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Lecture Physics A2: Electromagnetic Field and Wave - PhD. Pham Tan Thi

Lecture Physics A2: Electromagnetic Field and Wave - PhD. Pham Tan Thi present the content Maxwell’s equation, Gauss’s law for electric field, Faraday’s law, Ampère’s law with Maxwell’s correction, convert intergral form to differential form,... Electromagnetic Field and Wave Pham Tan Thi, Ph.D. Department of Biomedical Engineering Faculty of Applied Science Ho Chi Minh University of Technology Maxwell’s Equation Maxwell discovered that the basic principles of electromagnetism can be expressed i

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  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’ I Law Qinside ~ ⇢ ~ ~ E · dS = r·E = "0 "0 Gauss’ Law for Magnetism I ~ · dS B ~=0 ~ =0 r·B Faraday’s Law I ~ ~ d B @ ~ B E · dl = ~ =– r⇥E dt @t IAmpere’s Law ~ ~ d E @ ~ E B · dl = µ0 Ienclosed + µ0 "0 ~ = µ0 J~ + µ0 "0 r⇥B dt @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. I ~ ~ Qinside E · dS = "0 ~ Qinside E · dxdyˆ n= ~=n dS ˆ dS = n ˆ dxdy "0 ~ Qinside E · dxdycos✓ = "0 dx → Qinside dS Edxdy = dy → "0 2 Qinside ES = E(4⇡r ) = "0 Qinside E= Coulomb’s Law (4⇡r2 )"0
  5. Gauss’s Law of Magnetism Gauss’s law of magnetism states that the net magnetic flux through any closed surface is zero I ~ · dS B ~=0 The number of magnetic field lines that exit equal to the number for magnetic field lines that enter the closed surface → E I ~ = Qinside ~ · dS E "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. I ~ · d~l = d B ~· d~l = Edlcos✓ ~Ed E E dt = Edl (θ = 0) d B E(2⇡R) = dt d B e.m.f = dt d B W = F d = Eqd e.m.f = 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 I ~ ~ d E B · dl = µ0 Ienclosed + µ0 "0 dt 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 I ~ ~ d E B · dl = µ0 Ienclosed + µ0 "0 dt ~ · d~l = Bdlcos✓ = Bdl (when θ = 0) B I Z ~ ~ d E B · dl = µ0Bdl Ienclosed + = B(2⇡R) dt B(2⇡R) = µ0 I µ0 I B= 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 I ~ ~ d E B · d l = µ 0 "0 + µ0 Ienclosed dt For S1: I ~ · d~l = µ0 Ienclosed B For S2: I ~ · d~l = 0 B 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 I ~ ~ d E B · d l = µ 0 "0 + µ0 Ienclosed dt Displacement current Electric flux is defined as ~ ~ Qinside E = E · dS = µ0 For S2: I ✓ ◆ d Q dQdQ ~ ~ B · dl = µ0 Ienclosed + µ0 "0 = µ 0 "0 dt "0 dt dt
