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1 MIT OpenCouseWae Electomagnetc Felds, Foces, and Moton, Spng 5 Please use the followng ctaton fomat: Maus Zahn, Electomagnetc Felds, Foces, and Moton, Spng 5. (Massachusetts Insttute of Technology: MIT OpenCouseWae). (accessed MM DD, YYYY). Lcense: Ceatve Commons Attbuton-Noncommecal-Shae Ale. Note: Please use the actual date you accessed ths mateal n you ctaton. Fo moe nfomaton about ctng these mateals o ou Tems of Use, vst:

2 6.641, Electomagnetc Felds, Foces, and Moton Pof. Maus Zahn Lectue 15: Foce Denstes, Stess Tensos, and Foces I. Maxwell Stess Tenso A. Notaton _ F = τ, τ = T x + T y + T z x x x xx xy xz _ F = τ, τ = T x + T y + T z y y y yx yy yz _ F = τ, τ = T x + T y+ T z z z z zx zy zz T T xx xy xz Τ = T yx T yy T yz T zx T zy T zz f x = F d = τ d= τ n da = T n + T n + T n x da x x xx x xy y xz z τ τ S S n = T n + T n + T n = T n x xx x xy y xz z xn n n = T n + T n + T n = T n y yx x yy y yz z yn n τ n = T n + T n + T n = T z zx x zy y zz z zn n n f = τ d = τ n d = T n ds = j j S S Fd F = τ = T + T + T x y z x y z = T j x j B. EQS Stess Tenso 1 1 ε F = ρ E E E f ε + E E ρ ρ 1 = (εe) E (E E) ε + 1 ε E E ρ ρ Pof. Maus Zahn Page 1 of 17

3 (εe j ) 1 ε 1 ε F= E EE + EE ρ x j K K x x K K ρ E E j E= = x j x E 1 ε 1 ε F= εe E εe EE + EE ρ x j ( j ) j x j K K x x K K ρ E j 1 ε 1 ε F= ( εee j ) ε E j EE + EE ρ x j x K K x x K K ρ ε 1 EE j j x 1 1 ε F = (εee j ) εee ρ x j x K K EE K K ρ x j = δ j x j δ j = Konece Delta 1 = j 1 ε F= εee δ EE ε ρ = T x j j j K K ρ x j 1 ε T j = εee j δ E E ε ρ j K K ρ j C. MQS Stess Tenso F = J f B 1 H H µ + 1 ρ µ H H ρ ( ) ( ) 1 1 µ = H µh H H µ+ ρ H H ρ ( H) H = (H ) H 1 (H H) µ + ρ ρ H H F = µ (H ) H (H H) H H µ Pof. Maus Zahn Page of 17

4 F = µ H j x j H 1 (HH ) 1 HH µ 1 µ + ρ HH K K x K K K K x x ρ = (µhh ) H (µh ) µ H H 1 µ 1 µ j j K K H H + ρ H H x j x j x K K K K x x ρ B = 1 µhh K K x F= 1 µ µhh µhh ρ HH ( j ) K K K K x j x ρ = µhh 1 µ δ H H µ ρ = x j j j K K ρ x j T j T = µhh 1 δ H H µ ρ µ j j j K K ρ II. A-Gap Magnetc Machnes Coutesy of MIT Pess. Used wth pemsson. Pof. Maus Zahn Page 3 of 17

5 A. Genealzed Descpton π f = T n dzdy = w µ H H z zx x z x S π dz = w x= µ H H dz z x foce on a wavelength jz ( ), b ( z,t ) a z,t = Re Ae = Re Be jz π ( ) ( ) Re = Re A B π 1 a z,t b z,t dz= AB * 1 * π w µ * f = Re H H z z x π w µ = Re K H * x 1 s B coth d x snh d χ s = 1 B µ x coth d snh d χ Pof. Maus Zahn Page 4 of 17

6 s s 1 K s Η z = + j χ χ = H z = j j H K z χ = = j j s χ µ H x = µ + χ coth d snh d s Κ Κ = µ cothd j snh d j * s * +µ Re K H µ = Re j K Κ * + KK coth d x snh d =Re µ jk * s Κ snh d s f = πw µ * Re jk Κ (foce on each wavelength) snh d z s B. Synchonous Inteacton Coutesy of MIT Pess. Used wth pemsson. Pof. Maus Zahn Page 5 of 17

