Electrical Boundary Conditions. Electric Field Boundary Conditions: Magnetic Field Boundary Conditions: K=J s

Transcription

1 Electrical Boundar Condition Electric Field Boundar Condition: a n i a unit vector noral to the interface fro region to region 3 4 Magnetic Field Boundar Condition: K=J K=J 5 6

2 Dielectric- dielectric boundar condition Dielectric aterial are doinated b bound rather than free charge (E-field caue +ve and ve charge of olecule to eparate and for dipole throughout the aterial interior Conductor-dielectric boundar condition Therefore, the free charge denit and the urface current denit J are zero E n = E n B t B t The noral coponent of B i continuou acro the interface while the tangential coponent of E i continuou acro the interface 7 n D n 8 Conductor-free pace boundar condition D D E t t n E n 9

3 Electrodnaic (Not Yet) Electrotatic charge electrotatic field Stead current (otion of electric charge with unifor velocit agnetotatic field Tie varing current electroagnetic field 3 4 Changing Magnetic Field Current and Voltage In uar: Farada Law- Integral For B, H N S Current 5 6 Farada Law-Differential For 7 8

4 Tie Haronic field and their phaor repreentation In general, a phaor could be a calar or vector. If a vector E (,, z, t) i a tie-haronic field, the phaor for of E i E (,, z); the two quantitie are related a E = Re (E e jt ) If E = E o co(t -)a, we can write E a: E = Re (E o e -j a e jt ) E = E o e -j a phaor for 9 Notice that E jt jt Re( E e ) Re( jee ) t t E je t E Et j Mawell equation in ter of vector field phaor (E, H) and ource phaor (, J) in a iple linear, iotropic and hoogeneou ediu are: E H E H jh J v je E ds H ds S S L E dl j H ds H dl J ds je ds L v v dv Fro the table, note that the tie factor e jt diappear becaue it i aociated with ever ter and therefore factor out, reulting in tie independent equation Electroagnetic wave equation in free pace (coupling between E and H) Plane Wave Equation 3 4

5 Wave in General A wave i a function in both pace and tie. The variation of E with both tie and pace variable z, we a plot E a a function of t b keeping z contant and vice vera. 5 6 The poible olution in free pace i of the for: E ( z, t) Re( E e Re[ E e ) E e E co( t z) E co( t z) jt j( toz) o ] j( toz) o A negative ign in (t o z) i aociated with a wave propagating in the +z direction (forward traveling or poitive going wave) wherea a poitive ign indicate that a wave i traveling in the z direction (backward traveling or negative going wave) 7 8 A plane wave traveling in the poitive z direction What do Farada and Apere ean? B E. dl. d a changing agnetic field caue an electric field t H D. dl JC. d t a changing electric field/flu caue an agnetic field E or or ( z, t) Re[ E Re[ E e E ( z) e jt ] j( t z) ] co( t z) o o 9 Quetion : If we put thee together, can we get electric and agnetic field that, once created, utain one another? 3

6 Cro-breed Apere and Farada! D E H JC E... all in ter of E and H t t B H E... all in ter of E and H t t d d E H E... differentiate both ide dt dt t H E... curl of both ide t dh de d E dt dt dt dh E dt de d E E dt dt 3 Cro-breed Apere and Farada! D E H JC E... all in ter of E and H t t B H E... all in ter of E and H t t E E H E E... curl of both ide t t H E t H H H t t 3 Now oe iplification Travelling Wave E Y = E Y in(ωt-β) Take a tie-varing electric field, E, at a point E Y = E Y in(ωt) Add a econd one with a all phae difference, nearb E Y = E Y in(ωt) E Y = E Y in(ωt-) z E = (,E Y,) onl Align -ai with electric field and the -ai with the direction of (wave) propagation (a travelling wave propagating in the - direction, with onl a -coponent of E-field) 33 Now let have a lot of the, with a inuoidal variation of phae with direction. E Y = E Y in(ωt-β) 34 Plane Wave We will alo look for a plane wave olution where the field E Y i the ae (at an intant in tie) acro the entire z plane. Here i an aniation to ee what thi ean - looking at the z plane, down the direction of travel E Y = E Y in(ωt-β) Cro-breed Apere and Farada! i j k d d d de de E,, d d dz dz d E i j k d d d d E d E d E d E E,, d d dz dd dz d dzd de de dz d z E = (,E Y,) onl Look down here 35 And, a we have iplified down to E=(,E,), with E Y contant in the z plane, thi reduce to de E d 36

