Chapter 7. Time-Varying Fields and Maxwell s Equations

Transcription

1 Chapter 7. Time-arying Fields and Maxwell s Equations

2 Electrostatic & Time arying Fields Electrostatic fields E, D B, H =J D H 1 E B In the electrostatic model, electric field and magnetic fields are not related each other.

3 Faraday s law A major advance in EM theory was made by M. Faraday in 1831 who discovered experimentally that a current was induced in a conducting loop when the magnetic flux linking the loop changed. d electromotive force (emf): dt

4 Faraday s law Fundamental postulate for electromagnetic induction is d B dl dl d dt E C E C s S t E The electric field intensity in a region of time-varying magnetic flux density is therefore non conservative and cannot be expressed as the negative gradient of a scalar potential. The negative sign is an assertion that the induced emf will cause a current to flow in the closed loop in such a direction as to oppose the change in the linking magnetic flux Lentz s law B t d E dt B t

5 7-.3. A moving conductor in a static magnetic field Charge separation by magnetic force F qub m To an observer moving with the conductor, there is no apparent motion and the magnetic force can be interpreted as an inducted electric field acting along the conductor and producing a voltage. 1 u 1 B dl Around a circuit, motional emf or flux cutting emf 1 C ub dl

6 A moving conductor in a static magnetic field Example 7- (a) Open voltage? (b) Electric power in R (c) Mechanical power required to move the sliding bar 1 u B ax az ay ( a) dl u B dl ub h 1 C ubh ubh ( b) I Pe I R R R R 1 ( c) P F u, F mechanical force to counteract the magnetic force F F I dlba IB h N F F I M M M W mag x M mag ubh u Bh u Bh FM ax PM R R R W m P e P M

7 A moving conductor in a static magnetic field Example 7-3. Faraday disk generator 4 u B = a az ar dl r B C 3 dr B b rdr Bb

8 Magnetic force & electric force F q EuB When a charge q moves parallel to the current on a wire, the magnetic force on q is equivalent to the electric force on q. At the rest frame on wire F vb B q At the moving frame on charge FE q E ' I L L 1 v/ c / 1 v/ c ' E I To observer moving with q under E and B fields, there is no apparent motion. But, the force on q can be interpreted as caused by an electric field, E.

9 7-.4. A moving circuit in a time-varying magnetic field To observer moving with q under E and B fields, there is no apparent motion. But, the force on q can be interpreted as caused by an electric field, E. E E ub Now, consider a conducting circuit with contour C and surface S moves with a velocity u under static E and B fields. u Changing in magnetic flux due to the circuit movement produces an emf, : d dt B On the other hand, the moving circuit experiences an emf,, due to E : E C ' dl E ' Is it true that B E'

10 B?? E' E E ub From the Faraday's law of B Edl ds, C S t by replacing E with E = E - u B, B Edl ds ub dl t C S C u time variation at rest motional emf Note that E ' E dl C B d d B d dt dt s S Therefore, we need to prove that d dt B Bds ds ubdl t S S C

11 B?? E' Time-rate of change of magnetic flux through the contour d d 1 d lim t t d t d 1 dt dt B s S t t B s S B s S1 B t Bt tbt t H.O.T., t d B t 1 d d lim d d 1H.O.T. dt B s S s S t t t B s S B s u S1 In going from C 1 to C, the circuit covers a region bounded by S 1, S, and S 3 B d Bds Bds Bds S S 1 S ds dlut Bds Bds t ub dl 3 S S 1 C 1 t d B Bds ds u B dl dt S S t C emf = E dl C Therefore, the emf induced in the moving circuit C is equivalent to the emf induced by the change in magnetic flux d E' B dt : same form as not in motion.

12 A moving circuit in a time-varying magnetic field Example 7.3 Determine the open-circuit voltage of the Faraday disk generator B S b Bb t BdsB rdrd B t d dt b Compare! B E ' Edl d dl C s S t u B C 4 u B = a a a E' dl C 3 r B z rdr B b B E ' Bb rdr

13 B E ' Edl d dl C s S t u B C B d d B d dt dt s S (Example 7-4) Find the induced emf in the rotating loop under B() t a B sint y o (a) when the loop is at rest with an angle. y osin n B d s a B t a hw Bhwsintcos d a BScostcos, S hw : the area of the loop dt (b) When the loop rotates with an angular velocity ω about the x-axis 1 w 3 w a ubdl [( ) ( sin )] [( ) ( 4 sin )] C an ayb t axdx an ayb t axdx w ( B sin t sin t ) h B S sin t sin ( = t) B S(cos tsin t) B Scos t E' a a Compare! 1 n d d 1 B BSsin tbscost dt dt t B t a t S B Ssintcos B Ssintcost B Ssin t B E'

