ELE 3310 Tutorial 10. Maxwell s Equations & Plane Waves

Transcription

1 ELE 3310 Tutorial 10 Mawell s Equations & Plane Waves

2 Mawell s Equations Differential Form Integral Form Faraday s law Ampere s law Gauss s law No isolated magnetic charge E H D B B D J + ρ 0 C C E r dl r H dl S S S S J D ds B ds B D + Q 0 ds ds E electric field intensity (V/m) ρ volume density of free charge (C/m 3 ) H magnetic field intensity (A/m) J density of free current (A/m ) D electric flu density (C/m ) B magnetic flu density (T)

3 Faraday s law E B C E r dl S B ds This implies the electric field intensity in a region of time-varying magnetic flu density is non-conservative and cannot be epressed as the gradient of a scalar potential. V C r E dl dφ dt EMF (electromotive force) induced in a stationary closed circuit is equal to the negative rate of increase of the magnetic flu linking the circuit.

4 Ampere s law (modified) D r D H J + H dl J ds C + S To be consistent with the conservation of charge, etra term, D, is added. It is known as displacement current density (A/m ). This equation indicates that a time-varying electric field will give rise to a magnetic field, even in the absence of current flow. J is the density of free current, which includes both convection current ( ρu ) and conduction current ( εe ). Convection current is due to the motion of free charge distribution. Conduction current is due to the presence of E-field in a conducting medium.

5 Gauss s law and No isolated magnetic charge D ρ ρ is volume density of free charge (C/m 3 ). Total outward flu of the electric displacement (or simply, total outward electric flu) over any closed surface is equal to the total free charge enclosed in the surface. Isolated magnetic charge does not eist. There are no magnetic flow sources, and the magnetic flu lines always close upon themselves. S D ds Q B 0 B ds 0 S

6 Epression of E and H in potentials B A A E V 1 H A μ See page 37 of tetbook for reference V and A can be obtained by solving wave equations with appropriate boundary conditions. Then, E and H can be found. Relation of V and A is epressed by: A V με Recall: we set A 0 (eq. 6-0 of tetbook) previously. It is known as Lorentz condition for potentials (Lorentz gauge). This gauge enforces both V and A to propagate in speed of light.

7 Wave equations Non-homogeneous wave equations for potential scalar V and vector A V με V ρ ε A A με μj Solutions in free-space, for point source at origin (for given and J) V 1 ρ ( ) ( t R u) R, t dv A( R, t) 4πε V R μ J 4π ( t R u) dv Potential at a distance of R from the source at ttime R ut depends on the value of the charge density at an earlier time. It takes time R/u for the effect of and J to arrive the points R away from the origin. ρ V and A are called retarded scalar potential and retarded vector potential respectively. 1 wave speed in a medium : u (ms με V ( ) R -1 )

8 Wave equations 1 E E 0 1 H u H 0 u Homogeneous vector wave equations wave speed in a medium : u 1 με (ms -1 ) This set of equations are valid only for source-free region. (i.e.: ρ 0 and J 0 )

9 Time-Harmonic Fields (EM in frequency domain) ( y, z, t) Re E(, y, z) [ ] jωt e (, y, z) cos t E, E In this topic, sinusoid with only a single frequency is considered. In this case, E is epressed in terms of cosωt s.t.: ω Differentiating of a vector phasor w.r.t. t jω Integrating a vector phasor w.r.t. t 1 jω Therefore, Mawell s Equations (phasor form) become: E jωμh H D ρ B 0 J + jωεe

10 Time-Harmonic Fields (Wave equations) Lorentz condition for potentials becomes: Non-homogeneous wave equations for potential scalar V and vector A V + k V ρ ε A + k A μ J Solutions in free-space, for point source at origin (for given and J) Homogeneous vector wave equations A jkr 1 ρe V ( R) dv μ Je A 4πε R V 4π R V Epression of E and H 1 E V jωa H A μ jωμεv jkr ( R) dv E + k E 0 H + k H 0 ρ Wavenumber ω k ω με u This definition is only valid when product of ε and μ are real and positive.

11 Summery of the electromagnetic field equations E jωμh H J + jωεe Mawell s Equations D B J ρ 0 ρ ( E + B) F Q v Equation of continuity Lorentz s force equation These equations form the foundation of electromagnetic theory and can be used to eplain all electromagnetic phenomena

12 Plane Waves A uniform plane wave is a particular solution of Mawell s equations with E assuming the same direction, same magnitude, and same phase in infinite planes perpendicular to the direction of propagation (similarly for H). In practice, plane wave does not eist because practical wave sources are always finite in etent. If we are far enough away from a source, the wavefront becomes almost spherical; and very small portion of surface of a giant sphere is very nearly a plane. Solution of uniform plane wave characterized by E (uniform magnitude & constant phase) over plane surfaces perpendicular to z-ais is in this form: a [ ] j( ω t +φ ) ( ) k z z, t a Re E e E 0 0 The wave oscillate in -direction (i.e.: polarized in -direction), but propagate at z-direction. Phase ( φ) is a constant. We usually put it to zero for simplicity.

13 Plane Waves Uniform plane wave is usually epressed in phasor form : where E 0 is a constant vector, i.e.: it s constant over a plane. To be more general, uniform plane wave can be epressed by: where k E + k y + kz ω με E ( y, z) jkz ( z) E e 0, E k 0 jk y jk k i and i, where i, y or z, can be epressed in scalar product of two vectors, such that: E R is a position vector, describing an observation point in the space. R a + a y + a y z k is a wavenumber vector, direction of which (a n ) aligns with the propagating direction of the plane wave while magnitude of which (k) is wavenumber in a n direction. k ka a k + a k + a n e jk jk R (, y, z) E( R) E e z y 0 y z k z y z z

14 Plane Waves Since E jωμh, magnetic field intensity of a plane wave can be epressed in terms of E, such that: 1 H jωμ ( R) E( R) 1 H n η ( R) a E( R) jkan R ( ) ( a E ) e H R 1 η n 0 E are H are perpendicular to each other, and both are transverse to propagation direction. It is a transverse electromagnetic (TEM) wave. Eample: if E is polarized in a, and a n a z, H will be polarized in -a y. Intrinsic impedance of the medium (wave impedance) η E H ωμ k μ ε (Ω) Intrinsic impedance of free space μ0 η0 10π 377 ( Ω) ε 0

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