Field and Wave Electromagnetic

Field and Wave Electromagnetic Chapter7 The time varying fields and Maxwell s equation Introduction () Time static fields ) Electrostatic E =, id= ρ, D= εe ) Magnetostatic ib=, H = J, H = B μ note) E and D are not related to B and H for time static cases Example) A static E field in a conducting medium steady current. ( J =σ E) give rises to a static magnetic field:ampere's law. But E field can be completely determined from the static electric charge or potential distributions magnetic field is a consequence Electromagnetic Theory

Introduction () Time varying fields E and D are properly related to B and H ) modify E equation fundamental postulate leading to Faraday's law ) then modify the H equation to be consistent with the equation of continuity ρ cf ) ij = for static. but ij = for time varying t 3) id = ρ and ib = never changes. Electromagnetic Theory 3 Faraday s Law Michael Faraday 83, experimental law postulate Definition : the quantitative relationship between the induced emf and the rate of change of flux linkage Fundamental postulate for Electromagnetic Induction B E = Non-conservative field cannot be expressed C Edl i = S as the gradient of a scalar potential B ids Electromagnetic Theory 4

A Stationary Circuit in a Time Varying Magnetic Field () C d d Eidl = B ds (, since stationary ds = ) dt i S dt ight hand rule (counter clock wise) dφ dφ emf, v = Eidl =, Assume > dt dt v < driving current to flow in the direction of clock wise potential difference of gap between terminal and assume V = Eidl < V > V Electromagnetic Theory 5 A Stationary Circuit in a Time Varying Magnetic Field () Define v = C Eidl : emf induced in circuit with contour C ~ v : electromagnetive force driving current in the direction of right hand rule Meaning of contour integral Eidl = Eidl ( inside contour E = ) = Eidl = V V = V right hand Field between the terminal in the gap can be replaced with voltage source. But polarity of the change of the flux linkage v= V depends on Electromagnetic Theory 6

A Stationary Circuit in a Time Varying Magnetic Field (3) B e.g) > then v (current is in the direction of left hand rule) ie. V < Define Φ= S Bds i : magnetic flux crossing surface S[ Wb] dφ then v = This is valid even in the absense of a physical closed circuit dt note The emf induced in a stationary loop caused by a time-varying magnetic field is a transformer emf Electromagnetic Theory 7 Ex 7-) A Circular Loop of N Turns of Conducting Wire π r A circular loop of N turns, B= zb cos( )sin wt b Find the emf induced in the loop b π r π sol) each turn Φ= Bds i = zb cos sin wt) ( z π rdr) cf) dφ = π S ( i b 8b π z = ( ) B sin wt π N-turns NΦ dφ 8Nb π v= N = ( ) B cos wt [V] dt π b y x Electromagnetic Theory 8

Transformers () j mmf NI j j k k k N, N, i, i number of turns and the currents = Φ : the reluctance of the magnetic circuit Ni Ni =Φ (where Ni : mmf in the positive direction, Ni : mmf in the negative direction) l = μs l Ni Ni = Φ μs Electromagnetic Theory 9 Transformers () a) Ideal transformer i N μ, Ni = Ni = i N v N dt cf ) Faraday's law dφ = ( No negative sign, careful of sign of flux Φ) dφ v = N (But flux is in the reverse direction) = dt v N effective load seen by the source connected to primary winding N v v N N ( ) eff = = = L i N N i N N Impedance transformation ( Z) eff = N Z L v N Electromagnetic Theory

Transformers (3) b) eal transformer l Ni Ni = Φ μs μs μs Λ = NΦ= ( Ni NNi ), Λ = NΦ= ( NNi Ni ) l l di di di di v = L L, v = L L dt dt dt dt μs μs μs (where L = N, L = N, L = NN) l l l For an ideal transformer No leakage flux L = For a real transformer L, : = k L L k < ( k coefficient of coupling) L L Electromagnetic Theory Equivalent circuit Transformers (4), : winding resistance X, X : leakage inductive reactance c : power loss due to hysteresis and eddy current X c : nonlinear inductive reactance due to the nonlinear magnetization behavior of the ferromagnetic core Electromagnetic Theory

