Electromagnetic Field and Wave - Pham Tan Thi

Nội dung text: Electromagnetic Field and Wave - Pham Tan Thi

1. Electromagnetic Field and Wave Pham Tan Thi, Ph.D. Department of Biomedical Engineering Faculty of Applied Science Ho Chi Minh University of Technology
2. Maxwell’s Equation Maxwell discovered that the basic principles of electromagnetism can be expressed in terms of the four equations that now we call Maxwell’s equations: (1) Gauss’s law for electric fields; (2) Gauss’s law for magnetic fields, showing no existence of magnetic monopole. (3) Faraday’s law; (4) Ampere’s law, including displacement current;
3. Maxwell’s Equations Integral form: Differential form: Gauss’ Law Q ⇢ E~ dS~ = inside E~ = · " r · "0 I 0 Gauss’ Law for Magnetism B~ dS~ =0 B~ =0 · I r · Faraday’s Law d ~ E~ d~l = B ~ @B · dt E =– I r⇥ @t Ampere’s Law d ~ B~ d~l = µ I + µ " E ~ ~ @E · 0 enclosed 0 0 dt B = µ0J + µ0"0 I r⇥ @t Macroscopic Scale Microscopic Scale
4. Gauss’s Law for Electric Field The flux of the electric field (the area integral of the electric field) over any closed surface (S) is equal to the net charge inside the surface (S) divided by the permittivity ε0. Q E~ dS~ = inside · " I 0 Q E~ dxdynˆ = inside dS~ =ˆndS =ˆndxdy · "0 Q E~ dxdycos✓ = inside · "0 Q dx → inside dS Edxdy = dy → "0 Q ES = E(4⇡r2)= inside "0 Qinside Coulomb’s Law E = 2 (4⇡r )"0
5. Gauss’s Law of Magnetism Gauss’s law of magnetism states that the net magnetic flux through any closed surface is zero B~ dS~ =0 · I The number of magnetic field lines that exit equal to the number for magnetic field lines that enter the closed surface → E Q E~ dS~ = inside · " I 0
6. Faraday’s Law The electric field around a closed loop is equal to the negative of the rate of change of the magnetic flux through the area by the loop. dB E~E~ dd~~ll = Edlcos✓ E~ d~l = · · dt = Edl I (θ = 0) d E(2⇡R)= B dt d e.m.f = B dt d W = Fd = Eqd e.m.f = B dt W = Ed d V = Ed =e.m.f
7. Ampère’s Law with Maxwell’s Correction The line integral of magnetic field over a closed path is equal to the total current going through any surface bounded by the closed path d B~ d~l = µ I + µ " E · 0 enclosed 0 0 dt I 1. Time-changing electric fields induces magnetic fields 2. Displacement current Conduction currents cause Magnetic field ( motion of charged particles) Time changing electric fields also cause Magnetic field => Time changing electric fields is equivalent to a current. We call it dispalcement current.
8. Ampère’s Law with Maxwell’s Correction The line integral of magnetic field over a closed path is equal to the total current going through any surface bounded by the closed path d B~ d~l = µ I + µ " E · 0 enclosed 0 0 dt I B~ d~l = Bdlcos✓ = Bdl (when θ = 0) · d B~ d~l = µ BI dl = B+(2⇡R)E · 0 enclosed dt I Z B(2⇡R)=µ0I µ I B = 0 2⇡R
9. Ampère’s Law with Maxwell’s Correction The line integral of magnetic field over a closed path is equal to the total current going through any surface bounded by the closed path d B~ d~l = µ " E + µ I · 0 0 dt 0 enclosed I For S1: B~ d~l = µ I · 0 enclosed I For S2: B~ d~l =0 · I Two different situations in even one case!
