Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Magnetic monopole wikipedia , lookup
History of electromagnetic theory wikipedia , lookup
Electricity wikipedia , lookup
Electrostatics wikipedia , lookup
Faraday paradox wikipedia , lookup
James Clerk Maxwell wikipedia , lookup
Electromagnetism wikipedia , lookup
Lorentz force wikipedia , lookup
Computational electromagnetics wikipedia , lookup
Mathematical descriptions of the electromagnetic field wikipedia , lookup
Maxwell’s Equations January 16, 2013 1. Maxwell’s Equation, integral form 2. Maxwell’s Eqaution, differential form 3. Conservation of electric charge 4. Maxwell’s equations in static case 5. Maxwell’s equations in source-free dynamic case 6. Maxwell’s Eqaution in phasor form 7. The dependency of Maxwell’s equations 1 Maxwell’s equations in integral form 1.1 Faradya’s Law of induction The induced emf (electromotive force) in a closed circuit is equal to the negative of rate of change of magnetic flux pass through it. ‹ ˛ ∂ ∂Φ B =− B̄ · dS̄ Ē · d¯l = − ∂t ∂t S ∂S 1.2 Ampère’s circuital law with Maxwell’s Correction The mmf (magnetomotive force) in a closed circuit is equal to the rate of change of electric flux and current pass through it. ) ˛ ‹ ( ∂ D̄ ¯ ¯ · dS̄ H̄ · dl = J+ ∂t ∂S S 1.3 Gauss’s Law for electricity The electric flux pass through any closed surface is proportional to the enclosed electric charge. ‹ D̄ · dS̄ = ΦD = Q = ρv V S 1 1.4 Gauss’s Law for magnetism The magnetic flux pass through any closed surface is zero ‹ B̄ · dS̄ = 0 2 Maxwell’s equations in differential form 2.1 2.1.1 Vector Identities Gauss’s Divergence Theorem ˚ ‹ V̄ · dS̄ ∇ · V̄ dV = ∂V V 2.1.2 Stokes’ Curl Theorem ¨ ˛ V̄ · d¯l ∇ × V̄ · dS̄ = S 2.2 ∂S Faraday’s Law in differential form Apply Stoke’s Theorem ˛ ‹ ∂ ¯ Ē · dl = − B̄ · dS̄ ∂t S ∂S ‹ ∂ ∇ × Ē · dS̄ = − ∂t S −→ ‹ B̄ · dS̄ S Rearrange ‹ ( S ) ‹ ( ∂ B̄ ∇ × Ē · dS̄ = − · dS̄ ∂t S ) The integral kernel thus equal to each other ∇ × Ē = − 2.3 ∂ B̄ ∂t Ampère’s circuital law in differential form Apply Stoke’s Theorem ) ˛ ‹ ( ∂ D̄ · dS̄ H̄ · d¯l = J¯ + ∂t ∂S S ) ‹ ( ∂ D̄ ∇ × H̄ · dS̄ = · dS̄ J¯ + ∂t S S ‹ −→ The integral kernel thus equal to each other ∇ × H̄ = J¯ + 2 ∂ D̄ ∂t 2.4 Gauss’s Law for electricity in differential form Apply Gauss’s Divergence Theorem ‹ D̄ · dS̄ = Q ˚ ˚ −→ ∇ · D̄dV = Q = S ρv dV V The integral kernel thus equal to each other ∇ · D̄ = ρv 2.5 Gauss’s Law for magnetism in differential form Apply Gauss’s Divergence Theorem ‹ B̄ · dS̄ = 0 ˚ ˚ −→ ∇ · B̄dV = 0 = S 0dV V The integral kernel thus equal to