m = Mass flow rate The Lonely Electron Example 0a:

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1 The Lel Elect Exaple 0a: Mass flw ate l Liea velcit Hw fa ut f ptial eeg iteacti? Hge ucleus Bh : Uest the etu ccept. Liea etu istace eeg ( l ) l F ( tie ) ( tie ) + Like t use the peples ieas (if he ca it---u ca it!) Kew abut Culb s wk Kew abut Plack s eeg quatizati iea. E h f Agula etu Plack s cstat

2 The Lel Elect Exaple 0a: Mass flw ate l Liea velcit Hw fa ut f ptial eeg iteacti? Hge ucleus Bh : Uest the etu ccept. Liea etu istace eeg ( l ) l F ( tie ) ( tie ) - l Eeg Balace These te ae the squae f the elect s liea velcit. + Like t use the peples ieas (if he ca it---u ca it!) Kew abut Plack s eeg quatizati iea. E h f ( l ) ( ) ( ) ( l ) ( )( ) l P.E. Culb s law f fce betwee chages l ( Ze )(e) [ ] Distace betwee elect a ucleus ( ) () Kew abut Culb s wk l Agula etu ( l ) Eeg is eithe kietic ptetial Plack s cstat h l Bh s ce iea Liea velcit f tatig elect The balace stateet P.E. elect is has e electstatic uit f chage x 0 elects is Culb f chage

3 The Lel Elect Exaple 0a: Mass flw ate l Liea velcit Eeg Balace P.E. + ( ) Hw fa ut f ptial eeg iteacti? Hge ucleus Bh : ( l ) - ( Ze )(e) [ ] () P.E. ( ) balace stateet ( l ) - P.E. h l elect is has e electstatic uit f chage x 0 elects is Culb f chage als balace stateet P.E. h ( ) ( ) ( ) - This te is the squae f the elect s liea velcit,. l P.E. P.E. (-) - Ze (e) ( ) Ze ( ) -

4 The Lel Elect Exaple 0a: Mass flw ate l Liea velcit + Hw fa ut f ptial eeg iteacti? Hge ucleus Bh : h l elect is has e electstatic uit f chage x 0 elects is Culb f chage ( ) ( l ) - h ( ) ( ) ( ) - P.E. This te is the squae f the elect s liea velcit,. l P.E. P.E. (-) - Ze (e) ( ) Ze ( ) -

5 The Lel Elect Exaple 0a: Mass flw ate l Liea velcit + Hw fa ut f ptial eeg iteacti? Hge ucleus Bh : h l elect is has e electstatic uit f chage x 0 elects is Culb f chage P.E. ( ) ( ) - P.E. (-) P.E. l ( h ) ( ) ( ) - - Ze (e) ( ) h - ( ) ( ) Ze ( ) - h -3 ( ) - h (-)( ) ( ) -3 ( ) -3 Whe

6 The Lel Elect Exaple 0a: Mass flw ate l Liea velcit + Hw fa ut f ptial eeg iteacti? Hge ucleus Bh : h l elect is has e electstatic uit f chage x 0 elects is Culb f chage P.E. P.E. (-) P.E. ( l ) - - Ze (e) ( ) h - ( ) ( ) Ze ( ) - h -3 ( ) - h ( ) ( ) ( ) - h (-)( ) ( ) -3 ( ) -3 Whe Whe (-) Ze ( ) - - h ( ) - h ( Ze ) ( ) istace t the fist bital 0.58 A -3

7 The Lel Elect Exaple 0a: Mass flw ate l Liea velcit P.E. P.E. E () T (-) + Hw fa ut f ptial eeg iteacti? Hge ucleus Bh : 0.58 A - Ze (e) ( ) Ze ( - ) P.E. E () T - - ( ) h l elect is has e electstatic uit f chage x 0 elects is Culb f chage h ( ) ( ) h ( ) ( l ) Whe - Ze (e) ( )

8 The Lel Elect Exaple 0a: Mass flw ate l Liea velcit P.E. P.E. E () T E () T 0.58 A (-) + Hw fa ut f ptial eeg iteacti? Hge ucleus Bh : 0.58 A - Ze (e) ( ) Ze ( - ) + - P.E. ax i E () T E () T E () T - ( ) - h l elect is has e electstatic uit f chage x 0 elects is Culb f chage h ( ) ( ) h ( ) h ( ) ( ) ( - ( l ) Whe - Ze (e) ( ) P.E. -3 (- Ze - ) ) Whe E () T ( - h ( ) (-3) ( ) -4 ) ( - Ze (-) -3 )

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10 The Eistei Cstal Exaple 0b: h x 0 kg /sec h x 0 kg /sec C V -34 Uest classical echaics Eistei wke ut that the ttal eeg f the cstal is: What is its heat capacit? Eistei : Uest theaics - (e kt - ) T [ E ttal ] T [ 3N 0 Like t use the peples ieas (if he ca it---u ca it!) Kew Blta istibuti Kew Plack s eeg quatizati iea. Nube f scillati es (x,,z iectis) ] E ttal Ttal ube scillatig ats i cstal 3N 0 -(i) e i 0 kt E h f Plack s cstat - (e kt - ) Quatu uit f eeg