10. Ampère’s Law with Maxwell’s Correction The line integral of magnetic field over a closed path is equal to the total current going through any surface bounded by the closed path d B~ d~l = µ " E + µ I · 0 0 dt 0 enclosed I Displacement current Electric flux is defined as Qinside E = E~ dS~ = · µ0 For S2: d Q dQdQ B~ d~l = µ I + µ " = µµ0 " · 0 enclosed 0 0 dt " 0 d0tdt I ✓ 0 ◆
11. Recall: Maxwell’s Equations Integral form: Differential form: Gauss’ Law Q ⇢ E~ dS~ = inside E~ = · " r · "0 I 0 Gauss’ Law for Magnetism B~ dS~ =0 B~ =0 · I r · Faraday’s Law d ~ E~ d~l = B ~ @B · dt E =– I r⇥ @t Ampere’s Law d ~ B~ d~l = µ I + µ " E ~ ~ @E · 0 enclosed 0 0 dt B = µ0J + µ0"0 I r⇥ @t Macroscopic Scale Microscopic Scale
12. Convert Intergral form to Differential form Q dB E~ dS~ = inside E~ d~l = · " · dt I 0 I ⇢ d d E~ dS~ = dV E~ Ed~~l =d~l = B~ dS~B~ dS~ · "0 · · dtdt ZZZV I I Divergence theorem: the flux penetrating a Stokes’ theorem: the circulation of a field E closed surface S that bounds a volume V is around the loop l that bounds a surface S is equal to the divergence of the field E inside equal to the flux of curl E over S the volume ~ ~~ ~ ~ ~ E~ dS~ = ( E~ )dV E Edl =(dl =( E)dES)dS · r · · · r⇥r⇥ ZZZV I I ⇢ @B~ ( E~ )dV =0 E~ + dS =0 r · " r⇥ @t ZZZV 0 ! ~ ⇢ @B~ E = E~ =– r · "0 r⇥ @t
13. Divergence Operator Divergence at a point (x,y,z) is the measure of the vector flow out of a surface surrounding that point. E~ =ˆxEx +ˆyEy +ˆzEz @ @ @ =ˆx +ˆy +ˆz Fig. 3: No variation → Fig. 4: Zero divergence r @x @y @z zero divergence @E @E @E E~ = x + y + z r · @x @y @z ≡ (rate of change of E in x direction) ~ @Ex @Ey @Ez + (rate of change of E in y direction) E = 1 + + + (rate of change of E in z direction) r · @x @y @z Fig. 5 Fig. 1: Positive Fig. 2: Negative divergence at P divergence at P
14. Curl Operator in a Spherical Coordinate Curl is a measure of the rotation of a vector field. E~ =ˆxEx +ˆyEy +ˆzEz @ @ @ =ˆx +ˆy +ˆz r @x @y @z xˆ yˆ zˆ Fig. 2: How much is the @ @ @ Fig. 1: Non-zero curl E~ = curl? r⇥ 2 @x @y @z 3 Ex Ey Ez 4 5 @E @E @E @E @E @E =ˆx z y +ˆy x z +ˆz y x @y @z @z @x @x @y ✓ ◆ ✓ ◆ ✓ ◆ ≡ (how much does an object in y-z plane rotate) + (how much does an object in x-z plane rotate) + (how much does an object in x-y plane rotate) Circulation Curl = Area
15. Curl Operator in a Cylindrical Coordinate Curl is a measure of the rotation of a vector field. E~ =ˆ⇢E⇢ + ˆE +ˆzEz 1 @ 1 @ @ = + + r ⇢ @⇢ ⇢ @ @z 1 ˆ 1 ⇢ ⇢ˆ ⇢ zˆ E~ = @ @ @ r⇥ 2 @⇢ @ @z 3 E⇢ ⇢E Ez 6 7 4 5 ⇢ˆ @E @(⇢E ) @E @E zˆ @(⇢E ) @E ) E~ = z ˆ z ⇢ + ⇢ r⇥ ⇢ @ @z @⇢ @z ⇢ @ @z ✓ ◆ ✓ ◆ ✓ ◆
16. Differential form: Gauss’ Law for Magnetism B~ =0 ⟺ B~ dS~ =0 · r · I ~ 1 @rBr 1 @B @Bz B~ = Bout Bin =0 B = + + r · volume r · r @r r @ @z 1 @ µ I = a 0 =0 a @r 2a ✓ ◆ B = BA µ I B = 0 2⇡a
17. Differential form: Faraday’s Law @B~ dB E~ =– ⟺ E~ d~l = · dt r⇥ @t I @B~ 1 ⇢ˆ ˆ 1 zˆ curl of E~ = ⇢ ⇢ @t E~ = @ @ @ r⇥ 2 @⇢ @ @z 3 Circulation @B~ E0⇢ ⇢E E0z = 6 7 Area @t 4zˆ zˆ @B~5⇢ E~ = E = E~ (2⇡⇢) @B~ r⇥ ⇢ ⇢ @t 2 = ⇡⇢2 @t ~ @B~ ⇢ ~ 1 @B E~ = E = zˆ ⇢ @t 2 r⇥ 2 @t
