A charge q experiences a force in the electric and magnetic fields : Lorentz force equation.

Transcription

1 5. THE TATIC MAGNETIC FIELD Anlgy between electric nd gnetic fields E, electric field D, electric flux density D εe B μh 5- LOENTZ FOCE EQUATION H, gnetic field B, gnetic flux density A chrge q experiences frce in the electric nd gnetic fields F Fe + F q ( E + v B ) : Lrent frce equtin. Exple 5- An electrn hs n initil velcity v v y in unifr gnetic flux density B B. () hw tht the electrn ves in circulr pth. (b) Find the rdius r f the circulr pth in ters f electrn chrge e, electrn ss nd initil speed v. lutin () The gnetic frce n the electrn F e v B (5-4) F is lwys perpendiculr t v, hving cnstnt gnitude F. F cuses v t bend twrd the rigin due t e < 0, nd frces the electrn t ve n circulr pth. Electrgnetics 5- Prprietry f Prf. Lee, Yen H

2 (b) ewriting Eq. (5-4) F evb evb ρ ( e < 0) (5-5) ( ) At stedy stte, F is blnced with ss ccelertin f the electrn dv dv d d d v dt d dt d dt vω( ρ ) where v v d ρ d ω t, ω is ngulr velcity. (5-6) Equting Eq. (5-5) with Eq. (5-6) ω eb Using v v r eb r ω 5- BIOT-AVAT LAW A stedy current I is surce f the sttic gnetic field H. Bit-vrt lw reltes dh t I Idl dh 4π (5-7) where r r. r : Field pint r : urce pint r r : Distnce vectr : Unit vectr f Expnded by the bse vectrs t r, nt t r It is fr the surce t the field pint H is esured in peres per eter[ A / ] The nuertr dl dl' ' : It is evluted t r ( dl ) Trnsfred t r Mgnetic field intensity due t I in wire f negligible thickness H C Idl 4π (5-8) Electrgnetics 5- Prprietry f Prf. Lee, Yen H

3 Current I in ters f J nd J s Current density : J ρv v urfce current density : Js ρsv The differentil current eleent I dl Jdv J ds Using Eq. (5-9) in Eq. (5-8) H V J 4π s dv (5-9) (5-0) s ds H J 4π (5-) Exple 5- H f stedy current in stright wire Deterine H t pint P :( x, y,0) due t n infinitely lng stright cnducting wire f negligible thickness, lying lng -xis nd crrying stedy current I. lutin The current hs cylindricl syetry. The distnce vectr r r ρ ( ρ ) ( ) ' ' ρ ρ : ' The vectr differentil length dl d Electrgnetics 5-3 Prprietry f Prf. Lee, Yen H

4 Fr Eq. (5-8) Id ρ ( ) I ρd H 4 π 4 π r + ρ + 3 3/ I H πρ ( ρ + ) (5-) (5-3) The line current hs cylindricl syetry N chnge in I fter rttin f I but -xis. It ls hs trnsltinl syetry N chnge in I fter trnsltin f I lng -xis H hs the se syetries s I H is independent f nd s Eq. (5-3). A finite line current extending fr t Fr Eq. (5-) ( ρ + ) / 3/ Iρ d Iρ ρ H 4π 4π ρ + Using the ngles in Fig. 5-4 I H θ + θ 4πρ [ cs cs ] (5-4) This frul is true even fr θ r θ lrger thn 90. Electrgnetics 5-4 Prprietry f Prf. Lee, Yen H

5 Exple 5-3 At pint p : (, b,0) in the first qudrnt, find H f n infinite right-ngled wire crrying stedy current I. lutin Fr line current in y < 0, we use θ α, θ 0 nd ρ in Eq. (5-4) I H cs α + 4π [ ]( ) Fr the right-hnd rule Fr line current in x < 0, we use θ 0, θ β nd ρ b I H + csβ 4πb [ ]( ) Electrgnetics 5-5 Prprietry f Prf. Lee, Yen H

6 In Fig. 5-5 b cs α sin( α 90 ) + b cs β sin( β 90 ). + b The gnetic field intensity t p I b I H H + H π 4 b + b π + b I + b ( + b) 4πb ( ) ( ) Exple 5-4 Deterine H t pint p :( 0,0, b ) due t circulr wire f rdius centered t the rigin in xy -plne, crrying stedy current I. lutin In cylindricl crdintes r r b ρ ' dl d ' Fr Eq. (5-8) Idl πi d ' ( b ') ρ H C 0 3 4π 4π πib ρ' πi ' d + d π 4π (5-5) Electrgnetics 5-6 Prprietry f Prf. Lee, Yen H

