( ) ( ) ( ) 0. Conservation of Energy & Poynting Theorem. From Maxwell s equations we have. M t. From above it can be shown (HW)
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1 8 Conson o n & Ponn To Fo wll s quons w D B σ σ Fo bo n b sown (W) o s W w bo on o s l us n su su ul ow ns [W/ ] [W] su P su B W W s W A A s V A A : W W R o n o so n n: [/s W] W W 4 44
2 9 W : W F V R o n o so l n [/s W] σ σ Ps V V Ps Dss ow (o loss): W Ω Ω s P A V P Pow n olu nlos b su : W W n w Ponn quon P s ( ) ( ) ( ) W W Ps Psu Ts s Conson o n T on o nusol lon Fls In on u nsnnous l ( ) l ( ) l b ( ) R[ ( ) ] ( ) R ( ) [ ] R : Fls n lso b sb s n s I [ ] n ol sl R : os nnn boos (no ll) us nn o ss boos (no ll) us os R : W wll s o w osl ln ws n o w
3 W l o nn Ts s sl o u nlss o w σ s : B O n nl o l s l Ponn o o on ls n σ () σ () Fo () n () w (W) σ O s σ I n ol ( n ) n n s onbuon o ss ow us b o P σ. In o wos s ons s (ul n).
4 Ponn Vo Insnnous Ponn Vo s n s No: n ollowns I us s ls K o sn nsnnous ls.. n n ul ls o sn on ls.. onl sl nn W o w n s o on ls [ ] [ ] R R No : [ ] [ ] [ ] B A B A R R R [ ] R Now l s lul o T T n [ ] [ ] R R R R T T T T T T [ ] R Ws nsnnous Ponn o n s o -on ls s n b: [ ] [ ] [ ] R R R
5 A on o n nss Rll w n n n s W () ( ) Now l s lul o s qun.. W W () ( ) ( ) bu ( ) [ ( ) ] R n W () R[ ] R[ ] W * { ( ) ( )} 4 { R[ ] } 4 R[ ] 4 4 T s n b W W T T R[ ] 4 T. 4 4 ll W 4 Lon-Lon Dsson W ol oslln lon n nulus s ss n sn Ts lon osllo ol s on ll Lon ol. I s no ll ol o o s su bu w n o sons o ubon. A
6 wn Lon oul ol ws no nown nul ss ss s o o lons. T Lon ssuon ws n bsn o l l l nos o os n n s on bu wn l s l lons wll n Lon o n wll b sl o qulbu oson. n wo sln l s o nw o b w l s ull b ow s onl oson n w w o ol snus b n o ls o On l s l lon os bu w ssu nulus ns son s - - D n s son o Foo n nson on T s lso on wn ss: D Fon on F on D D T on (n) s sul o lon nn w o os lons l onl s bonl o o l. quon o oon: F F F on F oo F nl (l) o Q Q (ssun on ls) F (sn o oo o) oo Fon D (on o) n D Q D Q
7 4 L s n D & Q () Q o N s : lon ss s : s : s Q s son o ln non-oonous nl quon oluon o bo onss o wo s: oln ( ) n ul ( ) soluons Coln soluon w s nsn sons s soluon o Q oonous nl quon (.. on ) Coln soluon (nsn sons) s Pul soluon w s s s soluon s o ns o us. L us ssu -on soluons su s nl quon Q Q / w D / n / n subsu s n ou Cluln P & usbl Q / Q / Rll w I) Assu ols nl II) Assu no ouln bwn ols III) T N ols un olu. In o wos N s nub o ols un olu.
8 5 Polon () P s n b NQ P w Q s sso w ol [C]. NQ s nson o: C C Usn () QN P w () N Q P / W lul o N Q P / Rll P P χ χ χ N Q / W n N Q w s nson o: s Tn χ χ Co bo w son ( on) quon 7 Now suos N oluls un olu n olul s Z lon n lons olul bnn qun (sonn qun) n n onsn n N Q w Osllo sn n Z Rl n n s o ( ) n b
9 6 R I [ ] [ ] ( ) ( ) ( ) ( ) ( ) Rll sln o lons sub o o Q ws n b Q /. No sln o lons o qulbu s snusol w qun o sou I s no n (no on n ou nl ol).. D n Q / n () No s. T qun s ll sonn o ss. Ts ol s so sons No s no n ( ) n. I sonn qun s lso o ( s o s) n w s n o >. Wl bo onsons o no losss n s o s ( ) s n onuon losss sso w s. Rll susson o s onu n s on. Wn n s sn sonn qun s oo o s quon o oonous nl quon o l quns. Rsonn qun s n n b D n > (s o un) No: / w
10 7 W quon In ollown l quns nsnnous B () D σ s () Fo () w () Fo () w s σ (4) w A A A A n No A ( A) A A ( ) A. Lln s n o n Tn () n b wn s ( ) (5) uos u s nll oonous ( s nnn o ) n Us A Lw [q. ()] o n q. (5) W ( ) σ s O ( ) σ s Fo Guss Lw ll n W quon o l l: σ s (6)
11 8 W quon o n l: σ s σ () T on w quons: σ s σ s σ s Fo sou on w [s q. (5-ls )] σ s () () (4) I onu s lso o ( σ σ ) n s In s o on ls o sou bu loss u [q. (4)] w σ s ( ) σ s (6) [ ( σ s )] [ σ ] w σ σ σ σ s onu. s s Dn: ( ) σ w n snn l n n s o w Anuon onsn [N/] Ps onsn [/] Poon onsn [/] n [ σ ] Fo losslss s ( σ ) o q. (6) w 678 No ( ) σ o losslss s. Tn n n s o losslss u.
