Maxwell’s Equations

Chapter 7.

Time-Varying Fields and

Maxwell’s Equations

Electrostatic & Time Varying Fields

• Electrostatic fields

• In the electrostatic model, electric field and magnetic fields are not related each other.

0, 0,

1

 





   

  





E D

B H = J

D E

H B





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.

 

electromotive force (emf): d V

V dt

  

Faraday’s law

• Fundamental postulate for electromagnetic induction is

• 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.

d

V dt t

 

    

 E B

• 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

t

   

 E B

C C S

V d V dl dl d

dt t

 

      

 ^{E }  ^{E }   ^{B  s}

  

7-2.3. A moving conductor in a static magnetic field

• Charge separation by magnetic force

• 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.

• Around a circuit, motional emf or flux cutting emf

m

 q 

F u B

 

2

21 1

V   ^{u B }  dl

 

21 C

V    ^{u B }  dl

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

0 1 2 2 0 0

2 2 0

0 0

( )

( )

       

    



^ 

^



W

x z y

C

e

a V V V dl u B dl uB h

V uB h uB h

b I P I R

R R R

 

     

1 2 0

2 2 2

0 0

0

( )  , 

      

    

 



F u F F

F B a F F

F a

 mechanical force to counteract the magnetic force _m

N

W

M M M

mag x M mag

M x M

e M

c P

I dl IB h

u B h u B h

I uB h P

R R R

P P

A moving conductor in a static magnetic field

• Example 7-3. Faraday disk generator

    ^{ }

 

4

0 0

3 0 2

0 0

=

V

2

z r

C

b

V dl r B dr

B rdr B b





 

 

    

  

 



u B  a a  a



Magnetic force & electric force

0

B

 q 

F v B F

 q

E

 

 q   F E u B

 

 

0 1 /  0 / 1 /

    

L L v c v c

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

I

I

E'

 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

    E E u B

u

 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’.

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

'

dl  V

 ^{E }



B

d V

dt

  

Changing in magnetic flux due to the circuit movement produces an emf, V:

On the other hand, the moving circuit experiences an emf, V’, due to E’:

 Is it true that V

 V

    E E u B

  ^V  

C

dl

d

dl

t

     

 ^{E }   ^{B  s}  ^{u B }

  

time variation at rest

motional emf

 

E E E u B

by replacing with = - ,

u

  





 

From the Faraday's law of ,

C dl S d

t '

??

B E

V  V

     

^

B S

d d

V d

dt dt

'

   ^{E } 

E C

V dl

 

S S C

d d d dl

dt t

   

 ^{B  s}    ^{B  s}   ^{u B }

Note that

Therefore, we need to prove that

• Time-rate of change of magnetic flux through the contour

   

     

 

2 1

2 1

2 1

0

2 1

0

lim 1

lim 1

H.O.T.,

H.O.T.

S t S S

S S t S S

d d

d t t d t d

dt dt t

t t t t t

t d t

d d d d

dt t t

 

 

        

      



  

       

  

   

B s B s B s

B B B

B s B s B s B s

  

    u

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

'

    

E B

V V V d

dt

• Therefore, the emf induced in the moving circuit C is equivalent to the emf induced by the change in magnetic flux



 

= C

emf V dl

: same form as not in motion.

 

   

2 1 3

2 1

2 1 3

3 2 1

 0

    

       

    



   

  

  

B B s B s B s

s u B s B s u B

B s B s u B

   

  

  





V S S S

S S C

S S C

d d d d

d dl t d d t dl

d t

d d dl

dt t

'

??

B E

V  V

A moving circuit in a time-varying magnetic field

• Example 7.3

– Determine the open-circuit voltage of the Faraday disk generator

 

0 0

0 0 2 0

2

2

b t S

B

d B rdrd B t b

B b V d

dt

  



   

    



 

 

  ^{ }

' 0

3 0 2

0 0

=

2

E z r

C

b

V dl r B dr

B rdr B b





 

 

    

  

 



u B  a a  a



 

'

E C S C

V dl d dl

t

 

    







  

'

B E

V  V

Compare!

(Example 7-4) Find the induced emf in the rotating loop under

(a) when the loop is at rest with an angle .