18. Differential form: Ampere’s Law with Maxwell’s Corr ~ @E ~ ~ dE B~ = µ J~ + µ " ⟺ B dl = µ0Ienclosed + 0 0 0 @t · dt r⇥ I I I J: Current density J = = A ⇡R2 Circulation B~ = = µ J~ r⇥ Area 0 B(2⇡R) = µ J~ ⇡R2 0 B(2⇡R) I = µ enclosed ⇡R2 0 ⇡R2 I B = µ enclosed 0 2⇡R
19. Differential form: Ampere’s Law with Maxwell’s Corr ~ @E ~ ~ dE B~ = µ J~ + µ " ⟺ B dl = µ0Ienclosed + 0 0 0 @t · dt r⇥ I Start from: B~ = µ J~ r⇥ 0 Take divergence: ( B~ )=µ J~ r · r⇥ 0r · @⇢ @ @E~ J~ = = ("0 E~ )= "0 r · @t @t r · r · @t ! ( ⇢ :charge density) We need to add displacement current term
20. Differential form: Ampere’s Law with Maxwell’s Corr B~ = µ J~ r⇥ 0 1 ˆ 1 1 ˆ 1 ⇢ ⇢ˆ ⇢ zˆ ⇢ ⇢ˆ ⇢ zˆ B~ = @ @ @ = B~ = @ @ @ r⇥ 2 @⇢ @ @z 3 r⇥ 2 @⇢ @ @z 3 B⇢ ⇢B Bz 0 ⇢B 0 6 7 6 7 4 5 4 5 ⇢ˆ @0 @⇢B @0 @0 zˆ @⇢B @0 B~ = ⇢ ˆ + r⇥ ⇢ @⇢ @z @⇢ @z ⇢ @⇢ @ ✓ ◆ ✓ ◆ ✓ ◆ zˆ @⇢B µ I B~ = = 0 zˆ r⇥ ⇢ @⇢ 2⇡⇢2 Varying current causes a circulating magnetic field
21. How Electromagnetism is a Wave in Vacuum? @B~ @E~ E~ =– B~ = µ0"0 r⇥ @t r⇥ @t ~ ~ @( B) ~ @( E) E~ =– r⇥ B = µ0"0 r⇥ r⇥r⇥ @t r⇥r⇥ @t @ @@E~B~ ~ ~ 2 ~ 2 @ @B ( E) E = µµ00""00 ( B~ ) B~ = µ0"0 r r · r @t @@tt!! r r · r @t @t ! @2E~ @2B~ 2E~ µ " =0 2B~ µ " =0 r 0 0 @t2 r 0 0 @t2 1 8 where c= 3 10 m/s Wave equation is in form µ " ⇡ ⇥ p 0 0 d2y 1 d2y =0 7 Wb 2 2 2 µ =4⇡ 10 dx v dt 0 ⇥ A.m 2 ⇒ Both Electric and 12 C " =8.85 10 ⇥ N.m2 Magnetism are waves
22. E = E0cos(kx !t) B = B cos(kx !t) 0
23. Magnetism vs Electric @B~ Let’s work on one dimension E~ =– r⇥ @t ~ ~ ˆ @B @ E = [E0sin(!t kx)]j = [B sin(!t kx)]ˆi r⇥ r⇥ @t @t 0 ~ E~ = k[E0sin(!t kx)]ˆi @B r⇥ =[!B0sin(!t kx)]ˆi @t E0 ! 2⇡f kE0 = !B0 ⇔ = = = f B0 k 2⇡/ E 0 = c B0
24. Energy of an Electromagnetic Wave Start from the magnetic density: 2 1 2 1 E 1 8 ⌘B = B = c= 3 10 m/s 2µ 2µ c2 pµ0"0 ⇡ ⇥ 0 0 ✓ ◆ 1 2 "0 2 ⌘B = µ0"0E = E (Electric energy density) 2µ0 2 For an electromagnetic wave in free space, half of the energy is in the electric field and another half is the magnetic field ⌘ = ⌘E + ⌘B " " ⌘ = 0 E2 + 0 E2 2 2 2 Total energy: ⌘ = "0E
25. Power density Since waves are spread out in space and time, energy density is often a more useful concept than energy. The power of a wave should probably replaced with the more useful concept of its power density. The energy content of a wave fills a volume of space defines energy density as energy per volume u ⌘ = V The power density is the power per area P W S = A m2 
26. Poynting Vector Power density is defined as P 1 u 1 ⌘V S = = = A A t A t ⌘V ⌘(Al) ⌘(Act) S = = = = ⌘c At At At 1 E S = ⌘c = B2 µ B ✓ 0 ◆ 1 1 S = EB S = E~ B~ Poynting Vector µ0 µ0 ⇥ The magnitude of the Poynting vector represents the rate at which energy passes through a unit surface area perpendicular to the direction of wave propagation. The direction of the Poynting vector is along the direction of wave propagation.