7 ρ' is nt cnstnt ' x' cs y' ρ + sin ρ' ( ) ( + π) ρ' Thus I H + b 3/ (5-6) 5-3 AMPEE CICUITAL LAW Apere s circuitl lw is nlgus t Guss s lw. We cn deterine H by syetry in the current distributin. We cn derive Apere s circuitl lw fr Bit-vrt lw. The clsed line integrl f H is equl t the ttl current enclsed by the pth f integrtin. Hi d l I C Ttl current enclsed by C Ttl current crss the surfce bunded by C Cntinuity equtin fr stedy current i d s 0 :, clsed surfce (5-8) J Fig. 5-7 (), clsed heisphere Fig. 5-7(b), pen heisphere Perieter clsed pth C Ttl current enclsed by C Ttl current pssing thrugh the surfce Apere s circuitl lw C Hidl Jids C ight-hnd rule (5-9) : A clsed pth : The surfce bunded by C : ight thub pints t the directin f the surfce nrl ight fur fingers rtte lng C Electrgnetics 5-7 Prprietry f Prf. Lee, Yen H

8 Fr tkes s there Hids Jids is rbitrry surfce (5-0) Pint fr f Apere s circuitl lw H J (5-) T pply Apere s circuitl lw, the clsed pth C shuld be chsen such tht H is t lest cnstnt n C. Exple 5-5 Find H, by Apere s circuitl lw, due t n infinitely lng stright wire lying lng - xis, crrying stedy current I. lutin The line current hs cylindricl nd trnsltinl syetries. H hs the se syetries s I. H is independent f nd crdintes. Fr Bit-vrt lw Id l prduces dh nly in directin We expect H ( ρ) n the pth C H Electrgnetics 5-8 Prprietry f Prf. Lee, Yen H

9 On C f rdius ρ π H i l ( ) C i π 0 0 d H ρd H ρ d I errnging it I H πρ H The result is the se s Eq. (5-3). Much sipler clcultins in the cse f Apere s circuitl lw. A stright line current f finite length Cylindricl syetry but n trnsltinl syetry Bit-vrt lw Only H -cpnent We expect H H ( ρ, ). : H is cnstnt n C C y r y nt enclse the current N use f Apere s circuitl lw. Exple 5-6 An infinitely lng cxil cble crries unifr stedy current I. The rdii f the cylindricl surfces re, b nd c, respectively. Find H in the regin ρ> 0 by Apere s circuitl lw. Electrgnetics 5-9 Prprietry f Prf. Lee, Yen H

10 lutin The unifr current hs cylindricl nd trnsltinl syetries. Current eleents t nd, fr Bit-vrt lw Only H -cpnent We expect H ( ρ) H We chse circle centered t -xis s the pth f integrtin. In the regin 0 < ρ π Hidl H i ρd πh ρ C 0 ( ) The current enclsed by C πρ π I (5-) (5-3) Equting Eq. (5-) with Eq. (5-3) I ρ H H ( 0 π In ther regins, H is the se s Eq. (5-). But the enclsed current is I ( ρ b) c I c ρ b 0 ( c Equting the current with Eq. (5-) I πρ < ρ ) (5-4) (b ρ c ) ρ ) H ( b I c ρ H (b c πρ c b ρ ) (5-4b) ρ ) (5-4c) H 0 ( ρ c ) (5-4d) The cxil cble prduces n H utside. It is therefre free fr ny electrgnetic interferences. Electrgnetics 5-0 Prprietry f Prf. Lee, Yen H

11 Exple 5-7 Find H f tridl cil hving N turns f wire nd crrying stedy current I. The trid hs the inner rdius f nd the uter rdius f b. lutin The tridl cil hs cylindricl syetry but -xis H is cnstnt n C centered t -xis The cil penetrtes plne nly nce H ρ nd H -cpnents re linked with I The cil penetrtes 0 plne N ties H -cpnent is linked with NI. H >> H ρ, H An idel tridl cil hs H H inside the trid nd H 0 utside. In the regin < ρ< b π Hidl H iρd H ρ d πh ρ C C 0 The ttl enclsed current is NI NI H πρ r NI H H πρ ( < ρ< b) (5-5) In the regins ρ< nd ρ> b N net current is enclsed H 0 Electrgnetics 5- Prprietry f Prf. Lee, Yen H

