Classical Electrodynamics

Transcription

1 A First Look at Quatum Physics Classical Electroyamics Chapter Itrouctio a Survey Classical Electroyamics Prof. Y. F. Che

2 Cotets A First Look at Quatum Physics. Coulomb s law a electric fiel. Electric fiel a electric potetial. Discotiuity of electric fiel a potetial. Poisso a Laplace equatio.5 ree s theorem a ree fuctio.6 Electrostatic potetial eergy a eergy esity Classical Electroyamics Prof. Y. F. Che

3 The force betwee two poit charges is give by: qq F The electric fiel of a poit charge ca be efie via force: q q F E Classical Electroyamics Prof. Y. F. Che. Coulomb s law a electric fiel

4 Note : If there are may poit charges: i i q E Note : If the source is a istributio: E Compare the two equatios above: i i i q Note : The locality of the charge esity coul ot be precisely etermie. Classical Electroyamics Prof. Y. F. Che. Coulomb s law a electric fiel

5 . Electric fiel a electric potetial For Classical Electroyamics Prof. Y. F. Che

6 For ˆ ˆ ˆ R a a R a R R Cosequetly: R Ra ˆ Classical Electroyamics Prof. Y. F. Che. Electric fiel a electric potetial

7 E Note : The locality of the charge esity coul be precisely etermie. Classical Electroyamics Prof. Y. F. Che. Electric fiel a electric potetial

8 auss s law: Differetial form: E Itegral form: ˆ Q a E E 5 i E 5 E Classical Electroyamics Prof. Y. F. Che. Electric fiel a electric potetial

9 5 ii The electric fiel ca be epresse as the egative graiet of the electric potetial: E Note the curl of the graiet of ay well-behave scalar fuctio of positio vaishes. As a result we ca obtai: E Classical Electroyamics Prof. Y. F. Che. Electric fiel a electric potetial

10 . Electric fiel a electric potetial 5 iii With Stokes s theorem: E a ˆ E l l E 5 iv E epes o the cetral ature of the force betwee charges a o the fact that the force is a fuctio of relative istaces oly but oes ot epe o the iverse square ature. Classical Electroyamics Prof. Y. F. Che

11 . Discotiuity of electric fiel a potetial The tagetial compoet of the electric fiel is cotiuous: E a ˆ E l E t Et l E t Et The iscotiuity of the ormal compoet of the electric fiel meas the eistece of the charges at the bouary: Q E E a ˆ i Q E ˆ E ˆ A E E Q A i Sie Sie E E Classical Electroyamics Prof. Y. F. Che

12 . Discotiuity of electric fiel a potetial The iscotiuity of the electric potetial is ue to the eistece of the ipole layer a it ca be aalogous to the situatio of the capacitace. For the situatio of the sigle layer the electric potetial is cotiuous but the electric fiel is ot as show i the left figure. For the situatio of the ipole layer if the istace is limite to zero the the electric fiel ca be view to be cotiuous but the electric potetial is ot as show i the right figure. Electric fiel V/m 8 6 electric fiel electric potetial sigle layer Electric potetial V Electric fiel V/m ipole layer Positio m Positio m Classical Electroyamics Prof. Y. F. Che 6 - electric fiel electric potetial 8 6 Electric potetial V

13 . Poisso a Laplace equatio E This epressio is coveiet to be use i the situatio of free space or the charge istributio beig poit charge. E E E If =: : Laplace equatio : Poisso equatio Whe we eal with the problems ivolvig the bouary coitio or fiite regio usig Poisso equatio of Laplace equatio together with special mathematical techiques for eample ree fuctio is a coveiet way to solve the problem. Classical Electroyamics Prof. Y. F. Che

14 . 5 ree s theorem a ree fuctio ree s first ietity: let v f v f v f fv Itegrate the above equatio a use the ivergece theorem: a a ˆ a Classical Electroyamics Prof. Y. F. Che

15 . 5 ree s theorem a ree fuctio ree s seco ietity also kow as ree s theorem:... Iterchage a :... - a itegrate: a Classical Electroyamics Prof. Y. F. Che

16 To solve the Poisso equatio with the bouary coitio firstly we ca solve the impulse respose with the same bouary coitio: - With the replacemet of: i - ii iii a a iv a a Classical Electroyamics Prof. Y. F. Che. 5 ree s theorem a ree fuctio

17 a a Avace iscussios: i Free space meas o bouary coitio: - Classical Electroyamics Prof. Y. F. Che. 5 ree s theorem a ree fuctio

18 ii For Neuma bouary coitio: the simplest case is S N S N N a where S a S Note that the electric potetial for a poit charge is: Q If the total charge is epresse as the surface charge: a Q a Compare with the term of the re bo with the ree fuctio i the free space: - Classical Electroyamics Prof. Y. F. Che. 5 ree s theorem a ree fuctio

19 . 5 ree s theorem a ree fuctio As a cosequece the iterpretatio of the surface itegral of the re bo is the potetial ue to the surface charge esity give above. The iscotiuities i the electric fiel across the surface the lea to zero fiel outsie the volume V: E E E E Surface ˆ E E Classical Electroyamics Prof. Y. F. Che

20 iii For Dirichlet bouary coitio: D a D D Note that the electric potetial for a ipole is: Q Q With Taylor epasio:... Q Q If the total charge is epresse as the surface charge: a Q A efie ipole momet: a D a P ˆ Classical Electroyamics Prof. Y. F. Che. 5 ree s theorem a ree fuctio

21 . 5 ree s theorem a ree fuctio Dˆ a Compare with the term of the blue bo with the ree fuctio i the free space: - D D As a cosequece the iterpretatio of the surface itegral of the blue bo is the potetial ue to the ipole layer D give above. Classical Electroyamics Prof. Y. F. Che

22 . 5 ree s theorem a ree fuctio The iscotiuities i the electric potetial across the surface the lea to zero potetial outsie the volume V: D D D Surface D ˆ Classical Electroyamics Prof. Y. F. Che

23 . 6 Electrostatic potetial eergy a eergy esity For a poit charge the work oe o the charge is give by: W q If the potetial is prouce by other charges the the potetial is give by: N q j i j i j N So that the potetial eergy of the charge q i is: q q i j Wi The total potetial eergy of all the charges ue to all the forces actig betwee them is: N qiq j qiq j W j ji 8 i j i j i j It is uerstoo that i = j terms ifiite self-eergy terms are omitte i the ouble sum. j i i i j i Classical Electroyamics Prof. Y. F. Che

24 . 6 Electrostatic potetial eergy a eergy esity For a cotiuous charge istributio: W 8 5 E & W E E E E The potetial eergy epresse i 5 is efiitely oegative. This seems to cotraict our impressio from that the potetial eergy of two charges of opposite sig is egative. The reaso for this apparet cotraictios is that 5 cotais self-eergy cotributios to the eergy esity whereas the ouble sum i is ot. Classical Electroyamics Prof. Y. F. Che