11 The Eistei Cstal Exaple 0b: Eistei wke ut that the ttal eeg f the cstal is: C V - (e kt - ) C V [ ] C V [ T [ 3N 0 3N 0-34 h x 0 kg /sec h x 0 kg /sec Uest classical echaics E ttal ] ] [ T What is its heat capacit? Eistei : Uest theaics 3N 0 kt - ] [ (e - ) T [ [ 3N 0 (-) kt (e kt ( ) ][ Like t use the peples ieas (if he ca it---u ca it!) Kew Blta istibuti Kew Plack s eeg quatizati iea. Nube f scillati es (x,,z iectis) Ttal ube scillatig ats i cstal E ttal - (-)(e kt - ) ( ( ) ) T (e - ) - ( ] kt - 0 ) ] 3N 0 T -(i) e i 0 kt E h f Plack s cstat - (e kt - ) Quatu uit f eeg (e kt - ) )] C V kt kt [ (-) (e - ) - (e ) (-) 3N 0 ] [ ( ( ( ) kt (T) - ) )]

12 The Eistei Cstal Exaple 0b: Eistei wke ut that the ttal eeg f the cstal is: C V What is its heat capacit? Eistei : - (e kt - ) [ T [ -34 h x 0 kg /sec h x 0 kg /sec Uest classical echaics E ttal ] T [ - kt (e kt 3N 0 (e - ) ( ) ( ( ) kt ) )] Uest theaics 3N 0 C ] (-) V [ (-) - kt (e kt 3N ) ( kt ) 0 k (e - ) ( ) )] C V ][ [ ( Like t use the peples ieas (if he ca it---u ca it!) Kew Blta istibuti Kew Plack s eeg quatizati iea. Nube f scillati es (x,,z iectis) ] E ttal Ttal ube scillatig ats i cstal (T) - (T) - 3N 0 -(i) e i 0 kt E h f Plack s cstat - (e kt - ) Quatu uit f eeg Whe: u(t) kt C V [ [ ( 3N 0 k (e - ) (e ) ] u(t) - u(t) ( ( ) u(t) (T) - ) )]

13 The Eistei Cstal Exaple 0b: What is its heat capacit? Kew Plack s eeg quatizati iea. E h f C V - [ ] [ kt ] (e - ) T E ttal T Eistei : 3N 0 Kew Blta istibuti -(i) e i 0 kt - kt (e kt 3N 0 (e - ) ) ( ( ) kt ) )] [ ( C ] (-) V [ (-) (T) - Whe: u(t) kt [ ( 3N 0 k (e - ) (e ) ( u(t) ) ) )] C V ][ u(t) - u(t) ( (T) - B the 3 week i Nvebe, u shul be pepae t use MatLab t: ) buil a ata atix that ca ste the fllwig vect eleets as ws. (T, E ttal, E, ) ttal [ ] T [ ] E T ttal ) Plt a specifie ata eleets as ee pais f a specifie tepeatue age.

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15 Fist Oe liea iffeetial equati F(x) Usuall what u wat. Exaple : + a 0 -a f(x,) Usuall what u have. The value f the fucti is pptial t the value f the eivative f the sae x iput. F(x)? Seties u ca guess. F each value f a thee is a fail f cuves. Oe cuve f each value f c. F(x) e -ax [c ] the guess

16 Fist Oe liea iffeetial equati F(x) Usuall what u wat. Exaple : + a 0 -a f(x,) Usuall what u have. The value f the fucti is pptial t the value f the eivative f the sae x iput. F(x)? Seties u ca guess. F each value f a thee is a fail f cuves. Oe cuve f each value f c. F(x) e -ax [c ] -ax c (-a) the guess e a [] (a)[c e -ax ] -a -af(x) x -ac e -ax 6 plts i the a fail f cuves -a x [c e ] c 6 c c 3 c R

17 Exaple : 4 -a Five ebes F(x) f the itegal cuve fail whe a 3 c c c c - c - Each cuve is the geetic epesetati f the cespig sluti f the iffeetial equati. x e -a x [c ]

18 Exaple : 4 3 -a Five ebes F(x) c c f the itegal cuve fail whe a c 4 A specific sluti equies the eclaati f a e pai (iitial value) that is the cuve f iteest. A Iitial value pble + a 0 ( x, ) (0, ) c - c - Each cuve is the geetic epesetati f the cespig sluti f the iffeetial equati. x e -a x [c ] A Iitial value pble sluti [c 0 ] 0 c e -a x [c e -a 0 ] [c e 0 ] R

19 Fist Oe liea iffeetial equati F(x) Usuall what u wat. Exaple : -a f(x,) Usuall what u have. Seties u ca guess but + a 0 F(x)? e -a x [c ] F(x) the guess -ax c e (-a) a [] -ac (a)[c e -ax ] e -ax x -a -af(x) R