12 5-4 MAGNETIC FLUX DENITY Mgnetic flux density in free spce is defined s B μ H : Tesl [ T ] r webers per squre eter Wb / (5-6) μ is perebility f free spce 7 μ 4π 0 : Henrys per eter [ / ] H (5-7) Mgnetic flux is the ttl unt f the flux crssing surfce Φ Bi d s. (5-8) Dt prduct cnverts ds int n equivlent re in crss sectin f B. A differentil current eleent Clsed field lines Bit-vrt lw Arbitrry current Clsed field lines Liner prperty Flux lines f B lwys clse upn theselves Guss s lw fr B is theticl expressin fr the clsed flux lines Bi d s 0 (5-9) Pint fr f Guss s lw i B 0 (5-30) Mxwell s equtins fr sttic electric nd gnetic fields i D ρ v E 0 i B 0 H J In integrl frs C C Dids ρ dv V Eidl 0 Bids 0 Hidl Jids v (5-3) (5-3) Electrgnetics 5- Prprietry f Prf. Lee, Yen H

13 5-5 VECTO MAGNETIC POTENTIAL Chrge q Electric ptentil V Electric field E V Current V Vectr gnetic ptentil A Mgnetic flux density B A Vectr gnetic ptentil A Guss s lw i B 0 B A (5-33) Vectr identity i A 0 A in ters f I Bit-vrt lw μ Id l B ' 3 4 π C ecll 3 ewriting Bit-vrt lw μi B d 4π l C ' : r r (5-34) (5-35) (5-36) Using vectr identity ( VA) ( V ) A + V ( A ) μi dl μi dl B ( d ) 4 C ' π l 4π C ' 0 ewriting it μid l B C ' 4π Cpring Eq. (5-38) with Eq. (5-33) μid l A C ' 4π (5-37) (5-38) (5-39) Eq. (5-39) is uch sipler thn Bit-vrt lw becuse f n crss prduct. The vectr gnetic ptentil y be ulti-vlued functin A + ϕ is nther slutin 0, custrily Electrgnetics 5-3 Prprietry f Prf. Lee, Yen H

14 Fr J nd J s μ J A dv V ' 4π (5-40) A ' μj s ds ' 4π (5-4) A is esured in webers per eter [ Wb / ] r tesl eter [ T] eltin between A nd Φ Mgnetic flux Φ Bi d Φ ( A ) ids Aidl C i. (5-4) Clsed line integrl f A is equl t ttl gnetic flux Apere s circuitl lw in pint fr is derived fr Bit-vrt lw Curl f H H μ B μ A Using vectr identity, ( i ) H A A μ () Divergence ter Using Eq. (5-40) A A A ( i ) (5-43) μ J ' ' dv μ J A dv V ' 4 4 i i i (5-44) π π V ' Unpried crdintes Cnsider tw vectr identities J J + J J i i i i (5-45) 0 J ' ' i ' ij' + ' ij' ' i J' (5-45b) 0, cntinuity equtin fr stedy current Using ( / ) ' ( / ), cbine Eqs. (5-45) nd (5-45b) J' ' ' J i i (5-46) Electrgnetics 5-4 Prprietry f Prf. Lee, Yen H

15 Inserting Eq. (5-46) in Eq. (5-44), nd pplying divergence there μ J' μ J' A ' dv ds ' 0 4πV ' 4π ' i i (5-47) V ' + Extr regin utside V J ' 0 n ' Eq. (5-43) reduces t μ H A (5-48) ' () We next find Lplcin f A in ters f J by using Eq. (5-40) Fr the definitin, A A + A + A (5-49) ( ) ( ) ( ) x x y y Cbining Eqs. (5-49) nd (5-40), the x -cpnent μ J x μ Ax dv' J ' V ' 4 4 V ' x dv π π fr 0, r r r ' V δ, tiny regin centered t J cnstnt in δ x r r ' (5-50) Applying divergence there μjx μjx Ax dv' d ' 4π i V ' 4π s i (5-5) ' Excluding r r ' in V ' + s Fig. 5- J x 0 n Eq. (5-5) reduces t μ J μ J 4π μj x x Ax d ' li 4 0 π i s 4π x (5-5) 3 where we use ( / ) / nd '. iilr equtins fr J y nd J. The vectr Pissn s equtin A J (5-53) μ Inserting Eq. (5-53) in Eq. (5-48) H J Electrgnetics 5-5 Prprietry f Prf. Lee, Yen H

