Physics 2 - Lecture 2: Electromagnetic Field and Wave - Pham Tan Thi

Nội dung text: Physics 2 - Lecture 2: 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. 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
4. 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
5. 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
6. 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 ◆
7. 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
8. 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
9. 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
10. 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
11. 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
12. E = E0cos(kx !t) B = B cos(kx !t) 0
13. 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
14. 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.