7 Cro-breed Apere and Farada! So de d E E (in 3D) dt dt Becoe the D equation d E de d E d dt dt Plane wave equation for E decribe the variation in tie and pace of an electric plane wave With a -coponent onl (we have aligned the -ai with E) propagating in the -direction. There i an eactl equivalent equation for H Eliinate E, not H, fro the cobination of Apere and Farada. rather a wate of our tie. We can, however, infer that whatever behaviour we get for E will appl to H, although we do not et know the direction of H. 37 What have we here? Variation of E in pace (=direction of propagation) d E de d E d Magnetic pereabilit (4p 7 in vacuu, larger in a agnet) dt Conductivit ( in an inulator, uch larger in a conductor) Variation of E with tie dt Dielectric contant ( in a vacuu, larger in a dielectric) 38 Start with an inulator to ake life ea (=) d E de d E becoe d dt dt d E d E j ( t ) Look for a olution of the for E Ee Where and depend upon and the characteritic of the inulator d E d E E, E d dt d, what doe thi ean?? p p Reeber, p frequenc = p f, = = and v f wavelength dt 39 Still don t know what it ean Travelling wave of the for j ( t ) E E e E co t p It travel with a velocit v f p In a vacuu, = =4p -7, = = v 3 /... a failiar peed? In (eg) gla, = =4p -7, = r = r 8 v.43 /... light low down in gla 4 Thi i wh lene work V=3 8 / V=.43 8 / V=3 8 / 4 ( ) E (, Ee j t,) What i H up to? H de,,, Farada a E E,, je t d j ( t ) e j t H j t H (,, Hz) Hz Hze,,, jhze So and if t j t j ( t ) Hze Ee So Hand E are at 9 to one another... (,, Hz) and (, E,) Hand E are in tie-phae in a non-conductor Alo, Hz E E E, the intrinic ipedance ( Z i )of the ediu, i real for an inulator 4

8 Suar o far : Inulator H and E both obe e j(t-) H and E are in tie-phae E =Z i H i the characteritic ipedance Z i i real in an inulator Z i = 377Ω in free pace (air!) Z i 5Ω in gla Wave travel at a velocit v =/ 3 8 / in free pace Now a conductor Field lead to current Current caue Joule heating (I R) Lead to lo of energ Field till ocillate, but the deca Multipl the olution we have alread b a ter e -a? e -a HEAT! e -a in(ωt-β) HEAT! 43 HEAT! 44 Now a conductor In general: the electric field in a conductor a be epreed in the for: E (, t) Re( E ( ) e E e a Where E and E jt ) co( t ) E e a co( t ) were replaced in ter of their ag. and phae Now a conductor > d E de d E d dt dt Look for a olution of the for j ( t ) a E Ee e j t j E Ee e a For tidine, write a j. i called the propagation contant a j E E j E E j j, j j j t E E e e Eaple : Good Conductor j t, E E e e j j f a v 6 7 (S/) MHz / j 6.6 j 79 6 j.54 ( j) Coent : a=, o E and H are 45 out of (tie) phae v<< peed of light a =.54 5 >> rapid attenuation via e -a Let have a look at e -a Eaple : Good Conductor e -a μ μ μ 3μ.36=/e Aplitude fall b.36=/e in 6 i.e. the wave doen t get far in copper! Skin Depth : the depth of penetration into a good conductor (the wave will be attenuated b a factor e.368 a pf 48

9 Eaple : Good Conductor, E=Z i H. Intrinic Ipedance H de jt Farada a E, E,,,, E e t d j (,, ) t H j,,, t So and if H Hz Hz Hz e j Hze t j E H Z H z i z j j j Z i j j j Eaple : Good Conductor, E=Z i H. Intrinic Ipedance p j j j 4 i z z z z E Z H H H e H j E So Hz relate the agnitude of H and E p j e 4 p and E lead Hz b Siilaritie and difference between the propagation of unifor plane wave in free pace and conductive ediu Siilaritie: In both cae, the electric and agnetic field are unifor in the plane perpendicular to the direction of propagation. The electric and agnetic field are perpendicular to each other, and to the direction of propagation i.e.no coponent of either the electric or the agnetic field i in the direction of propagation. Difference: Free Space Conductive Mediu E, H vector are in phae, the E, H vector are not in phae, the intrinic wave ipedance o i a real intrinic wave ipedance i a conuber. ple nuber. The phae velocit = c (peed of The phae velocit i le than the light. peed of light. For a plane wave of a given freq., o The =p/ i horter than o i longer than in the aterial ediu. Doe not attenuate in agnitude a it It eponentiall attenuate, with propagate. the kin depth b = /a 5 5 Polarization of plane wave For a wave propagating along the z ai, the electric field a be epreed a having two coponent in the and direction: E = (A a + B a ) e -jz where the aplitude A and B a be cople. A ja Ae, B B e. If A and B have the ae phae angle (a = b). In thi cae, the and coponent of the electric field will be in phae E ( A a E ( A a B a ) e j( za) jb B a )co( t z a). If A and B have different phae angle. In thi cae, E will no longer reain in one plane: E A co( t a z) E B co( t b z) The locu of the end point of the electric field vector will trace out an ellipe once each ccle Elliptical polarization 3. If A and B are equal in agnitude and differ in phae angle b p/, the ellipe becoe a circle Circular Polarization The tip of the E vector follow a line Linear polarization 53 54

10 - If one take a naphot of a circularl polarized wave at an intant then he will ee the picture below. - The E-field vector doe not change in agnitude but it direction twit in pace. - An oberver itting in the path of the wave will ee the E- field vector rotate in a circular trajector at hi location a the wave pae b