16 Exple 5-8 Find H t p :( 0,0, ), using vectr gnetic ptentil, due t n infinite current sheet J J lying in xy -plne. y lutin Fr Eq. (5-4) μ J da 4π y ds' : r r ( ) ( ) x x + y y + We d nt fix r t r yet. / At r A μ J y x' y' 4π dx dy μ B A 4 J y π x' y' dx dy (5-54) The curl in the integrnd x y y x y / ( ) ( ) 0 x x + y y + 0 3/ 3/ ( x x ) ( y y ) ( x x ) ( x x ) ( y y ) (5-55) x Electrgnetics 5-6 Prprietry f Prf. Lee, Yen H

17 Inserting Eq. (5-55) int Eq. (5-54) μ J B A x ' ' 3/ 4π x y μ J dx dy ( ) ( ) x x + x y + x x dxdy ( ) ( ) ( ) ' ' 3/ 4π x y x x + x y + 0, integrnd is n dd functin (5-56) ince the integrtin liit is fr t, we cn reduce Eq. (5-56) t μ J x B ' ' 3/ 4π x (5-57) y ( x ) + ( y ) + dxdy Evluting it in cylindricl crdintes μ J ρ dρ d μ J π B π x x ' 0 ' 0 3/ / 4π ρ ( ρ ) + 4π ( ρ ) + ρ 0 μ J x The gnetic field intensity J H x ( > 0) (5-58) By substituting fr in Eq. (5-57) J H x ( < 0) (5-58b) ρ Electrgnetics 5-7 Prprietry f Prf. Lee, Yen H

18 5-6 THE MAGNETIC DIPOLE A gnetic diple is sll clsed lp crrying stedy current. eferring t Fig. 5-3 A clsed current lp f rdius in xy -plne. p is field pint lcted t π / plne. p is surce pint lcted n the current lp epeting Eq. (5-39) μi A 4π C dl (5-59) Using ixed crdinte syste r sin θ + cs θ y r' cs ' + sin ' x y ( ) r r' cs ' + sin θ sin ' + cs θ The inverse f x y r r' + sinθsin ' / Assuing >>, nd using binil expnsin / + sinθsin ' + sin θ sin ' (5-60) Expnding d l by, θ nd t p. dl' d ' ( cs sin cs cs θ sin ) d θ + θ + (5-6) ewriting Eq. (5-59) using Eqs. (5-60) nd (5-6) μ I π A + sin θsin d ' cs sin θ cs cs sin θ θ + π μ I 4π π sin sin d θ 0 ( ) r μ I π A sin θ 4π (5-6) Utiliing crss prduct, the vectr ptentil f current lp μ 4π A : Mgnetic diple ent (5-63) I I (5-64) π Electrgnetics 5-8 Prprietry f Prf. Lee, Yen H ight-hnd rule; right thub indictes the surfce nrl while right fingers fllw the directin f I.

19 B fr the curl f A μ 4π B ( csθ + sinθ ) (5-65) 3 θ It hs the se fr s E f n electric diple( p nd ε nd / μ ). It is fr field pttern ( >> ). 5-7 MAGNETIC MATEIAL An t Psitively chrged nucleus + Negtively chrged electrns Orbiting rund the nucleus Mgnetic diple. Orbiting electrns Electrn spin Nucler spin, negligible Electrgnetics 5-9 Prprietry f Prf. Lee, Yen H