12 9 Tn w n o n W quon o sl oonns o [ ] w As n l -oonns o l l us ss ollown: ) ( ) ( ) ( ) ( ) ( ) ( T nl quons o o oonns o l sl oluons o W quon To n soluons o w ssu n us son o bls nqu o W n w so s ll onsn quon. oluons
13 4 A B ( ) C ( ) D sn( ) os A B C ( ) D sn( ) os A B C ( ) D sn( ) os ± ll ln w soluons sn ll snn w soluons os o T o soluon osn ns on obl n boun onon. Fo l o ws onn n - n -ons n ln lon - on w : [ C os( ) D sn( ) ] [ C os( ) D sn( ) ] w u A B s osl ln w n nn o ) s nl ln w (o To s s no ollown R[ ( ) ] [ C os( ) D sn( ) ] C os( ) D sn( ) A os Fo ou o o [ ]
14 4 L s lo os( ) o n s Insn To ollow on W us s Z Z n s w us A os( ) Z onsn w ( Z ) V Z Z onsn V s ll s lo oluon o W quon n ou F bu Loss u Rll w quon o loss u ws n b σ () [ ] w σ ( ) On n q. () ( ) ( ) ( ) ( ) n so o o n
15 4 On n w oos soluon o o () n us son o bls o sow W onsn quon Tn s n b D C B A D C B A D C B A sn os sn os sn os onnl unons sn nu ln ws n bol osn n sn sn nu snn ws Cos o sn o Rll w ±. W oul qull n ± n w ou os: w on soul w oos ls lon -s s lon -s ls lon --s s lon -s ls lon -s ows lon -s ls lon --s ows lon -s
16 4 Fo osl ln w (-s) n ss ( w no n o nl sou o n) w us w s s os u n. n o sn o osl ln w n ss s w ou o o nn o u Tln ws o os ln o n ln nn ws os sn ( ) o os o n ( ) o os o n nsn ws o os o n Anu ln ws o os ln o n ln Anu snn ws os os os sn ( ) ( ) ( ) sn( ) sn( ) o os n n ( ) sn( ) os( ) os( ) sn( ) o os n n No : os os ( ) os( ) os( ) os( ) sn( ) sn( ) ( ) os( ) sn( ) sn( ) W quon n Clnl Coons Pousl w sol w quon ss o losslss n sou on n nul oon uos boun onon ( ol onson) o obl qus us o sol w quon n lnl oons. ow o w o bou s?
17 44 In lnl oons Tn [ ] Bu n wl Tn ow o w sol o. In o wos w s? No ws obn b usn Usn bo n w (W quon n losslss sou on) W n s onsn In lnl oons n ψ ψ ψ ψ n T us o n n lnl oon n wll sul n l nl quons: â â â
18 45 w ψ ψ ψ ψ ψ w ψ ( ) o No nl quons o quons wl nl quon o n oul l nl s no oul T soluons o os usul n onsun T n T os (T n T w s o WRT -on) boun lu obls n wll b ons. Fo n sson o ψ ( ψ ) w ψ ψ ψ ψ ψ w () ψ ψ () L ψ ( ) ( ). ubsu () n () n w : ( ) ( ) ( ) ( ) ( ) D bo ss b n w : W ( ) s onsn (4) n w s onl unon o o o s (w unons o n ) us qul o onsn ( ) o ll lus o w us w s no onsn Tn q. (4) n b wn s
19 46 ( ) ( ) No n bo w s onl unon o o o s us qul o onsn ( ) n sl o ous s w s w L us lso n quon n lnl oons) s onsn usn onsn quon w s ( ) W ( ) ( ) n ( ) (onsn quon o w onsn. Abo s lssl Bssl Dnl quon. ( ) ( ) ( ) ( ) u T soluon o ψ ψ w ψ ( ) ( ) s n b sls soluons o ψ w n () () ( ) ( ) W onsn quon ()
20 47 oluons o nn w A ( ) B sn( ) o n b os C D Tln w oluon o nn w A B sn o n b os C D Tln w oluon o () () Tln w ( ) A ( ) B ( ) o nn w ( ) C ( ) D Y ( ) () () ( ) ( ) nl unon o s n nl unon o son n ( ) Bssl unon o s n Y ( ) Bssl unon o son n ± K () () T unons os L sn L Y l soluons. W on s us n n obl ns on obls n (ull boun onons). ll (Bssl D. q.) s n b As n l ons ll lnl wu. T soluon ns o u < s n b: ψ n [ C ( ) D Y ( )] [ A os B ( )] [ C D ] sn No ns u soluon n us b snn ws soluon n us b o n soluon n us b ln ws. Fuo sn Y s snul n D
