Chapter 7. Time-Varying Fields and Maxwell's Equations

Chapter 7. Time-Varying Fields and Maxwell’s Equations Electrostatic & Time Varying Fields

• Electrostatic fields

ED 0,    BH=J0, DE  1 HB  • In the electrostatic model, electric field and magnetic fields are not related each other. Faraday’s law

• A major advance in EM theory was made by M. Faraday in 1831 who discovered experimentally that a current was induced in a conducting loop when the magnetic flux linking the loop changed.

d electromotive force (emf): V   V dt Faraday’s law

• Fundamental postulate for electromagnetic induction is dB B V  V  EE dl  dl   d sE  dtCC S t t • The electric field intensity in a region of time-varying magnetic flux density is therefore non conservative and cannot be expressed as the negative gradient of a scalar potential.

• The negative sign is an assertion that the induced emf will cause a current to flow in the closed loop in such a direction as to oppose the change in the linking magnetic flux  Lentz’s law

d B V  E dt t 7-2.3. A moving conductor in a static magnetic field

• Charge separation by magnetic force

FuBm  q 

• To an observer moving with the conductor, there is no apparent motion and the magnetic force can be interpreted as an inducted electric field acting along the conductor and producing a voltage.

2 VdluB 21 1

• Around a circuit, motional emf or flux cutting emf

VdluB 21 C A moving conductor in a static magnetic field

•Example 7-2

(a) Open voltage V0 ? (b) Electric power in R (c) Mechanical power required to move the sliding bar

1 ()a V V VuB  dl  a u  a B a dl   uB h 012C 2 xz 0 y 0 2 VuBh uB h ()bI 00  P IR2  0  W RRe R

()cP MMFuF , M mechanical force to counteract the magnetic force Fm 1 FBaFFIdl IBh  N  mag2 x0 M mag 22 uB h uBh u2  Bh IP0 Fa 00   W RRRMx M

 PPeM A moving conductor in a static magnetic field

• Example 7-3. Faraday disk generator

4 VdlrBdruB =  a  a a  00C 3   zr  2 0 Bb Brdr 0  V 0 b 2 Magnetic force & electric force FEuB q 

When a charge q0 moves parallel to the current on a wire, the magnetic force on q0 is equivalent to the electric force on q0. At the rest frame on wire At the moving frame on charge

' FvBB  q0  FEE  q0

I 22 L L001/ vc  /1/ vc

E'

I

 To observer moving with q0 under E and B fields, there is no apparent motion. But, the force on q0 can be interpreted as caused by an electric field, E’. 7-2.4. A moving circuit in a time-varying magnetic field

 To observer moving with q0 under E and B fields, there is no apparent motion. But, the force on q0 can be interpreted as caused by an electric field, E’.

EEuB 

Now, consider a conducting circuit with contour C and surface S u moves with a velocity u under static E and B fields.

Changing in magnetic flux due to the circuit movement produces an emf, V: d   V dt B On the other hand, the moving circuit experiences an emf, V’, due to E’:

E'dl V C E '

 Is it true that VVBE ' VVBE ' ??

EEuB  

B From the Faraday's law of Esdl d , CS t by replacing EEEuB with =  -  , u B EsuBdl d    dl V CSCt

time variation motional emf at rest dd Note that Vdl E VdB   Bs E ' C dt dt S d B Therefore, we need to prove that Bsdd s uB  dl dtSS t  C VVBE ' ?? • Time-rate of change of magnetic flux through the contour

dd 1  Bsdttdtdlim B  s21 B s  dt dtSSt 0  t 21  S  B t BBtt  t   t H.O.T., t d Bt 1   u Bsdd slim Bs  dd21  Bs H.O.T. dtSS tt 0 t   S21  S 

• In going from C1 to C2, the circuit covers a region bounded by S1, S2, and S3

BBsBsBs dddd 0  VSSS213 213 emf = V  E dl ddlts u Bs d  Bs d  t uB   dl C 321SS  C 21 d Bt Bsdd s  uB  dl dtSS t   C • Therefore, the emf induced in the moving circuit C is equivalent to the emf induced by the change in magnetic flux

d VV V : same form as not in motion. EB' dt A moving circuit in a time-varying magnetic field