A Moving Conductor in a Static Magnetic Field F = qu B m Charge Seperation Coulomb force of an attraction F and F will balance each other to be in equilibrium. m e Magnetic force per unit charge F q m Fm = u B, V = E dl, E = q V = ( u B) dl The emf generated around the closed loop is V ' = ( u B) dl flux cutting emf C Electromagnetic Theory 3 Ex 6-5) A Metal Bar Sliding Over Conducting ails B = zb ˆ, constant u a) V = V V = ( u B) dl ' = ( xu ˆ zb ˆ ) ( ydl ˆ ) ' C = ub h V ( ubh) b) I =, Pl = I = c) mechanical power ' F = I dl B = xib ˆ h m ' ( I : negative direction to dl ) ub h Pm = F u = Fm u = Electromagnetic Theory 4

A Moving Circuit in a Time Varying Magnetic Field () F = q( E+ u B) m To an observer moving with C, the force on q can be interpreted as caused by an electric field E ', E' = E+ u B B E' dl = ds + ( u B) dl C S C General form of Faraday law the emf induced in the moving frame of reference motional emf due to the motion of the circuit in B transformer emf due to the time variation Electromagnetic Theory 5 A Moving Circuit in a Time Varying Magnetic Field () The time rate of chage of magnetic flux, dφ d = B ds dt dt S = lim B( t+δt) ds B( t) ds Δ t Δt S S Bt () cf) B( t+δ t) = B() t + Δ t+ H. OT.. Taylor's series d B B ds = ds lim B ds B ds H. O. T. dt S + + S Δ t Δt S S assuming side surface S as the area swept out by the conductor in time Δt ds = dl u Δt 3 from divergence theorem V Bdv= B ds B ds+ B ds S S S S C 3 S3 B ds B ds = Δ t ( u B) dl d B B ds ds ( u B) dl dt = S S C d dφ V ' = E ' i dl = B ds = C dt S dt 3 Electromagnetic Theory 6

Maxwell s Equation () static E = id = ρ, D= ε E H = J ib =, B = μ H Time varying B E = t id = ρ D H = J + t ib = Electromagnetic Theory 7 Maxwell s Equation () Note Continuity equation ij = : for steady state current ρ ij = : time varying current t Vector identity ρ i( H) = = ij contradiction ij = t ρ D i( H ) = = ij + = i J +, where ρ = id D H = J + t Cf) Lorentz force equation, F = q( E+ u B) Displacement current density. [A/m ] Time varying electric field and induced magnetic field coupling Electromagnetic Theory 8

Integral Form of Maxwell s Equation Cf) Differential form Point function B E = Apply stokes's theorem over open surface S with contour C B ( E) ids= ds S i S B dφ Edl i = ds= c i : Faraday's law s dt D H idl = I + ds c i : Ampere's circuital law s 3 id= ρ Dids = Q : Gauss law s 4 ib = Bids= : No isolated magnetic charge s Electromagnetic Theory 9 Ex. 7-5 (a) Displacement current = conduction current conduction current current on the wire. Apply circuit theorem dvc ic = C = CV ωcosωt dt Displacement current. eminding D H = J + t A Assuming the area A, plate separation d, permitivity μ, then C = ε d Assume E is uniform in the dielectric (ignoring fringing effects) then v c V E =, D= εe = ε sinωt d d D A id = ids = ε Vωcosωt = CV ωcosωt = i A d C Electromagnetic Theory

Ex. 7-5 (b) Magnetic field intensity reminding Ampere's law D D H = J +, H dl = I + ds i C i S surface S with ring C surface S with ring C D=, Hidl = π rh C ( Symmetry H I = around the wire along the contour C) constant Jids = ic = CV ωcosωt S no conduction current, but displacement current CV I = id, Hφ = ωcosωt π r φ φ Electromagnetic Theory Potential Functions () Vector magnetic potential, A B= A (Solenoidal nature of B) Vector identity ib=, i( A) = ecall Faraday's law Curl free B A E = E = ( A) E+ = Vector identity ( V ) = and reminding E = V for electromagnetics A A E+ = V for time varying i.e) E = V [ V / m] Electromagnetic Theory

Potential Functions () A Cf) Static = E = V Time varying E is induced by charge distribution ρ and time varying magnetic field time varying current, J B also depends on A E, B are coupled ρ μ J V = dv ', A dv ' 4πε = v' 4π : From the static condition v' These are solution of poisson equation ρ V and A μ J ε = = The time-retardation effects associated with the finite velocity of propagation is neglected Electromagnetic Theory 3 Potential Functions (3) Quasi-static fields - ρ and J vary slowly with time - the range of interest is small compared to the wavelength cf) Frequency is high, and is large compared to wavelength : time-retardation effect must be included. From the equations A D B= A, E = V, H = J + ( B= μ H, D = ε E) A A= μj + με V ecalling vector identity A= ( ia) A Electromagnetic Theory 4