20 Dignetic teril Orbiting electrns + Electrn spin Zer gnetic diple ent In n pplied B Orbiting electrns + Electrn spin A net gnetic diple ent ppsite t B Centrifugl r centripetl frce N chnge in rbit but chnge in velcity Prgnetic teril An inherent sll diple ent nd distributin f diple ent Diples lign t n externl B Zer net diple ent A net diple ent Ferrgnetic teril A lrge gnetic diple due t electrn spin Align in dins nd dins Grwing f the din prllel t B Zer net diple ent A lrge diple ent 5-7- Mgnetitin Mgnetitin is the gnetic diple ents per unit vlue M li Δv 0 j NΔv j Δv j : N, nuber density f ts r diples. Units f peres per eter. The ediu is gnetied if 0 M. eferring t Fig plne : Interfce between gnetic teril nd free spce qure lp f : A gnetic diple : Unit vectr f, n ngle θ t the surfce nrl The teril is sliced int hyptheticl lyers f thickness ( /) sinθ. M induces the Mgnetitin surfce current density () ide view in Fig. 5-4() Diples centered in lyer Current flwing ut f the pper in free spce Current flwing int the pper in lyer Nn-er net current in free spce nd lyer Zer net current in lyers belw lyer Electrgnetics 5-0 Prprietry f Prf. Lee, Yen H

21 () Frnt view in Fig. 5-4(b) A diple ppers t be rectngle f sin θ. Diples centered in shded re cntribute t the current crss x 0 plne. urfce current density in x directin ( ) J I N sin θ N sin θ s Ttl nuber f diples in sinθ. Mgnetitin surfce current density Js M n : M N (5-66) n, unit surfce nrl t the interfce. Electrgnetics 5- Prprietry f Prf. Lee, Yen H

22 M induces the gnetitin current density J eferring t Fig. 5-5 The xy -plne is n interfce between gnetic teril ( 0) nd free spce ( 0) A clsed lp C is in xy -plne. J flws in xy -plne due t M. s Nner net surfce current crss C A net current fr the inside t the interfce Cntinuity equtin Mgnetitin surfce current crss dl n C Δ I J dl ( M ) dl is perpendiculr t C nd dl i i : ρ s s ρ n Using vectr identity ( A B) ic Ai ( B C) s ( n ρ ) Δ I Mi dl Mi dl : n ρ ρ is prllel t dl. >. Ttl surfce current crss C : I s Mi d l C (5-67) Fr cntinuity equtin : Is J i d s (5-68) Current crss Equting Eq. (5-67) with Eq. (5-68) nd pplying tkes there J ids ( M) i ds r J M (5-69) The curl f M is equl t the gnetitin current density in the teril, which then prduces n internl gnetic field ccrding t Apere s circuitl lw in pint fr. Electrgnetics 5- Prprietry f Prf. Lee, Yen H

23 Exple 5-9 A cylindricl pernent gnet f rdius nd height b hs unifr cnstnt gnetitin M M. Find gnetitin current density nd gnetitin surfce current density. lutin Fr Eq. (5-69) J M 0 Fr Eq. (5-66) Js M n M 0 (tp surfce) J M ( ) 0 (btt surfce) s ( ρ ) J M M (side surfce) s The pernent gnet is equivlent t cylindricl sheet with surfce current density M Perebility Free current density J. Externl gnetic flux density B e. Mgnetitin M. Mgnetitin current density J. Internl gnetic flux density B i. Mgnetic field H is redefined t be relted nly t J. B in the teril is given by μh. Ttl gnetic flux density in teril B J + J J + M : J is free current density μ J is induced current density. errnging it B M J μ We redefine H in teril ediu s B H M μ r ( ) B μ H + M (5-70) (5-7) Apere s lw in teril ediu is, therefre, H J : J is free current density (5-7) We cn use the se fr f Apere s lw regrdless f the teril edi. Electrgnetics 5-3 Prprietry f Prf. Lee, Yen H

24 By integrting Eq. (5-7) ver surfce ( H) ids Ji ds Applying tkes there, integrl fr f Apere s lw Hi d l I : I is ttl current enclsed by C (5-73) C In hgeneus, istrpic nd liner gnetic teril χ, gnetic susceptibility (5-74) M χ H : Inserting Eq. (5-74) in Eq. (5-7) B μ ( + χ ) H μμ rh : μ, perebility f free spce (5-75) : μ r, reltive perebility r B μh : μ, perebility [ H / ] (5-76) Dignetic teril 6 4 : μr, χ ~ 0 ~ 0 Prgnetic teril 5 3 : μr, χ ~ 0 ~ 0 5 Ferrgnetic teril : μ r >> χ ~ 0 ~ 0 Tw fundentl lws fr gnetsttic fields i d l I : Apere s lw (5-77) H C Bi d s 0 : Guss s lw (5-78) In pint frs H J (5-79) i B 0 (5-80) Cnstitutive reltin between B nd H B μh (5-8) ince bth divergence nd curl f H re specified, the gnetic field cn be uniquely deterined in regin f spce, ccrding t Helhlt s there. Exple 5-0 An infinitely lng stright cnducting wire lies lng -xis in free spce nd crries stedy current I. The wire is cvered by n infinitely lng cylindricl shell f perebility μ, nd rdii nd b, respectively. Find () H nd B in the regin ρ> 0 (b) gnetitin surfce current densities J s nd J s t ρ nd ρ b (c) B in the regin ρ> 0 using J nd J. s s Electrgnetics 5-4 Prprietry f Prf. Lee, Yen H