•Example 7.3 – Determine the open-circuit voltage of the Faraday disk generator

2 bt b Bsd  B rdrd  B t S 0000 2 d Bb2 V   0 B dt 2

Compare! B VdldEsuB  dl E ' CSCt 4 VdlrBdruB =  a  a a  Ezr'0C 3     2 0 Bb Brdr 0 0 b 2

VVB  E ' B dd VdldE ' EsuB  dlVd  Bs CSCt B dt dt S

(Example 7-4) Find the induced emf in the rotating loop under Ba()tBt yo sin

(a) when the loop is at rest with an angle . BsdBthwBhwt  asin a  sin cos   yo n 0 d   VBStShw cos cos ,   : the area of the loop a dt 0  (b) When the loop rotates with an angular velocity ω about the x-axis

13ww VdlaaBtadxaaBtadx uB [( )   ( sin )]   [( ) ( sin )]  anyxnyxC 242200  w 2(BtthBSt sin sin ) sin sin ( = t) 2 00

22 VVVBSEaa'0 (cos t  sin t )  BS  0 cos 2 t

Compare!  1 tsinsinsBattSBSt    cos  BStcos t  BStin 2 n 00 2 0 dd 1  VBStBStB  00sin 2  cos 2 VV dt dt 2 BE' 7-3. How Maxwell fixed Ampere’s law?

• We now have the following collection of two curl eqns. and two divergence eqns. B EHJ ,  = t

DB ,  =0

• Charge conservation law  the equation of continuity  J  t • The set of four equations is now consistent with the equation of continuity?

Taking the divergence of  HJ = , HJ 0  H  0 from the vector null identit y J does not vanish in a time-varying situation and this equation is, in general, not true. Maxwell’s equations

• A term ∂ρv /∂t must be added to the equation.

 D HJ 0    J   D   tt  D HJ t E B J  000t • The additional term ∂D/∂t means that a time-varying electric field will give rise to a magnetic field, even in the absence of a free current flow (J=0).

• ∂D/∂t is called displacement current (density). Maxwell’s equations

• Maxwell’s equations Continuity equation B E   t J  D t HJ=  t Lorentz’s force

D  B=0 F  qE() B

• These four equations, together with the equation of continuity and Lorentz’s force equation form the foundation of electromagnetic theory. These equations can be used to explain and predict all macroscopic electromagnetic phenomena.

• The four Maxwell’s equations are not all independent – The two divergence equations can be derived from the two curl equations by making use of the equation of continuity Maxwell’s equations

• Integral form & differential form of Maxwell’s equations

Differential form Integral form Significance

B Bd E  Edl  dS  Faraday’s law t CStdt

D D HJ =   Hsdl I d Ampere’s law t CSt

D   DsdQ Gauss’s law  C

B=0 Bsd  0 No isolated S magnetic charge Maxwell’s equations

•Example 7-5 – Verify that the displacement current = conduction current in the wire.

iC : conduction current in the connecting wire

dC iCC 110 CVcos t dt  A C  1  d E  C : uniform between the plates d C  Vt0 sin V  DE  0 sin t d

D V0 idD   s cos  tACVti  10 cos C A td Maxwell’s equations

•Example 7-5 – Determine the magnetic field intensity at a distance r from the wire

Two typical open surfaces with rim C may be chosen:

(a) a planar disk surface SS12, b a curved surface

For the surface S1, D = 0 CV HJsdl2coscos rH d i C V t   H10 t  C 10 CS1 2 r

 Since the surface S2 passes through the dielectric medium,

no conducting current flows through S2  displacement current D 2 rH idi  s  DCA t CV Ht10cos  2 r 7-5. Electromagnetic Boundary Conditions

B Edl  dS 0 when  h 0 CSt

 EE12tt

DD  Hssdl  J d;  d 0 when h 0 CStt  S 

 aHHJHHJnsttsn212   12 

DaDDdS d    whenh 0 sVns2 12

 DD12nns B dS 0  s

 BB12nn0  1122H n H n

  Electromagnetic Boundary Conditions

Both static and time-varying electromagnetic fields satisfy the same boundary conditions:

EE12tt  The tangential component of an E field is continuous across an interface.