21 48 ψ n ( )[ A os B ( )][ C D ] C sn T l ous o u ( > ) us b ln n bo n n b o n n () ψ ou ( ) B ( )[ A os B sn ][ C D ] () W ( ) s osl ln w No ollown lons o nl unons o s n son n. () () ( ) π π π 4 π π π 4
22 49 Fls os T Pln w n Uno ln ws Fl s oon o s- oons o s ul l onuon o n boun lu obl. n l onuons (os) ss wll quons (w quon). Ts usull o s os. In T o n on n s onsn n lol ln nnn o. Ts ln s ll qus Pln. In nl qus lns no lll wo n ons lon o o w Ps Fon o T w I qus lns lll (.. s onon o lns o T o s) n w s w ln w. In o wos qus sus lll ln sus. I n on o lll ln qus sus l s qulu ln sus ( lu s s o ln) w s w uno ln w. In s s l s no unon o oons u qulu n qus ln
23 5 Ps ons o ln w W non w o w o w n b w o Cons ollown ln w: wn s onsn n n D o sou on Rll ( F ) F F Tn [ ] bu Usn w n lso sow I n lso b sown (W) n ( ) n u: s R
24 5 L s ssu suons o w n bo n n n s Rlon bwn n o ln ws Fo w â s un o lon. W n sson o n b wn s w / η η / s u nns n n w n n [ Ω] η 77 s s nns n. / π l sson o n s o n b oun o b η -ls o L (l-n )
25 5 Fsnl Rlon & Tnssson Cons T s o Pnul Polon: T n s n ln Pln o nn s ln Inn ws K Rl ws K Tns ws K θ Anl o nn θ Rlon nl θ Tns nl θ θ θ θ θ As s l s o s o ( θ nul o ln o nn) o T (l l s nss o oon on) o σ olon w sn sn os θ θ w snθ n θ n osθ n osθ Tn No lso n n. ( snθ osθ ) n w n n snθ n sn θ n os Fo η ( snθ osθ ) w ( osθ snθ ) η θ
26 5 Fo Rl w w w n os sn θ θ sn n os sn θ θ T l n n [ ( sn os ) ] [ ] θ θ [ osθ θ sn ] [ ( snθ osθ ) ] [ ] η Tns n snθ n snθ osθ n osθ n snθ osθ No [ ] [ ] osθ θ sn snθ osθ η θ θ θ θ n n sn θ n sn θ n osθ.. n os [ ] [ ] θ θ θ W now l B.C. ln n qun nnl n o b onnuous (wo oo l) ( ) ( ) nnl nnl ( ) nnl ( ) nnl No nnl oonns lon n snθ snθ snθ snθ snθ snθ osθ osθ osθ η η η
27 54 T bo s s o 4 quons n 4 unnowns ( θ θ ). I n b u o quons n unnown. On s s on w θ θ snθ θ θ qul) snθ s s nll s Lw o Ron (.. nn & l nls on nll s Lw o Ron snθ snθ n snθ n snθ n snθ n snθ Ts ss nnl oonn o oon o oss n s onnuous. T us o wo nll s lw (bo) wll u ou s o 4 quons n 4 unnowns o quons n unnowns: () osθ ( ) osθ () η η () n () n b sol o η osθ η osθ η osθ η osθ η osθ η osθ η osθ θ θ θ Usn η / n ulln o n boo b w / osθ / osθ / osθ / osθ osθ osθ osθ osθ Rll osθ n osθ osθ osθ n osθ osθ osθ osθ osθ osθ
28 55 n () n sll () No () n () lon n nssson on (Fsnl l ons) o T o olon. Two In Pobl W ons T o nul olon. T Fsnl lon ons n n b wn s: A () A A D () A D C A (slb nss s ) () A C (4) D C (5) D C C w (6) n osθ A A C D A ( ) Us (5) n () n
29 56 () C A C () C A Usn () n A A C A C In sl nn (W) w n sow T () (q. - 55) w () T No u n s n n. Tn () n () n b wn s T Tn T n osθ T Fo sson o w s n T ;.. s no lon o slb. Ts s ll onon. Rll n n osθ n osθ osθ osθ η osθ η osθ W ns o T (nssson on) un onon.
30 57 No w n ( T T T ( ). Rll. n un onon ) w w n ls n. Ts ss un onon slb onl nss s on ln w. A nol nn θ θ n onon (no lon o slb) n b η osθ η osθ wll sl o η η. No un onon w / [ ] / s ll ou l. T w n w w w wll l s / Fnl Rs: ou soul su (sl su) os su s l n Bws nls.
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On Fractional Operational Calculus pertaining to the product of H- functions
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