq. (4.39) becomes

d E - - [d, H a 9 T S ~ S - S d T (4.57)In a t r ans fo rm a t i on unde r cons t an t t em pera t u re ,

[d , f H a . d m - - A F -}- f z r (4.58)w h i ch show s t ha t t he w ork pe r fo rm ed on t he body pa r t l y con t r i bu t e s t ot he f r ee ene rgy and pa r t l y goes i n t o en t ropy p roduc t i on . A s m en t i oned ,the problem i s to kn ow the genera l p r inc ip les cont ro l l ing th i s subdiv i s ionin a mag net i c sys tem w i th hys te res is . W he n the sys tem i s cyc li ca lly dr iven


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1 2 4 CHAPTER 4. M agn etic W ork and Thermodynamicsth rough the same sequence o f s ta tes , the f ree energy and the en t ropy a reper iod ic in t ime , and one immedia te ly conc ludes tha t

/z0 ~ H a ' d m = ~ TSiS ( 4 . 5 9 )cycle cycle

This re sult is identica l to Eq. (2.13). In fact , the h ea t relea sed in a cycle isexac tly equa l to the hea t in te rna l ly p rod uced , fo r the en t ropy m us t r e tu rnto the sam e valu e a t the en d of the cycle . In a gener ic noncyclic t ransfo rm a-t ion, how ever , Eq. (4 .59) can no lon ger be appl ied an d so me g enera l iza t ioni s r equ i red . In Cha p te r 13, we wi l l show the fo rm taken by th i s genera l i za -t ion in Pre isach systems.As a final r em ar k i t m ay be no ted tha t the evo lu t ion equ a t ion cons id -ered in Sect ion 2 .3 , Eq. (2 .33), can be d er ive d in the f ra m ew ork of no neq ui-l ib r ium the rmo dynam ics . In fact , fo r a sys tem desc r ibed by the con juga tevar iables H and X, Eq. (4 .58) becom es

f H a X AF + f T iS (4.60)The ra te o f en t rop y p ro duc t ion cr in a t r ans form at ion wh ere the s ys tempasses th rough a sequence o f the rmodynamic s ta tes charac te r ized bydif ferent values of X is then

H d X d F _ H d X OF d X _ d X cOGLTc r (4.61)d t d t d t 3 X d t d t 3 Xw h e r e G L ( X ; H , T i s the Landau f ree energy in t roduced in Sect ion 2 .1 .4 .Equ a t ion (4 .61) has the sam e s t ruc tu re as Eq . (4 .55) , wi th the en t ropyp r o d u c t i o n e x p r e s se d i n t e rm s o f t h e p r o d u c t b e t w e e n t h e fl ux d X / d t a n dt h e th e r m o d y n a m ic f or ce - 3 G L / 3 X . Eq uat ion (2 .33) represe nts the l inearpheno m enolog ica l l aw, ana logous to Eq . (4.56) and cons i s ten t w i th ther e q u i r e m e n t o f p o s it iv e e n t r o p y p r o d u c t i o n , t h a t m a y b e a s s u m e d t odesc r ibe re laxa t ion toward equ i l ib r ium.

4 . 3 B I B L I O G R A P H I C A L N O T E SM agn etosta t ic ene rgy is d iscusse d in deta i l in [B.55, B.63]. The t re a tm entof mag ne t ic w ork in the rm ody nam ics i s a lway s a de lica te po in t. Pa r t i cu la refforts a t c lar i ty and logical r igor are m ad e in [B.6, B.39]. The discu ss iong iven in Sec t ion 4 .1 .2 emphas izes the ro le o f magne t iz ing cur ren t s anddoes no t d i scuss the fo rm taken by energy re la t ions when work i s a l so


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4.3 BIBLIOGRAPHICALNOTES 1 2 5done to move pe rmanen t magne ts wi th respec t to each o the r . Th is a spec tis d iscusse d in [B.9 , Ch apt er 2] and in Ref. 4 .1 .

Classical the rm ody nam ics i s d i scussed in m any tex ts . See fo r exam ple[B.35, B.36, B.39, B.51]. For the in terpre ta t ion of ther m od yn am ic re la t ionsin stat ist ical m ech anics , see [B.32, B.40, B.42, B.43, B.46 , B.53, B.54]. A ninsp i r ing shor t p resen ta t ion o f s ta t i s t i ca l the rmodynamics can be foundin [B.124]. An e legan t d i scuss ion o f the rm ody nam ic re la tions and the rmo-dyn am ic po ten t ia l s wi th e mp has i s on the sym m et ry (H,X) - (T,S) men-tioned in Section 4.2 is given in [B.51].No neq ui l ib r ium the rm ody nam ics i s d i scussed in [B.38 , B .41 , B .45] . InRef . 4 .2 , the subject i s ad dre sse d w ith pa r t icular a t tent io n to the form at ionof d iss ipat ive s t ructures in sys tems that are kept far f rom equi l ibr ium.Some aspec t s o f none qui l ib r ium the rm ody nam ics in the p resence o f hys -teres is are c onsid ered in Ref. 4 .3 . In [B.29], ent rop y pro du ct io n in hystere t icsys tems i s ana lyzed in the f ramework o f the Pre i sach hys te res i s mode l .4.1 H. Zijlstra , "P erm ane nt Mag nets: The ory," in [B.98, Vol. 3], 37-105.4.2 G. Nicolis, "Irreversible Th erm odyn am ics," Rep . Progr. Phy s. 42 (1979), 225-268.4.3 Y. Hu o and I. Muller, "No nequilibrium Therm odynam ics of Pseudoelasticity,"C o n t i n u u m M e c h . T h e r m o d y n . 5 (1993), 163-204.

mechanical Magnetic Work and Thermodynamics

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