(b) When the loop rotates with an angular velocity ω about the x-axis

  ^{ }

 

0

0

sin sin

cos cos , : the area of the lo cos

op

y o n

a

d B t hw B hw t

V d B S t S hw

dt

 

  



   

     



       

 

0 0 0

0 0

t sin cos sin 1 sin 2

2

1 sin 2 cos 2

2

n cos

B

t t S B S t B S t B S t

d d

V B S t B S t

dt dt

  t 



 

 

      

  

      

B  a

 

0 0

[( ) ( sin )] [( ) ( sin )]

2 2

2( sin sin ) sin sin ( = t)

2

a C n y x n y x

w w

V dl a a B t a dx a a B t a dx

w B t t h B S t

   

       

         

 



 



2 2

' 0 (cos sin ) 0 cos 2

E a a

V V V^  B S t  t  B S t Compare!

 

'

E C S C

V dl d dl

t

 

    







  

_

( )t  _yB_o sint

B a

'

B E

V  V

7-3. How Maxwell fixed Ampere’s law?

• We now have the following collection of two curl eqns. and two divergence eqns.

• Charge conservation law  the equation of continuity

• The set of four equations is now consistent with the equation of continuity?

, = , =0

t



    



  

E B H J

D B

 

t



    J 



   

does Takin

not g the dive

vanish in r

a time-varying situation and this equation is, i

gence of = ,

0 0 from the vector null

n general, not true

ident y

.

it



       



 

J

H J

H J H

  



Maxwell’s equations

• A term ∂ρ

/∂t must be added to the equation.

• 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).

  ⁰

t

t t

 

 

   

               

 







 

D H J D

H J J D

   

0 0 0

B E

   ^ t

  

J 

Maxwell’s equations

• Maxwell’s equations

• 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

=

=0

t t



   



  



 



E B

H J D

D B





t



    J 



( )

F ^ q E ^{ }  B

Continuity equation

Lorentz’s force

Maxwell’s equations

• Integral form & differential form of Maxwell’s equations

Differential form Integral form Significance

Faraday’s law

Ampere’s law

Gauss’s law

No isolated magnetic charge

t

   

 E B

= t

  

 H J D



  D 

 B  =0

C S

B d

dl dS

t dt

 

    

 ^E  



C

dl I

d

t

  

 ^{H }   ^{D  s}



C

d  Q

 ^{D s }



S

d  0

 ^{B s }



Maxwell’s equations

• Example 7-5

– Verify that the displacement current = conduction current in the wire.

1 1 0

1

0

cos

: uniform between the plates sin

C C

C

i C d C V t

dt C A

d

E d

D E V t

d

  





  

 





  

0

cos

cos

D C

A

i d V t A C V t i

t  d    

  

   ^{D  s}       

0

sin

C

V t

 ^ 

: conduction current in the connecting wire

i

Maxwell’s equations

• Example 7-5

– Determine the magnetic field intensity at a distance r from the wire

1,  2

Two typical open surfaces with rim may be chosen:

(a) a planar disk surface b a curved surface C

S S

1

1 0 1 0

2

2

,

2 cos cos

2

2

C S C

dl rH d i C V t H C V t

r

S

S displacement current rH

 



    





     



 

D

H J s



For the surface S = 01

Since the surface passes through the dielectric medium, no conducting current flows through

1 0 cos

2

D C

i A d i

t

H C V t

 r  



   



 



7-5. Electromagnetic Boundary Conditions

1t 2t

E E

 

 

 

2 1 2

1 2

1 2 1 1 2 2

0

0

0

n n s

n n n n

n s

s V

s

dS d whe

D D

B B B

H d

H

n h

S







 



   

 

  

  



 



 



D a  D D





0 0

C S

dl B dS when h

t

      

 ^E  



; 0 0

C

dl

J d

d when h

t t

 

   

              

 ^{H }  ^{D  s}  ^{D  s}



   

2 1 2

n s t t sn

a H H J H  H J

    

Electromagnetic Boundary Conditions

1t 2t

E  E

1n 2n s

D  D  

1t 2t sn

H  H  J

1n 2n

B  B

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

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

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

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

 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 

1 1

1 2

2 2

1 1

1 2

2 2

1 2 1 1 2 2

1 2 1 1 2 2

t

t t

t

t

t t

t

n n n n

n n n n

E E D

D

H H B

B

D D E E

B B H H









 

 

  

  

  

  

 

    

 

 

D E H , B

In a perfect conductor (  

Medium 1(dielectric) Medium2 (perfect metal)

1 2

1 2t

1 2n

1 2n

0 0

0

0

0 0

t t

t s

n s

n

E E

H J H

D D

B B



 

 

 

 

Boundary conditions at an interface between dielectric and perfect conductor

2

0

,

0

E D H B

    

Boundary conditions

• Table

7-4. Potential functions

• Vector potential

• Electric field for the time-varying case.

