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Notes 4 Maxwell's Equations ECE 3317 Applied Electromagnetic Waves Prof. David R. Jackson Fall 2020 Notes 4 Maxwell’s Equations Adapted from notes by Prof. Stuart A. Long 1 Overview Here we present an overview of Maxwell’s equations. A much more thorough discussion of Maxwell’s equations may be found in the class notes for ECE 3318: http://courses.egr.uh.edu/ECE/ECE3318 Notes 10: Electric Gauss’s law Notes 18: Faraday’s law Notes 28: Ampere’s law Notes 28: Magnetic Gauss law . D. Fleisch, A Student’s Guide to Maxwell’s Equations, Cambridge University Press, 2008. 2 Electromagnetic Fields Four vector quantities E electric field strength [Volt/meter] D electric flux density [Coulomb/meter2] H magnetic field strength [Amp/meter] B magnetic flux density [Weber/meter2] or [Tesla] Each are functions of space and time e.g. E(x,y,z,t) J electric current density [Amp/meter2] 3 ρv electric charge density [Coulomb/meter ] 3 MKS units length – meter [m] mass – kilogram [kg] time – second [sec] Some common prefixes and the power of ten each represent are listed below femto - f - 10-15 centi - c - 10-2 mega - M - 106 pico - p - 10-12 deci - d - 10-1 giga - G - 109 nano - n - 10-9 deka - da - 101 tera - T - 1012 micro - μ - 10-6 hecto - h - 102 peta - P - 1015 milli - m - 10-3 kilo - k - 103 4 Maxwell’s Equations (Time-varying, differential form) ∂B ∇×E =− ∂t ∂D ∇×HJ = + ∂t ∇⋅B =0 ∇⋅D =ρv 5 Maxwell James Clerk Maxwell (1831–1879) James Clerk Maxwell was a Scottish mathematician and theoretical physicist. His most significant achievement was the development of the classical electromagnetic theory, synthesizing all previous unrelated observations, experiments and equations of electricity, magnetism and even optics into a consistent theory. His set of equations—Maxwell's equations—demonstrated that electricity, magnetism and even light are all manifestations of the same phenomenon: the electromagnetic field. From that moment on, all other classical laws or equations of these disciplines became simplified cases of Maxwell's equations. Maxwell's work in electromagnetism has been called the "second great unification in physics", after the first one carried out by Isaac Newton. Maxwell demonstrated that electric and magnetic fields travel through space in the form of waves, and at the constant speed of light. Finally, in 1864 Maxwell wrote A Dynamical Theory of the Electromagnetic Field where he first proposed that light was in fact undulations in the same medium that is the cause of electric and magnetic phenomena. His work in producing a unified model of electromagnetism is considered to be one of the greatest advances in physics. (Wikipedia) 6 Maxwell’s Equations (cont.) ∂B ∇×E =− Faraday’s law ∂t ∂D ∇×HJ = + Ampere’s law ∂t ∇⋅B =0 Magnetic Gauss law ∇⋅D =ρv Electric Gauss law Questions: When does a magnetic field produce an electric field? When does an electric field produce a magnetic field? When does a current flow produce a magnetic field? When does a charge density produce an electric field? 