14 7-3. How Maxwell fixed Ampere s law? We now have the following collection of two curl eqns. and two divergence eqns. B E, H= J t D, B= Charge conservation law the equation of continuity J t The set of four equations is now consistent with the equation of continuity? Taking the divergence of H= J, H J H from the vector null J does not vanish in a time-varying situation and this equation is, in general, not true. identity

15 Maxwell s equations A term ρ v / t must be added to the equation. D H J J D t t D H J t E B J t The additional term D/ t means that a time-varying electric field will give rise to a magnetic field, even in the absence of a free current flow (J=). D/ t is called displacement current (density).

16 Maxwell s equations Maxwell s equations B E t D H= J t D B= Continuity equation J t Lorentz s force F qe ( B) These four equations, together with the equation of continuity and Lorentz s force equation form the foundation of electromagnetic theory. These equations can be used to explain and predict all macroscopic electromagnetic phenomena. The four Maxwell s equations are not all independent The two divergence equations can be derived from the two curl equations by making use of the equation of continuity

17 Maxwell s equations Integral form & differential form of Maxwell s equations Differential form Integral form Significance B E t H = J D t C B d Edl ds S t dt C D Hdl I ds S t Faraday s law Ampere s law D D d s C Q Gauss s law B= B d s S No isolated magnetic charge

18 Maxwell s equations Example 7-5 erify that the displacement current = conduction current in the wire. i C : conduction current in the connecting wire dc ic C1 C 1 cost dt A C1 d C E : uniform between the plates d DE sint d sin C t D t d id d s cost AC 1 cost i A C

19 Maxwell s equations Example 7-5 Determine the magnetic field intensity at a distance r from the wire Two typical open surfaces with rim C may be chosen: (a) a planar disk surface S, b a curved surface S 1 For the surface S, D = C 1 C 1 Hdl rh JdsiC C1 cost H cos t S 1 r Since the surface S passes through the dielectric medium, no conducting current flows through D rh id d s i A t C 1 H cost r C S displacement current

20 7-5. Electromagnetic Boundary Conditions C B Edl ds when h S t E E 1t t D D Hdl J ds ; ds when h t t C S S a H H J H H J n 1 s 1t t sn s s DdS d a D D when B ds D B D 1n n s n 1 B H H 1n n 1 1n n s h

21 Electromagnetic Boundary Conditions Both static and time-varying electromagnetic fields satisfy the same boundary conditions: E E 1t t H H J B The tangential component of an E field is continuous across an interface. 1t t sn The tangential component of an H field is discontinuous across an interface where a surface current exists. B 1n n The normal component of an B field is continuous across an interface. D D 1n n s The normal component of an D field is discontinuous across an interface where a surface charge exists.

22 Boundary conditions at an interface between two lossless linear media Between two lossless media () with =, and S =, J s = E D1 t 1 E D 1t t t D B E, H H B H B 1t t 1t 1 t D D E E 1n n 1 1n n B B H H 1n n 1 1n n

23 Boundary conditions at an interface between dielectric and perfect conductor In a perfect conductor ( infinite, for example, supercondiuctors), E D, H B Medium Medium 1(dielectric) (perfect metal) E 1t t H J H D B E 1t s t D 1n s n 1n n B

24 Boundary conditions Table

25 7-4. Potential functions ector potential B A T B Electric field for the time-varying case. A E AE t t A E t A E /m t Due to charge distribution Due to time-varying current J

26 Wave equation for vector potential A From,, or t t t t t t t t t A H B A D E D H J A A J A A A A J A A J A J A (Lorenz condition, or gauge) Non-homogeneous wave equation for vector potential A 1 traveling wave with a velocity of (# Show that the Lorentz condition is consistent with the equation of continuity. Prob. P.7-1)

27 Wave equations for scalar potential A A A From, v v v v A E and E t t t t t ρ ε ρ ε ρ με ε ρ με ε for scalar potential t A The Lorentz condition uncouples the wave equations for A and for. The wave equations reduce to Poisson s equations in static cases.