7 K s s s j(ω = K sn ω t z = Re jk e t-z) s s K = t z z U K sn ω ( ' δ) ; z'= t =K sn (ω + ) ( δ U t z ) =Re jk e j(ω +U) t j δ e Κ s = jk s e jω s t o j δ j(ω ) Κ = jk e e +U t o πw µ s jω s t -j δ -j( ω +U ) t f = Re snh d j z ( jk ) e ( jk e ) e πw µ s -j (ω -ω ) = KKRe je δ e j s -U t snh d Fo tme aveage foce ω = ω +U s (synchonous condton) Usually ω = ω s =U πw µ f = K s K z snh d sn δ Coutesy of MIT Pess. Used wth pemsson. Pof. Maus Zahn Page 6 of 17

8 III. Electostatc Machne π wπ z Tzx f = w π dz = π ε EzE x x= x= E = j z 1 πw * f = Re ε E E z z x w * ) E x = π Re ε ( j s 1 coth d s Dx snh d = ε 1 D x coth d snh d s + coth d snh d ε E x = ε * * s Re ε coth d j ε E x = Re j snh d + =Re +j * ε s snh d Pof. Maus Zahn Page 7 of 17

9 πw ε f = Re * snh d j z s s s cos (ωst z ) ( ( )) = = cos ω t z' - δ ; z' = z Ut j(ω +U )t = e s jω s t = e e j δ πwε z snh d j ω ω -U t f = Re j s e -j δ e ( s - ) ω = ω + U s πwε z s f = sn δ snh d I. Devaton of the Koteweg-Helmholtz Foce Densty fo Incompessble Meda fom the Quasstatc Poyntng s Theoem A. Poyntng s Theoem B E = t D H = J f + t D = ρ f B= ( ) ( E H =H E) E ( H) B D = H E E J f t t Pof. Maus Zahn Page 8 of 17

10 B. Powe In Quasstatc Electc Ccuts Fa away fom the ccut elements E= E = Φ H = J f Jf = P = (E H) da n S = + ( Φ H) da S = ( Φ H) d Φ H = H Φ Φ ( ( ) ( ) = J f Φ = J ( f Φ) H) Pof. Maus Zahn Page 9 of 17

11 P n = (J f Φ) d = N S Jf Φ da = =1 S Jf da N = I =1 I C. Electoquasstatcs (EQS) Ohmc Meda: J f ' = σ E ' = J f ρ f v J f = σ E + ρ f v D ε x, y, z E = ( ) ( ) E ( ε ( ) ) E H d = E H da = I = x,y,z E d t E (σ E + ρ v d S I = ε ( x, y, z ) E (ε ( ) ) d + σ ε (x, y, z ) t x, y, z E d + ρ E f ) f E v d 1 (x,y,z t = 1 ε ε (x, y, z ) d + σ E d + ρ f ) E E v d 1 ε (x, y, z ) E ε ε (x, y, z ) ( x,y,z ) t E d = t ε (x,y,z) d ε (x, y, z ) E 1 t d ε (x, y, z ) = E t 1 ε (x,y,z) E d + t ( ε (x,y,z )) d Pof. Maus Zahn Page 1 of 17

12 Theoem: d α d = α d + (α v ) d dt t Consevaton of mass: α = ρ mass densty d ρ dt ρ d = = t d + (ρ v ) d ) ( v ) = ρ + (ρ v ) = = ρ + (v ρ+ρ t t Incompessble: dρ = ρ + (v ) ρ = v = dt t d dt ε d = t ε d + ( ε v ) d = ε ε + ( ε v ) = + ( v ) ε + ε v = t t ε = (v ) ε t 1 1 ε ( ) t ε (x, y, z ) E d = ε (x, y, z ) E d x,y,z t E + ( v ) ε (x,y,z) d 1 I = 1 ε (x,y,z) E d + σ E d + ρ E E f ε vd t Enegy Stoed (W E ) Rate Powe Foce Densty Dsspated P E Wo Rate = Mechancal Powe 1 F= ρ f E E ε (foce pe unt volume) nt/m 3 Pof. Maus Zahn Page 11 of 17

13 f= Fd foce (nts) D. Magnetoquasstatcs J' f =J, f E'=E + v B J'=J f f = σ E' = σ (E + v B) B = µ ( x, y, z )H ( H ) d = E H da = I = H ( µ ( ) ) E x, y, z H d S t P dsspated = E' J ' f d = E' J f d Jf (v B) = Jf (B v) = (Jf B) v E' vxb J f d x, y, z H H (x,y,z H t µ µ (x,y,z) t x,y,z ) = µ ( ) µ ( ) H 1 1 = ( ) µ (x,y,z) H µ x, y, z t 1 µ (x, y, z ) H 1 1 = µ (x, y, z ) H t µ ( x, y, z ) t µ(x, y, z ) ( µ x, y, z ) H = µ (x, y, z) µ (x,y,z) H t µ (x, y, z ) t d µ d = µ + (v ) µ = ( v = ) dt t 1 H t µ (x, y, z ) H 1 = t µ (x, y, z ) H H µ v Pof. Maus Zahn Page 1 of 17