each other ∇ · B̄ = 0 3 Conservation of electric charge Consider the follow equations ∂ D̄ ∇ × H̄ = J¯ + ∂t ∇ · D̄ = ρv ( ) ∇ · ∇ × V̄ = 0 Faraday’s Law Gauss’s Law Div of curl is zero Take the div of Faraday’s Law ( ) ∂ D̄ ∇ · ∇ × H̄ = ∇ · J¯ + ∂t ( ) ∂ 0 = ∇ · J¯ + ∇ · D̄ ∂t ∂ρv 0 = ∇ · J¯ + ∂t i.e. The rate of current transfer out of a volume eqaul to decreasing rate of charge in that volume ∂ρv ∇ · J¯ = − ∂t 3 4 Maxwell’s equations in static case Static ⇐⇒ ∂ =0 ∂t { ∇ × Ē = 0 ∇ × H̄ = J ∇ · B̄ = 0 ∇ · D̄ = ρv ∇ · J¯ = 0 ∇ × Ē = 0 ∇ · D̄ = ρv ∇ × H̄ = J ∇ · B̄ = 0 ∇ · J¯ = 0 E & D field are generated by ρv H & B fields are generated ( by J ) J & ρv are independent ∇ · J¯ = 0 E , H are decoupled, the electrostatic field and magnetostatic field are independent. 5 Maxwell’s Equations in source-free dynamic case Source free ⇐⇒ J = ρv = 0 ∂ B̄ ∂t ∂ D̄ ∇ × H̄ = ∂t ∇ · B̄ = 0 ∇ · D̄ = 0 E & H are coupled, they are not independent. This coupling generate the phenomenon of EM wave propagation ∇ × Ē = − 6 Maxwell’s equations in phasor form 6.1 Phasor Review Apply Euler’s Eqaution to sinusoidal term [ ] [( ) ] V (t) = V0 cos (ωt + ϕ) = ℜe V0 ej(ωt+ϕ) = ℜe V0 ejϕ ejωt The phasor form is thus V0 ejϕ i.e. V (t) = V0 cos (ωt + ϕ) • Original time-domain form is real number • Phasor form is complex number 4 ←→ Ve = V0 ejϕ 6.2 Phasor Differentiation and Integration [( ) ] ∂ ∂ V (t) = V0 cos (ωt + ϕ) = −ωV0 sin (ωt + ϕ) = ℜe jωV0 ejϕ ejωt ∂t ∂t Therefore ∂ V (t) ←→ jω Ve ∂t ˆ ˆ V (t)dt = 1 V0 cos (ωt + ϕ) dt = V0 sin (ωt + ϕ) = ℜe ω [( ) ] 1 jϕ jωt V0 e e jω Therefore ˆ V (t)dt ←→ 6.3 1 e V jω Maxwell’s Equations in Phasor Form ∇ × Ē = −jω B̄ ∇ × H̄ = jω D̄ + J¯ ∇ · B̄ = 0 ∇ · D̄ = ρv ∇ · J¯ = −jωρv The equations are now complex and time-independent. 7 The dependency in Maxwell’s Equations There are 4 equations, but the other equations can be derived form the 2 curl equations with some vector identities. ∇ × Ē = −jω B̄ ∇ × H̄ = jω D̄ + J¯ ∇ · B̄ = 0 ∇ · D̄ = ρv ∇ · J¯ = −jωρv 7.1 Conservation of charge derived from Ampere’s Law Already shown previously 5 7.2 Gauss’s Law for electricity derived from Ampere’s law & Conservation of Charge Apply div of curl is zero into Ampere’s Law ∇ × H̄ = jω D̄ + J¯ ( ) ( ) ∇ · ∇ × H̄ = ∇ · jω D̄ + J¯ −→ 0 = jω∇ · D̄ + ∇ · J¯ Apply Conservation of charge ∇ · J¯ = −jωρv 0 = jω∇ · D̄ − jωρv i.e. ∇ · D̄ = ρv 7.3 Gauss’s Law for magnetism derived from Faraday’s Law Apply div of curl is zero into Faraday’s Law ∇ × Ē = jω B̄ ( ) ( ) ∇ · ∇ × Ē = ∇ · jω B̄ −→ ( ) 0 = ∇ · jω B̄ = jω∇ · B̄ i.e. ∇ · B̄ = 0 6