7 Charge Density ∆Q dQ ρv ( xyz, ,) = lim = ∆→V 0 ∆V dV + + + + + + + + + + + + ρv ( xyz,,) ( xyz,,) dV dQ Non-uniform cloud of charge density Example: Protons are closer together as we move to the right. 8 Current Density Vector 2 J = current density vector A/m J ∆I + + + ∆S E Medium σ ∆=ISJ ∆ Current flow is defined to be in the direction that positive charges move in. 9 Current Density Vector (cont.) Ohm’s law Material σ [S/m] Silver 6.3×107 JE= σ Copper 6.0×107 Copper (annealed) 5.8×107 Gold 4.1×107 Aluminum 3.5×107 Zinc 1.7×107 J Brass 1.6×107 σ Nickel 1.4×107 E Iron 1.0×107 Tin 9.2×106 Steel (carbon) 7.0×106 Steel (stainless) 1.5×106 http://en.wikipedia.org/wiki/Electrical_resistivity_and_conductivity 10 Current Density Vector (cont.) Current through a tilted surface: J nˆ + + + ∆S E Medium σ ∆=I( J ⋅ nSˆ) ∆ 11 Current Density Vector (cont.) ∆=I( J ⋅ nSˆ) ∆ Note: The direction of the unit normal vector = ⋅ determines whether the current is I∫ J nˆ dS measured going in or out. S nˆ J S 12 Law of Conservation of Electric Charge (Continuity Equation) ∂D ∇×HJ = + ∂t ∂D ∇⋅( ∇×HJ) =∇⋅ +∇⋅ ∂t ∂ 0 =∇⋅JD +( ∇⋅ ) ∂t Flow of electric ∂ρ Rate of decrease of electric ∇⋅J =− v current out of volume ∂ charge (per unit volume) (per unit volume) t 13 Continuity Equation (cont.) ∂ρ ∇⋅J =− v ∂t Integrate both sides over an arbitrary volume V: ∂ρ ∫∫∇⋅J dV = − v dV VV∂t Apply the divergence theorem: S ∫∫∇⋅JJdV = ⋅ nˆ dS VS V nˆ Hence: ∂ρ ∫∫J ⋅=−nˆ dVv dV SV∂t 14 Continuity Equation (cont.) ∂ρ ∫∫J ⋅=−nˆ dSv dV SV∂t S Physical interpretation: V nˆ ∂ρ ∂ i=−=−v dVρ dV (This assumes that the out ∫∫v surface is stationary.) VV∂∂tt ∂Q ∂Q i = − encl or i = encl out ∂t in ∂t 15 Continuity Equation (cont.) ∂Q i = encl in ∂t J Qencl This implies that charge is never created or destroyed. It only moves from one place to another! 16 Maxwell’s Equations (cont.) Time -Dependent ∂∂BD ∇×E =− ∇×HJ = + ∇⋅ B =0 ∇⋅ D = ρ ∂∂tt v Time -Independent (Statics) ∇×ED =0 ∇⋅ = ρv ∇×HJ = ∇⋅ B =0 Decouples E and HE ⇒ comes from ρv and HJ comes from Note: Regular (not script) font is used for statics, just as it is for phasors. 17 Maxwell’s Equations (cont.) ∂ Time-harmonic (phasor) domain → jω ∂t ∇×E =− jBω ∇×H = J + jDω ∇⋅B =0 ∇⋅D =ρv 18 Constitutive Relations The characteristics of the media relate D to E and H to B Free Space DE= ε0 (ε0 = permittivity) BH = µ0 (µ0 =)permeability -12 ε 0 8.8541878× 10 [F/m] -7 µ0 = 4π × 10 [H/m] (exact* ) *Prior to 2019 1 c = c ≡× 2.99792458 108 [m/s] (exact value that is defined) µε00 19 Constitutive Relations (cont.) Definition of the Amp*: *Prior to 2019 # 1 I d Two infinite wires carrying DC currents # 2 I Fx2 x 2 I µ0 −7 From