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 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; Maxwell’s Equations Integral form: Gauss’ Law I Qinside ~ ~ E · dS = "0 Differential form: ⇢ ~ r·E = "0 Gauss’ Law for Magnetism I ~ · dS ~=0 B Faraday’s Law I d B ~ ~ E · dl = dt IAmpere’s Law d E ~ ~ B · dl = µ0 Ienclosed + µ0 "0 dt Macroscopic Scale ~ =0 r·B ~ @ B ~ =– r⇥E @t ~ @ E ~ = µ0 J~ + µ0 "0 r⇥B @t Microscopic Scale 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 ~=n dS ˆ dS = n ˆ dxdy dx dy → dS → Qinside E= (4⇡r2 )"0 Qinside ~ E · dxdyˆ n= "0 Qinside ~ E · dxdycos✓ = "0 Qinside Edxdy = "0 Qinside ES = E(4⇡r ) = "0 Coulomb’s Law Gauss’s Law of Magnetism Gauss’s law of magnetism states that the net magnetic flux through any closed surface is zero I ~ · dS ~=0 B The number of magnetic field lines that exit equal to the number for magnetic field lines that enter the closed surface → E I ~ · dS ~ = Qinside E "0 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 = E d B dt ~Ed ~· d~l = Edlcos✓ E = Edl (θ = 0) E(2⇡R) = e.m.f = e.m.f = d B dt W = F d = Eqd W = Ed d V = Ed = e.m.f d B dt d B dt 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 Time-changing electric fields induces magnetic fields Displacement current Conduction currents ( motion of charged particles) Time changing electric fields cause Magnetic field also cause Magnetic field => Time changing electric fields is equivalent to a current We call it dispalcement current 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 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 + µ0 Ienclosed dt For S1: I ~ · d~l = µ0 Ienclosed B For S2: I ~ · d~l = B Two different situations in even one case! ... magnetic fields Displacement current Conduction currents ( motion of charged particles) Time changing electric fields cause Magnetic field also cause Magnetic field => Time changing electric fields... number of magnetic field lines that exit equal to the number for magnetic field lines that enter the closed surface → E I ~ · dS ~ = Qinside E "0 Faraday’s Law The electric field around a closed... J~ + µ0 "0 r⇥B @t Microscopic Scale 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