Potential Functions (4) V A ( ia) A= μj με με A V A με = μ J + A+ με i - we only designated A= B but we are free to choose ia - vector A will be specified by giving A and ia V V - let ia+ με =, for static = ia= Lorentz gauge for potentials Electromagnetic Theory 5 Potential Functions (5) V cf) For static, ia = A =, = then vector poisson equation A= μ J - Then nonhomogeneous wave equation for vector potential becomes A A με = μ J : Vector potential wave equation

Potential Functions(6) Scalar potential wave equation A A E = V, id= ρ iε V + = ρ ρ V V + ( ia) =, ia= με ε V V με = ρ ε Electromagnetic Theory 7 Boundary Condition () Electric field's boundary condition B Eidl = ds D ds = ρdv c i... s i s... v From equation B i ds when Δ h, since area S s E = E ( E Δw E Δ w= ) t t t t Electromagnetic Theory 8

Boundary Condition () From equation ( Dids = Din + Din) Δ S = ni( D D ) Δ S = ρ s sδs n i( D D ) = ρ, D D = ρ s n n s Magnetic field's boundary conditions D H idl = c J + ids s HiΔ w+ Hi( Δ w) = JsnΔw, H t Ht = Jsn ie.) n ( H H) = Js cf) n & J are perpendicular to each other s Electromagnetic Theory 9 Boundary Condition (3) note) The tangential component of the H field is discontinuous across an interface where a free surface current exists if both media have finite conductivity, currents are defined by volume current density surface currents do not exists H = H t t i.e) discontinuous only for interface with an ideal perfect conductor or super conductor. i B= B = B n n Electromagnetic Theory 3

Interface Between Two Lossless Linear Media Linear media permitivity : ε, permeability : μ Lossless σ= Assume, at interface, no free charge ρ = no surface currents J = S S D ε B μ E = E =, H = H = μ t t t t t t Dt ε Bt D = D ε E = ε E, B = B μ H = μ H n n n n n n n n Electromagnetic Theory 3 Interface between a Dielectric and Perfect Conductor () Good conductor perfect conductor Interior of perfect conductor (surface charge only) : E ( E, D) ( B, H) are zero in the interior of a conductor cf) In static case, E, D may be zero, but H, B may not be zero. E =, H =, D =, B = Electromagnetic Theory 3

Interface between a Dielectric and Perfect Conductor () E t =, Et = n ( H H ) = J, H = s t H t = Jsn, if Jsn = H t = n i ( D D ) = ρ, D =, D = ρ B s n n s =, B = n n note) n : outward normal from medium At an interface between a dielectric and a perfect conductor : E is normal to and points away from(into) the conductor surface ρs E = E n = ε : H is tangential to the interface with a magnitude of H = H t = Js cf ) direction n ( H H ) = J s Electromagnetic Theory 33 Wave Equation and Their Solutions () A Wave equation : A με = μ J V ρ V με = ε Solution : i Assume an elemental point charge at time t, ρ() t Δν located at the origin of the coordinates. i Spherical coordinates. i V depends only on. and t because of spherical symmetry. (No dependence on φ) i Except at the origin, V V με = Electromagnetic Theory 34

Wave Equation and Their Solutions () New variable V(, t) = U(, t) U U U(, t) = U + = U + U U U U U + = + + U U U(, t) U(, t) με =, i.e) με = Any function of ( t με ) or of ( t+ με ) will satisfy the differential equation f( t με ) is a wave equation which travels away from the origin Electromagnetic Theory 35 Wave Equation and Their Solutions (3) f( t+ με ) is a wave equation which travels to the origin physical nonsense U(, t) = f( t με ) the function at +Δ at a later t+δt. U( +Δ, t+δ t) = f t+δt ( +Δ) με = f( t με ) if Δ t =Δ με. Δ Δ = lim = u = : velocity of propagation Δt με Δ t Δt με V(, t) = f t u Determine f t u Electromagnetic Theory 36

Wave Equation and Their Solutions (4) ecall potential function induced by a static point charge ρ() t Δν at the origin ρ t Δv' ρ() t Δv' u Δ V( ) =, Δf t = 4πε u 4πε ρ t u V(, t) = dv' 4πε : etarded scalar potential v' Scalar potential at a distance from the source at time t Depends on the value of charge distribution at an earlier time etarded vector potential J t μ u At (, ) = dv' 4π v ' t u Electromagnetic Theory 37 Source Free Wave Equation If the wave is in a simple (linear, isotropic and homogeneous) non conducting medium. i.e) ε, μ ( σ=) H E E = μ, H = ε ie =, ih = From vector identity E E = μ ( H) = με cf) A B C= B( AC i ) C( AB i ) = B( AC i ) ( ABC i ) ( ie) ( i ) E = E E E με = if με = u E H E =, H = : Homogeneous vector wave equation u u Electromagnetic Theory 38