H12ttsnHJ  The tangential component of an H field is discontinuous across an interface where a surface current exists.

B12nn B  The normal component of an B field is continuous across an interface.

DD12nns  The normal component of an D field is discontinuous across an interface where a surface charge exists. Boundary conditions at an interface between two lossless linear media

Between two lossless media () with  = 0, and S = 0, Js = 0

D1t 1 EE12tt  D22t 

B B1t 1 DEH  ,  HH12tt  B22t  

D12nnDEE 1122 n  n

B BHH  12nn 1122 n n Boundary conditions at an interface between dielectric and perfect conductor

In a perfect conductor (  infinite, for example, supercondiuctors),

EDHB22220, 0

Medium2 Medium 1(dielectric) (perfect metal)

EE12tt 0  0

HJ12tts H 0

DD12nns   0

BB12nn  0  0 Boundary conditions

• Table 7-4. Potential functions

• Vector potential BA  T  B 0

• Electric field for the time-varying case.

A EAE 0 tt A E  V t A  E V   V/m t

Due to charge distribution 

Due to time-varying current J Wave equation for vector potential A

A From HB AD,  E  V , t D HJ   t A AJ   V  tt  2 2 V A AA  J   2 or tt 2A V 2 (Lorenz  condition, or gauge) A 2 JA 0 tt   2A (# Show that the Lorentz condition is consistent 2A J with the equation of continuity. Prob. P.7-12) t 2  Non-homogeneous wave equation for vector potential A 1 traveling wave with a velocity of  Wave equations for scalar potential V

A ρ From EV  andE   v , t ε

A ρv V  t ε ρ V 2V  AA v με ttε

2 2  V ρ V με v for scalar potential V t2 ε

V A  0 t  The Lorentz condition uncouples the wave equations for A and for V.

 The wave equations reduce to Poisson’s equations in static cases. Gauge freedom A • Electric & magnetic field EBA V , =  t • Gauge transformation

If AA , B remains unchanged. AA  E  VV        tt  tt  Thus, if VVV is further changed to , E also remains same. t

 Gauge invariance  E & B fields are unchanged if we take any function (x,t) on simultaneously A and V via:

AA 2 V   A  0 2 0  2 VV t t t

 The Lorentz condition can be converted to a wave equation. 7-6. Solution of wave equations

• The mathematical form of waves

f (x, t =0) t = 0 t = t0

up x = upt0

x 22ff1 fxt, fxutp 222 0 : wave equation xutp Wave equation

• Simple wave – http://navercast.naver.com/science/physics/1376

xt yxt,cos A A cos kx t  T 22 2,fk  , u  Tk p Solution of wave equations from potentials

2  V ρ : Nonhomogeneous wave equation 2V με v R 2 for scalar electric potential t ε d'

 t First consider a point charge at time t, (t)v’, located at a origin. Except at the origin, V(R) satisfies the following homogeneous equation ( = 0):  221 V Since V2  R for spherical symmetry VR , ,  VR RR 2 1 2 VV  22R 0 RR R  t 1 Introducing a new varible, VRt , URt , R 22UU R 1  0,URt Ut or UR  utu ,  22 pp Rt up  R  Thus, we can write in a form of VRt ,  Vt  u p Solution of wave equations



The potential at R for a point charge  t is, R  t  d' VR t 4 R   

R  tRu/   VRt,  Vt   p Vt  Ru/ p  u p 4 R R Now consider a charge distribution over a volume V’ . d R 1  Rt,/ Rup  VRt,  d  V 4 V  R V’    R,t  J Rt,/ Rup  A Rt, d   Wb/m 4 V R

 The potentials at a distance R from the source at time t depend on the values of  and J at an earlier time (t- R/u)  Retarded in time

 Time-varying charges and currents generate retarded scalar potential, retarded vector potential. Source free wave equations

Maxwell’s equations in source-free non-conducting media (ε, μ, σ=0). HE EHEH ,  = ,   0,  =0 t t

• Homogeneous wave equation for E & H.