14 I = 1 µ (x,y,z) H d + P dsspated t Enegy densty W M + v J f B 1 H µ d FM = foce densty Mechancal Powe 1 W M = µ (x, y, z ) H d, P dsspated = E ' J f d = E ' J f ' d = σ E ' d Total Magnetc Enegy F = J M f 1 B H µ foce densty. Compessble Meda A. Electoquasstatcs (EQS) Ohmc meda: J' = σ E' Polazaton dependent on mass densty (ρ) alone, electcally lnea D = ε ( ρ) E EQS Gallean Tansfomaton: J = σ E + ρ f v ( E H) d = E H da = S I = E t ε ( ρ) E d E (σ E + ρ v) d f E ε ρ 1 ( )E ρ ρ 1 ε ( )E = ( ) ( ) ε ( )E = ε ρ E t ε ρ t ε ( ρ ) t Pof. Maus Zahn Page 13 of 17

15 ρ ρ E 1 = E t ε ( ρ ) ε ( ) t ρ 1 ε ( ) ε ( ) E ε ρ E = 1 ( ) ε ( ρ ) E t t + ε ( ρ) E ρ +1 ε( ) ( ) ε ρ t = 1 ε ( ) 1 ε (ρ) ρ E + E t t ε( ρ ) ρ ρ = ε( ) t ρ t ; ρ + (ρ v) = t (Consevaton of mass) ε( ρ ) ε(ρ) = ( (ρ v)) t ρ 1 1 ε ρ I = t ε ( ρ ) E d E t ( ) d σ E d ρ E v f d 1 ε ρ E ( ) d = 1 ε E ( ρ v ) t ρ d 1 ε 1 ε = E ρ v d + ρ v E d ρ ρ 1 ε 1 ε 1 ε = ρ ρ E v n da S + v ρ E ρ E ρ ρ d I = 1 ε ( ρ ) E d + σ E d t 1 ε ρ E v n da ρ S 1 1 ε + v ρ f E E ε+ ρ ρ E d Pof. Maus Zahn Page 14 of 17

16 whee ε ρ = ε ρ electc enegy 1 ε ( ) W E = ρ E d, P dsspated = σ E d (powe dsspated) 1 1 ε F E = ρ E f E ε + ρ E foce densty ρ 1 ε ρ ρ S E v n da = because as S, E da W E I = + P dsspated + FE v d t Mechancal Powe B. Magnetoquasstatcs (MQS) MQS Gallean Tansfomaton: J'=J, f f E ' =E + vxb B= µ ( ρ) H ( ExH ) d= ExH da = I = H t µ ( ρ ) H d S E' v B J f d ' P = dsspated E' J f d = E' Jf d Jf (v B) = Jf H t µ ( ρ )H = (B v) = (Jf B) v ( ) H µ ρ = ( ) t ( )H µ ρ µ ρ 1 µ ρ = t µ ρ 1 µ ( ρ ) H ( ) t 1 µ ( ρ) ( ) H 1 1 µ ( ρ ) H t µ ( ρ) Pof. Maus Zahn Page 15 of 17

17 1 ( ) µ ( ρ ) ( ) µ ρ µ ρ H + H t µ ρ t = 1 ( ) ( ) 1 µ ρ µ ρ H + H t t = 1 ( ) ( ) + d µ ρ µ ( ρ ) d = µ ( ρ) v = dt t ( ) ( ) µ ρ µ ρ ρ ρ = ; + ρ t ρ t t µ ( ρ ) µ ρ = ( ) ( ρ t ρ ( v)) ( v) = 1 1 µ ( ρ ) I = ( ) H d H d t Pdss µ ρ t ( ) d 1 µ ρ H = 1 H µ t ρ (Jf B) v d ( ρ v) d 1 1 µ µ = H ρ v d + ρ v H d ρ ρ S = 1 ρ ρ µ 1 µ 1 µ H v n da + v ρ ρ H H ρ ρ d 1 ( ) I = µ ρ H 1 µ d + P dss ρ H v n da t S ρ 1 1 µ + v J B H µ + ρ f ρ H d Pof. Maus Zahn Page 16 of 17

18 whee µ ρ = µ magnetc enegy ρ W d, P ' M = 1 µ ( ρ) H dsspated = E' J f d = E' Jf d Powe dsspated F 1 M = J f B H µ + 1 ρ µ H foce densty ρ 1 µ ρ H v n da = because as S, S ρ H da I = W M + P dsspated + FM v d t Mechancal Powe C. Conclusons Foce denstes 1 1 ε EQS: F E = ρ f E E ε + ρ E ρ 1 1 µ M f H µ + ρ E MQS: F = J xb ρ Pof. Maus Zahn Page 17 of 17