28 Gauge freedom Electric & magnetic field Gauge transformation A E, B= A t If AA, B remains unchanged. A A E t t t t Thus, if is further changed to, E also remains same. t Gauge invariance E & B fields are unchanged if we take any function (x,t) on simultaneously A and via: A A t A t t The Lorentz condition can be converted to a wave equation.

29 7-6. Solution of wave equations The mathematical form of waves f (x, t =) t = t = t f x, t f xupt u p f 1 f x u t p x = u p t x : wave equation

30 Wave equation Simple wave x t y xt A A kx t T f, k, up T k, cos cos

31 Solution of wave equations from potentials t με ρ ε First consider a point charge at time t, (t)v, located at a origin. Except at the origin, (R) satisfies the following homogeneous equation ( = ): 1 Since R for spherical symmetry R,, R R R R t up v : Nonhomogeneous wave equation for scalar electric potential 1 R R R R t 1 Introducing a new varible, R, t UR, t R U U R UR, tu t or URupt, up R Thus, we can write in a form of R, t t u p 1 d' t R

32 Solution of wave equations t t R t The potential at R for a point charge is, R, t 4 R R t u p tr/ u Now consider a charge distribution over a volume. p tr/ u 4 R 1 R, t R/ up R, t d 4 R R, t R/ up ARt, d Wb/m 4 J R p R d d' R R R,t The potentials at a distance R from the source at time t depend on the values of and J at an earlier time (t- R/u) Retarded in time Time-varying charges and currents generate retarded scalar potential, retarded vector potential.

33 Source free wave equations Maxwell s equations in source-free non-conducting media (ε, μ, σ=). H E E, H=, E, H= t t Homogeneous wave equation for E & H. E E E = E H t E E H t t 1 E E up t In an entirely similar wa 1 H y, H up t

34 Review The use of Phasors Consider a RLC circuit di 1 L Ri idt t i t I t dt C, cos Phasor method (exponential representation) j jt jt cos Re Re s j jt jt Re Re s t t e e e i t I e e I e I s s e j j Ie (Scalar) phasors that contain amplitude and phase information but are independent of time t. If we use phasors in the RLC circuit, di Re jt, Re Is jt jise idt e dt j 1 R jl Is s C 1 it it () Re s / R jl e C

35 Time-harmonic Maxwell s & wave equations ector phasors. E H xyzt,,, ReE xy,, z jt xyz,,, t Re H x, y, z e e jt Time-harmonic (cos t) Maxwell s equations in terms of vector phasors D B E t B= D H= J t E E jh H H J j E

36 Time-harmonic Maxwell s & wave equations Time-harmonic wave equations (nonhomogeneous Helmholtz s equations) ( R, t) ( ) ( ) ( R) k ( R) ( R, t) t k wave number = u R j R ( Rt, ) ( R) A A( R) k A( R) J A J t p

37 Time-harmonic retarded potential Phasor form of retarded scalar potential R 1 4 R e R jkr d cos t kx e e e j tkx jt kx Phasor form of retarded vector potential. A R 4 J R e R jkr d Wb/m R When kr 1 jkr kr e 1 jkr... 1 R, static expressions(eq & Eq. 5-) AR

38 Time-harmonic retarded potential Example: Find the magnetic field intensity H and the value of β when = 9 E H z z E 9 1 (1/s) a y 9 zt, a 5cos1 t z /m 5e j z y 1 E j H z 1 j x y z a a a x y z j z 5e 1 a 5e 5 e a H ( z) a j 1 1 Ez H ay H xay 5 e j j z rad/m c j z j1z Hzax 5e ax.398e H a a jz jz x x x x z j z 9 zt, x 5e x.398cos1 t1z j z

39 The EM Waves in lossy media If a medium is conducting (σ ), a current J=σE will flow D H= J t H je j E j ce, j c complex permittivity j j When an external time-varying electric field is applied to material bodies, small displacements of bound charges result, giving rise to a volume density of polarization. This polarization vector will vary with the same frequency as that of the applied field. As the frequency increases, the inertia of the charged particles tends to prevent the particle displacements from keeping in phase with the field changes, leading to a frictional damping mechanism that causes power loss. This phenomenon of out of phase polarization can be characterized by a complex electric susceptibility and hence a complex permittivity. Loss tangent, δ c

40 The EM Waves in lossy media Loss tangent, δ c H j ce j E j j c tan c, c : loss angle E H= J j E t j c J Good conductor if >>1 E Good insulator if <<1 Moist ground : loss tangent ~ 1.81

41 The electromagnetic spectrum Spectrum of electromagnetic waves