Time Harmonic Fields Maxwell's equations - linear differential equations - sinusoidal time variation of source functions at given frequency - E, H are sinusoidal with the same frequency Time harmonic steady state sinusoidal Phasors : Amplitude and phase information independent of time cf) j t e ω : time dependent factor Electromagnetic Theory 39 Time Harmonic Electromagnetics () Vector phasors of field vectors : depend on space coordinates jωt Exyzt (,, ; ) = e Exyze (,, ), where Exyz (,, ): vector phasor : complex quantity jωt Exyzt (,, ; ) = e jω E( xyze,, ) where jω E( x, y, z): vector phasor Exyz (,, ) jωt Exyztdt (,, ; ) = e e jω Exyz (,, ) where : vector phasor jω i.e) jω, ( jω), t t j ω Electromagnetic Theory 4

Time Harmonic Electromagnetics () Maxwell's equations Vector field phasors ( E, H) Source phasors ( ρ, J), Simple (linear, isotropic and homogeneous) media E = jωμ H, H = J + jωε E jωt ρ Assuming e ie =, ih = ε Time harmonic wave equation for V and A ρ Non-homogeneous helmholtz's equations V k V ω, : A+ k A= μ J wave-number u + = ε where k = ωμε k = ω με= Electromagnetic Theory 4 Time Harmonic Electromagnetics (3) A V cf ) A με = μ J ( ia + με = ia + jωμεv = ) V ρ V με = ε E E με = H H με = Phasor solution ω jω ( t ) j u u ρe jωt ρe 4πε v' 4πε v' jk V(, t) = dv' V( ) e = dv' e ρe V( ) = dv' 4πε [V] v' jk μ Je A ( ) = dv' 4π [Wb/m] v' jωt Expressions for the retarded scalar and vector potentials due to time harmonic sources Electromagnetic Theory 4

Time Harmonic Electromagnetics (4) jk k cf) e = jk +...: Taylor series expansion. ω π f π k = = =, u = fλ u u λ jk ρ if k = π e =, then V ( ) = dv ' λ 4 πε static potential v' Procedure for determining the electric and magnetic fields due to time harmonic charge and current distributions. Find phasors V( ) and A( ) A. Find phasors E ( ) = V jω A cf) E= V t B ( ) = A 3. Find instantaneous Et (, ) Electromagnetic Theory 43 Source-free Fields in Simple Media () Source free fields in simple media E = jωμh H = jωε E ie = ih = E+ k E = Homogeneous vector Helmholtz s equation H + k H = and k = ωμε Principle of duality : Source free Maxwell's equations in a simple media are invariant under the linear transformation E μ E' = ηh, H' =, η = η ε Electromagnetic Theory 44

Source-free Fields in Simple Media () If simple medium is conducting i.e) σ J = σe σ H = ( σ + jωε) E = jω ε + E = jωεc E jω σ and εc = ε j [F/m] : complex permitivity ω cf) out of phase polarization : power loss to overcome a fractional damping mechanism caused by the inertia the charged particle finite conductivity ohmic losses Complex permitivity ε = ε' jε'' [F/m], where ε '' : out of phase polarization and finite conductivity c equivalent conductivity σ = ωε'' representing all losses Electromagnetic Theory 45 Source-free Fields in Simple Media (3) Complex permeability : out of phase component of magnetization μ = μ' jμ'', where μ' μ'' for ferromagnetic materials μ = μ' Complex wavenumber k c = ω με = ω μ( ε' jε'') : in a lossy dielectric c Loss tangent ε'' σ tan δc =, where δc : loss angle ε' ωε ε '' loss tangent ε ' Good conductor & Good insulator σ ωε : Good conductor σ ωε : Good insulator Electromagnetic Theory 46

Source-free Fields in Simple Media (4) Cf) Electric hertz vector, π π A= με, V = iπ : combine the vector and scalar potential and satisfy the Lorentz condition A E = V V ia + με = combine continuity equation with J and ρ ρ P ij + =, J =, ρ = ip Single vector equation π P π με = ε π P E = ( iπ) με = π ε π H = ε t Electromagnetic Theory 47 The electromagnetic spectrum