E EE 22 EH= t

2E EH    t t2 2 2 1  E E 22 0 utp  2 2 1  H In an entirely similar way,  H220 utp  Review –The use of Phasors

Consider a RLC circuit di 1 LRiidtVtitI ,   cos  t    dt C  0

Phasor method (exponential representation)

Vt Vcos t Re  Vej0 ejt  Re Ve jt  00   s  itRe Iejjt e Re Ie jt  0 s   VVe j0 s 0  (Scalar) phasors that contain amplitude and phase information j IIes  0 but are independent of time t.  di I If we use phasors in the RLC circuit,  RejIe jtjt , idt Re s e  s   dt j 1 Rj L  Iss  V C    1  it it() Re Vs / R  j L  e C Time-harmonic Maxwell’s & wave equations

• Vector phasors.

jt Exyzt,,, ReExy, , z e  jt Hxyz,,,t  Re H x,,y ze 

• Time-harmonic (cos t) Maxwell’s equations in terms of vector phasors

D     E B  E  t EH j  B=0 H 0 D HJ=  HJ   j E t Time-harmonic Maxwell’s & wave equations

Time-harmonic wave equations (nonhomogeneous Helmholtz’s equations) 2VRt(,)  (,)Rt 2VRt(,)   t2  2 2 ()R VR()  j VR ()   2  k wave number =  up   22VR() kVR ()    2A 22AA(R)()k R  J 2AJ    2  t  Time-harmonic retarded potential

• Phasor form of retarded scalar potential

1  Re  jkR costkx  VR  d  V 4 V  R eeejtkx  jt kx   • Phasor form of retarded vector potential.

 J Re  jkR ARd   Wb/m V  4  R  R • When kR 2 1  kR22 ejkR jkR 1   ...  1 2 VR,A R  static expressions(Eq. 3-39 & Eq. 5-22) Time-harmonic retarded potential

• Example: Find the magnetic field intensity H and the value of β when = 90 9 Eazt,5cos10y  t z  V/m

aaaxyz 9  10 (1/s) 1  Hz  jxyz0   jz  jz 05e 0 Eaze y 5 1    aaa55()eeHzjz   jz   1   x xxx H z        E jz00  j 0    2 11   j z EHaaz5yxyH 2 e jjz   0 3  310   rad/m 000c

 jz  j10 z Hazexx5  a 0.0398 e 0

  jz 9 Hazt,xx 5 e  a 0.0398 cos 10 t  10 z 0 The EM Waves in lossy media  • If a medium is conducting (σ≠0), a current J=σE will flow  D   HJ=  HE  jj   EE  jc , t j   complex permittivity  j j c   • When an external time-varying electric field is applied to material bodies,  small displacements of bound charges result, giving rise to a volume density of polarization. This polarization vector will vary with the same frequency as that of the applied field. • As the frequency increases, the inertia of the charged particles tends to prevent the particle displacements from keeping in phase with the field changes, leading to a frictional damping mechanism that causes power loss. • This phenomenon of out of phase polarization can be characterized by a complex electric susceptibility and hence a complex permittivity.

• Loss tangent, δc The EM Waves in lossy media

• Loss tangent, δc   E  HJ=  j   E  tj    HE jjc   E j  j c J    tan , : loss angle  cc  c

 • Good conductor if >>1 E 

 • Good insulator if <<1 

• Moist ground : loss tangent ~ 1.8104@1kHz, 1.810-3 @10GHz The electromagnetic spectrum

• Spectrum of electromagnetic waves