25 lutin The current nd the shell hve bth cylindricl nd trnsltinl syetries. Bit-vrt lw predicts nly H -cpnent. H ( ρ) H () Applying Apere s lw in the regin 0 < ρ πρ H I I H H πρ ( 0 <ρ ) (5-8) Iμ B μ H πρ (5-8b) In the regin ρ b π ρ H I I πρ H ( ρ b) (5-83) Iμ B μ H (5-83b) πρ In the regin π ρ H I I πρ ρ b H ( ρ b) (5-84) Iμ B μ H (5-84b) πρ Electrgnetics 5-5 Prprietry f Prf. Lee, Yen H

26 (b) Fr Eqs. (5-74) nd (5-75) At At μ χ μ M H H (5-85) ρ, inserting Eq. (5-83) int Eq. (5-85) μ I M μ π Mgnetitin surfce current density t ρ μ I J M s n : μ n ρ π ρ b, inserting Eq. (5-83) int Eq. (5-85) μ I M μ πb The surfce current density t ρ b μ I J M s n μ n ρ πb (5-86) (5-87) (c) In Fig. 5-6(b), we replced the cylindricl shell with the gnetitin surfce current densities J s nd J s lcted t ρ nd ρ b, respectively, in free spce. In 0 <ρ< B J μ r d I C μ B i l, the circultin lw fr B Using cylindricl syetry πρ B I μ Thus Iμ πρ B ( 0 <ρ ) In B <ρ< b d I + πj C μ B i l s urfce current t ρ enclsed by C Electrgnetics 5-6 Prprietry f Prf. Lee, Yen H

27 In Using Eq. (5-86) μ I πρ B I + π μ μ π Iμ B πρ B ( <ρ< b) ρ> b d I + πj πbj C μ B i l s s In the ppsite directins Using Eqs. (5-86) nd (5-87) μ I μ πρ B I + π πb I μ μ π μ πb Iμ πρ B ( ρ> b) B We hve the se result s in prble (). This ethd f equivlent current is gd fr B, but nt fr H, hwever Mgnetic Bundry Cnditins Applying Apere s lw t rectngulr lp c Hidl H Δw H Δ w + Hidl + Hidl C t t b d Jsn Δw Free surfce current density As Δ h ges t er, bundry cnditin fr H H H J t t sn (5-88) (5-89) Electrgnetics 5-7 Prprietry f Prf. Lee, Yen H

28 J sn fllws the right-hnd rule, ight thub pints t the directin f J sn while right fur fingers rtte lng the directin f C. Fr er surfce current density, the tngentil cpnent f H is cntinuus crss the interfce H H (5-90) t t The tngentil cpnent f B is discntinuus B B t t μ μ Applying Guss s lw t the bx nd letting Δ h pprch er B B n n Using B μh μ H μ H n n (5-9) (5-9) (5-93) Exple 5- Given n interfce between free spce nd n irn plte f μ 000μ, the gnetic flux density B is t.0 t the surfce nrl in free spce. Find the directin f B in the irn plte. Electrgnetics 5-8 Prprietry f Prf. Lee, Yen H

29 lutin Fr the bundry cnditin B B sin fr tngentil cpnent μ B t μ n ( ) ( ) B cs fr nrl cpnent In the irn plte B ( μ/ μ ) B sin( ) B cs ( ) t tn θ 7.46 B n θ 86.7 In Fig. 5-9, the gnitude B is exggerted fr visibility. Electrgnetics 5-9 Prprietry f Prf. Lee, Yen H