ECE 3318: Fx2 = µπ=4 × 10 [A/m] 2πd 0 Definition of I =1 Amp: −7 Fdx2 =2×= 10[ N/m] when 1[ m] 20 Constitutive Relations (cont.) Free space, in the phasor domain: DE= ε0 (ε0 = permittivity) B = µH0 ()µ0 = permeability This follows from the fact that aV ( t) ⇔ aV (where a is a real number) 21 Example Given the following electric field in free space: 1 E (t) =θˆ( Ecos( ωφθ t −+ kr )) sin 0 00r Find the magnetic field. k0= ω µε 00 In the phasor domain: ∇×E =− jBω ∇×H = J + jDω 1 − φ ˆ jk00 r j E= θθ Ee0 e sin r ∇⋅B =0 ∇⋅D =ρ ∇×E =− jBω v So 1 HE= ∇× − jωµ ∇×E =− jHωµ0 0 ∂∂θ 1 ( Eφφsin ) ∂Eθ 11∂∂EE(rE ) 1∂ (rEθ ) ∇×Er = ˆ − +θφ rr− + ˆ − θ θ φ θφ θ rsin ∂ ∂ rsin ∂ ∂ r rr ∂∂ 1 HE= ∇× (no φ variation) − jωµ0 22 Example (cont.) φ 1 − ˆ j00 jk r E= θθ( Ee0 ) e sin 1 r HE= ∇× − jωµ0 1 ∂ (rEθ ) ∇×E =φˆ rr∂ 11φ − = φθˆ j 00− jk r φ 1 − H e E00sin ( jk) e j00 jk r − jrωµ ∂ r Ee0 e sinθ 0 1 r = φˆ rr∂ − k jφ 1 − jk r ∂ jk0 r = φθˆ 0 00 1 φ ( e ) HE0 esin e ˆ j 0 = φθeE0 sin ωµ0 r ∂ rr 1 φ − = φθˆ ej 00 Esin (− jk) e jk r r 00 ˆ k0 1 H =φE00sinθω cos( t−+ kr φ) ωµ0 r 23 Example (cont.) ∂B ∇×E =− Alternative approach (in the time domain directly): ∂t ∂D ∂B ∇×HJ = + ∇×E =− ∂t ∂t ∇⋅B =0 ∇⋅D =ρv ∂ ˆ 1 (rEθ ) ∂B ∇×E =φ = −∇×E rr∂ ∂t 1 ∂r E00cos(ω t−+ kr φθ) sin 1 r = φˆ rr∂ ∂cos ωφt −+ kr ˆ 1 ( ( 0 )) = φθE0 sin rr∂ 1 =φθˆ Esin( − k)( − sin ( ωφ t −+ kr )) r 00 0 24 Example (cont.) ∂B = −∇×E ∂t 1 ∇×E =φθˆ Esin( k)( sin ( ωφ t− kr + )) r 00 0 So All fields must be pure sinusoidal waves in ∂B 1 the time-harmonic =−φˆ kE sin θω( sin ( t−+ kr φ)) steady state. ∂tr00 0 11 B =−φˆ kE sin θ(− cos( ωt −++ kr φ)) C( r,, θφ) 00r ω 0 ˆ k0 1 H =φE00sin θω( cos( t−+ kr φ)) ωµ0 r 25 Example (cont.) 1 E (t) =θˆ( Ecos( ω t −+ kr φθ)) sin 00r ˆ k0 1 H =φE00sin θω( cos( t−+ kr φ)) ωµ0 r z This describes the far-field radiation from a small vertical dipole antenna. y x 26 Material Properties In a material medium: DE= ε (ε = permittivity) Bµ = H ()µ = permeability ε = εε0 r εr = relative permittivity = µ µµ0 r µr = relative permittivity 27 Material Properties (cont.) Where does permittivity come from? V0 + − Ex - - + - + + - + + - + - - + - + - + - + Water D≡+ε 0 EP Molecule: i + + q 1 ppi = i d Pp≡ ∑ i ∆V ∆V p= qd - −q 28 Material Properties (cont.) D≡+ε 0 EP χ Linear material: PE= εχ The term e is called the 0 e “electric susceptibility.” Note: χ > 0 for most materials so e DE=ε00 + εχe E =εχ0 (1 + e ) E Define: εχre≡+1 Then DE= εε0 r 29 Material Properties (cont.) Teflon ε r = 2.2 Water ε r = 81 (a very polar molecule, fairly free to rotate) Styrofoam ε r = 1.03 Quartz ε r = 5 Note: εr > 1 for most materials: εr≡+1, χχ ee > 0 30 Material Properties (cont.) Where does permeability come from? Iron B Because of electron spin, atoms tend to acts as little current loops, and hence as electromagnetics, or bar magnets.
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