20 Fist Oe liea iffeetial equati F(x) Usuall what u wat. Exaple : + a 0 s x (x) (x) g(s) s + c s 0 F this exaple: p(x) a g(s) 0 s x e ax e as 0 s + c e ax 0 + c s 0 -a f(x,) Usuall what u have. F(x)? Seties u ca guess but (x) g(x) a slve ule still has t wk. ax e 0 (0) (x) e (x) e ax + c e -ax t x a t t 0 the sluti e -a x [c ] F(x) the guess -ax c e (-a) a [] -ac (a)[c e -ax ] e -ax x -a -af(x) R

21 Fist Oe liea iffeetial equati slve F(x) Usuall what u wat. Exaple : + a 0 -a f(x,) Usuall what u have. F(x)? s x (x) (x) g(s) s + c s 0 F this exaple: p(x) a g(s) 0 (x) g(x) Seties u ca guess but a slve ule still has t wk. ax e (0) (x) e (x) e ax t x a t t 0 s x e ax e as 0 s + c e ax 0 + c s c e -ax c F(x) -ax e [c ] (-a) a [] x e -ax -ac (a)[c e -ax ] e -ax -a -af(x) Nee e ee pai t fi value f C. I egieeig els C has uits. I this el F(x) has sae uits as C. has uits f C / the uits f x x is iectl pptial t egative F(x) R

22 Fist Oe liea iffeetial equati slve F(x) Usuall what u wat. Exaple : - x X e -x x + X s x (x) (x) g(s) s + c s 0 s x s 0 e -s s s + c f(x,) Usuall what u have. F(x)? e -x Usuall what u ee. F this exaple: p(x) -x g(s) x (x) g(x) - + c e -x e -x x (x) e (x) e t x t 0 (x) e -x p(t) t t x -x t t c e -x F(x) + - c e -x Nte: Iitial ee pai (iitial values )epe the applicati R

23 Fist Oe liea iffeetial equati slve F(x) Usuall what u wat. Exaple : - x X e -x x + X s x (x) (x) g(s) s + c s 0 s x s 0 e -s s s + c f(x,) Usuall what u have. F(x)? e -x Usuall what u ee. - + c e c c (x) e (x) e 3 t x t 0 (x) e -x p(t) t t x -x t t 0 F(x) - c e -x Iitial values: x 0, ( 0, ) + Nte: Iitial ee pai (iitial values )epe the applicati F this exaple: p(x) -x g(s) x (x) g(x) F(0) e -x e -x x + - c e -x Nee e ee pai t fi value f C. c + R

24 Fist Oe liea iffeetial equati slve Exaple 3: - x x + F(x)? s x (x) (x) g(s) s + c s 0 F this exaple: p(x) x g(s) (x) g(x) -x () e t x (x) e (x) e -x t 0 - t e -x t x t 0 e -t () t + c S, this itegal is ve iptat but it has t aaltical aswe R

25 Fist Oe liea iffeetial equati slve Exaple 3: - x x + F(x)? s x (x) (x) g(s) s + c s 0 F this exaple: p(x) x g(s) (x) g(x) -x () e t x (x) e (x) e -x t 0 - t e -x Nte: t x t 0 e -t () t Ggle ERF + c The ef fucti shws up a lt i taspt pbles. The etue values f the e fucti ae ste i lk up tables. The MATLAB fucti is calle ef() use MATLAB t plt ef(x) values f x 0 t x 00. S, this itegal is ve iptat but it has t aaltical aswe Nee t use a ueical eth This itegal is s ppula it has bee put it a lk-up table as ajuste values kw as the E fucti. t x t 0 e -t t [ ] The e fucti [F(e)] R

26 Eule s iffeetial equati slve ueical eth Exaple 4: F(x ) + (x - x ) F(x ) + (h ) [-x + 4] + F(x)? F(x ) + [-x + 4 ] h x Eule s eth is als kw as Taget lie eth a/ the tw te Tal Seies eth. Eule s eth whe the chage i the iepeet vaiable is cstat. (as peicte b the Eule eth) h (x - x ) 0. + ata sua ( 0.0,.0 )

27 Eule s iffeetial equati slve ueical eth Exaple 4: F(x ) + (x - x ) F(x ) + (h ) [-x + 4] + x 0. F(x)? F(x ) + [-x + 4 ] h x Eule s eth is als kw as Taget lie eth a/ the tw te Tal Seies eth. Eule s eth whe the chage i the iepeet vaiable is cstat. (as peicte b the Eule eth) F(0).0 + [-0 + 4(.0) ] (0.).5 h (x - x ) 0. + ata sua ( 0.0,.0 ) ( 0.,.5 )

28 Eule s iffeetial equati slve ueical eth Exaple 4: F(x ) + (x - x ) F(x ) + (h ) [-x + 4] + x 0. F(x)? F(x ) + [-x + 4 ] h x F(0.).5 + [-0.+ 4(.5) ](0.).9 Eule s eth is als kw as Taget lie eth a/ the tw te Tal Seies eth. Eule s eth whe the chage i the iepeet vaiable is cstat. (as peicte b the Eule eth) F(0).0 + [-0 + 4(.0) ] (0.).5 x 0..9 h (x - x ) 0. + ata sua ( 0.0,.0 ) ( 0.,.5 ) ( 0.,.9 ) R

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