  11. Recall: Maxwell’s Equations Integral form: Differential form: Gauss’ I Law Qinside ~ ⇢ ~ ~ E · dS = r·E = "0 "0 Gauss’ Law for Magnetism I ~ · dS B ~=0 ~ =0 r·B Faraday’s Law I ~ ~ d B @ ~ B E · dl = ~ =– r⇥E dt @t IAmpere’s Law ~ ~ d E @ ~ E B · dl = µ0 Ienclosed + µ0 "0 ~ = µ0 J~ + µ0 "0 r⇥B dt @t Macroscopic Scale Microscopic Scale
  12. Convert Intergral form to Differential form I I Qinside ~ · d~l = d B ~ ~ E · dS = E "0 dt ZZZ I I ~ ~ ⇢ d ~d ~~ ~ E · dS = dV ~ ~ E ·E~ ~ dl ·=dl = BdS BdS V "0 dt dt 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 I I ZZZ ~ · dS ~= ~ E ~d~l· = ~ ·E d~l(r ~⇥ E)dS ⇥ E)dS = (r ~ E (r · E)dV V ZZZ ! ~ ⇢ @ ~ B (r · E )dV = 0 ~+ r⇥E dS = 0 V "0 @t ~ ⇢ @ ~ B r·E = ~ =– r⇥E "0 @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. ~ =x E ˆEx + yˆEy + zˆEz @ @ @ r=xˆ + yˆ + zˆ Fig. 3: No variation → Fig. 4: Zero divergence @x @y @z zero divergence ~ = @Ex + @Ey + @Ez r·E @x @y @z ≡ (rate of change of E in x direction) ~ @Ex @Ey + (rate of change of E in y direction) r·E =1 + + (rate of change of E in z direction) @x @y 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. ~ =x E ˆEx + yˆEy + zˆEz @ @ @ r=x ˆ + yˆ + zˆ @x @y @z 2 3 xˆ yˆ zˆ Fig. 2: How much is the ~ =4 @ r⇥E @ @ 5 Fig. 1: Non-zero curl @x @y @z curl? Ex Ey Ez ✓ ◆ ✓ ◆ ✓ ◆ @Ez @Ey @Ex @Ez @Ey @Ex =x ˆ + yˆ + zˆ @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. ~ = ⇢ˆE⇢ + ˆE + zˆEz E 1 @ 1 @ @ r= + + ⇢ @⇢ ⇢ @ @z 2 3 1 ˆ 1 ⇢⇢ˆ ⇢zˆ ~ = 6@ @ @ 7 r⇥E 4 @⇢ @ @z 5 E⇢ ⇢E Ez ✓ ◆ ✓ ◆ ✓ ◆ ~ = ⇢ˆ r⇥E @Ez @(⇢E ) ˆ @Ez @E⇢ + zˆ @(⇢E ) @E⇢ ) ⇢ @ @z @⇢ @z ⇢ @ @z
  16. Differential form: Gauss’ Law for Magnetism I ~ =0 r·B ⟺ ~ · dS B ~=0 ~ 1 @rBr 1 @B @Bz ~ = r·B Bout Bin =0 r·B = + + r @r r @ @z volume ✓ ◆ 1 @ µ0 I = a =0 a @r 2a B = BA µ0 I B= 2⇡a
  17. Differential form: Faraday’s Law I @ ~ B d B ~ =– r⇥E ⟺ ~ · d~l = E @t dt 2 3 ~ 1 ˆ 1 ~ = @ B ⇢⇢ˆ ⇢zˆ curl of E ~ = 6@ @ @ 7 @t r⇥E 4 @⇢ @ @z 5 Circulation @B~ E0⇢ ⇢E E0 z = Area @t zˆ ~⇢ zˆ @ B ~ r⇥E = E = ~ E(2⇡⇢) ~ @B ⇢ ⇢ @t 2 2 = ⇡⇢ @t ~⇢ 1 @ ~ B ~ = @ B ~ = r⇥E zˆ ⇢ E 2 @t @t 2
  18. Differential form: Ampere’s Law with Maxwell’s Corr I @ ~ E d E ~ = µ0 J~ + µ0 "0 r⇥B ⟺ ~ ~ B · dl = µ0 Ienclosed + @t dt I I J: Current density J = = A ⇡R2 ~ Circulation r⇥B = = µ0 J~ Area B(2⇡R) ~ = µ 0 J ⇡R2 B(2⇡R) Ienclosed 2 = µ0 ⇡R ⇡R2 Ienclosed B = µ0 2⇡R
  19. Differential form: Ampere’s Law with Maxwell’s Corr I @ ~ E d E ~ = µ0 J~ + µ0 "0 r⇥B ⟺ ~ ~ B · dl = µ0 Ienclosed + @t dt Start from: ~ = µ0 J~ r⇥B Take divergence: ~ = µ0 r · J~ r · (r ⇥ B) ! @⇢ @ ~ @E ~ r·J = = ~ = ("0 r · E) r· "0 @t @t @t ( ⇢ :charge density) We need to add displacement current term
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