ECE 107: Electromagnetism

Transcription

1 ECE 107: Electromagnetism Set 7: Dynamic fields Instructor: Prof. Vitaliy Lomakin Department of Electrical and Computer Engineering University of California, San Diego, CA

2 Maxwell s equations Maxwell s equations B E = t D H= + J t D = ρ B =0 nˆ ( E E ) = nˆ ( D D ) = ρ nˆ ( H H ) = J nˆ ( B B ) = D= εe B= µ H Goals of this chapter To close the loop between dynamics and statics To explain effects introduced by time variations To extend some results from static to dynamics s s E dl = d dt H dl = d dt B ds = 0 D ds = Q D= εe B= µ H B ds D ds + I 2

3 Faraday s law Faraday s law: Magnetic flux: E dl C Electromotive force voltage: V emf induces currents in a wire loop. The currents produce the flux that opposes the inducing flux (Lenz s law) allows for transformers, V emf generators, motors, etc.! Its existence is a foundation of Electromagnetics = d dt S B ds B S s Vemf Φ= d dφ d = = d dt dt B s S 3

4 Faraday s law (2) Transformers: Current à mutual flux Flux à voltage V 1 = N 1 dφ dt, V 2 = N 2 dφ dt V 1 V 2 = N 1 N 2 Energy conservation à I 1 V 1 = I 2 V 2 I 1 I 2 = N 2 N 1

5 Faraday s law (2) Electric generator: Rotating loop in a magnetic field à EMF Electric motor: Opposite to the generator An AC current is sent to the coil in a constant magnetic field à coil spins

6 Faraday s law (2) Eddy currents: Time varying magnetic field à Electric field à (Eddy) current E= B J = σe t Examples: Eddy current waist separator Eddy current break Transcranial magnetic stimulation

7 Ampere s law Ampere s law: H dl = I + d C dt S Displacement current: d Displacement current density: D ds d I = J S d ds= d dt D s S d Jd = D dt Displacement current is required to establish consistent equations of Electromagnetics. Its introduction resulted in Maxwell s equations To explain its importance, consider a capacitor under an ac voltage. 7

8 Charge-current continuity relation Start with the Ampere s law Apply divergence Continuity relation: ( H) =0 (vector identity) D H= + J t The amount of charge change equals the current needed to compensate the change For statics The current and charge are independent J v ds Kirchhoff s current law S = D t + J J = t J v = t ρ v S J v ds = t V ρ v i I = = 0 i 0 D =ρ v Gauss law I = t Q 8

9 Complex permittivity (1) Modified Ampere s law Ampere s law Constitutive relation Current J = Modified Ampere s law Complex permittivity Loss tangent H= jωd+ J J c conductivity J c =σe D= εe + J i impressed given source H = jωεe + J c + J i H = jωεe + σe + J i H = jω ε jσ ω E + J i jσ j εc = ε = ε jε = εc e δ ω tan δ = ε ε = σ ( ωε) ε c 9

10 Complex permittivity (2) Maxwell s equations in conducting media Differential equations E = jωµ H H = jωε c E + Ji D = ρ i B = 0 Boundary conditions ˆn 12 ( E 2 E 1 ) = 0 Continuity relation ˆn 12 (ε c2 E 2 ε c11 E 1 ) = ρ is ˆn 12 ( H 2 H 1 ) = J is ˆn 12 (µ 2 H 2 µ 1 H 1 ) = 0 J = jωρ i is 10

11 Electromagnetic potentials (1) Fields radiated in free space by a charge distribution ρv(,) r t and current distribution Jv(,) r t (recall that ρ v and J are related J = ρ ) 1 H= A µ E = V retarted potential A t V (R,t) = 1 4πε A(R,t) = µ 4π V retardation due to causality ρ v ( t R v p d v V R J v (t R v p ) d v R v ) v v t 8 v = c= 3 10 m s p 11

12 Electromagnetic potentials (2) Frequency domain fields radiated by a charge distribution ρ v (r) and current distribution (recall that and are related ) J v ρ v J v = jω ρ v ω 2π k = = wavenumber v λ p H = 1 µ A E = V jω A V (R) = 1 4πε A(R) = µ 4π V ρ v e jk R d v V R J v e jk R d v R 12

13 Radiation from a short dipole (1) Consider an electric dipole p Its time derivative results in current d dt p = Il ẑ jω p = Il ẑ When l = λ then I is uniform = ql zˆ Physically, the dipole is represented by two wires Vector potential 13

14 Radiation from a short dipole (2) Expressions for the radiated fields η 0 = µ 0 When ω 0 the expressions reduce to the expression due to a static dipole with p= Il ( jω) Magnetic field is present only in the dynamic case No electric field is in the ϕ direction, hence the field is TM to z The short dipole is one of the simplest antennas ε 0 = 120π characteristic impedance of free space 14

15 Radiation from a short dipole (3) Far field approximation: 2 3 The terms ~1 ( kr),1 ( kr) can be neglected Far field approximation fields R or? λ kr? 1 Characteristics of far field fields of ANY antenna Relation between E and H is simply through the impedance of free space similar to the case of transmission lines The phase behavior is like for TLs with z replaced by R 2 The field decays but only as 1 R (compare to 1 R decay 3 for a static charge and 1 R decay for an electric dipole) 15