 

  T

B A

 

 

0

V/m

t t

t t

V V

   

             

 

 

 

 



 



E A E A

E A

E A

 ^{    B 0} 

Due to charge distribution 

Due to time-varying current J

Wave equation for vector potential A

 

2 2

2

2 2

2

From , ,

or

V t

t

t V t

V

t t

V

t t

t

  

 

  

 





 

  

           

  

       

   

          

 

 

             

   

            

    





H B A D E A

H J D

A J A

A A A

A

J A

A

J A

J A





Non-homogeneous wave equation for vector potential A

traveling wave with a velocity of 1



(# Show that the Lorentz condition is consistent with the equation of continuity. Prob. P.7-12)

Wave equations for scalar potential V

 

2 2

2 2

     



  

        

 

        

 

    

 A

A A





 

From

,

v

v

v

E V A and E

t

V t

V V

t t

V V

t

ρ ε ρ

ε

ρ με

ε

με ρ

ε

V 0

 ^ t

    

  

  A 

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

• Electric & magnetic field

• Gauge transformation

 Gauge invariance

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

If , B remains unchanged.

Thus, if is further changed to , also remains same.



 



  

     

             

   

 A A

A A

E

E

V V

t t t t

V V V

t

, =

V t

    



E A B A





  

  



V V

t

 ^ 0

  

A 

 V

t

2 2

2 0

  ^ 

  

t

The Lorentz condition can be converted to a wave equation.

7-6. Solution of wave equations

• The mathematical form of waves

x t = t₀

t = 0

x = u_pt₀ u_p

f (x, t =0)

   

1 0 : wave equation ,

p

f x t p f f

x u t

f x u t     

 

 

Wave equation

• Simple wave

– http://navercast.naver.com/science/physics/1376

 

 

2 2

2 , , _p

x t

y x t A A kx t

T

f k u

T k

  



  

 



 

 

    

   

Solution of wave equations from potentials

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):

   

   

   

2 2

2

2 2

2 2

2 2

2 2

Since 1 for spherical symmetry , ,

1 0

Introducing a new varible, , 1 ,

0 , or , 1

Thus, we can write

p p

p

V R V V R V R

R R

V V

R R R R t

V R t U R t R

U U R

U R t U t U R u t u

R t u

 



 

  

     

    

      



 

 

           

 in a form of

 

p

V R t V t R u

 

   

2 2

2

    

 V

V t

με ρ

ε

: Nonhomogeneous wave equation for scalar electric potential

 

 ^d'

R

Solution of wave equations

Now consider a charge distribution over a volume V’ .

 

 R

V’

d

R

R

 

p

V R t V t R u

 

   

  



/ 4

p p

t R u V t R u

R







  

   

 

   

The potential at R for a point charge is,

4

t V R t

R





 

 





 

  

 

R

    _{ }

    _{ }

, /

, 1 V

4

, /

, Wb/m

4

p V

p V

R t R u

V R t d

R R t R u

R t d

R



 

 







  



  





 ^J

A

 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).

• Homogeneous wave equation for E & H.

, = , 0, =0

t

 ^ t  ^

      

 

H E

E H  E  H

= ^t

 

H E

 

2 2

2 2

2 2

2 2

t

In an entirely similar wa

1 0

y, 1 0

p

p

t

u t

u t

 

   



   

 

     

 



 E E

H H

E H E

 

 E   E E

Consider a RLC circuit

Phasor method (exponential representation)

     

     

0

0 0

0

cos Re Re

Re Re

j j t j t

s

j j t j t

s

V t V t V e e V e

i t I e e I e

 

  

 ^{ }

    

 

   

Review –The use of Phasors

   

 

1 , cos  

 



L di Ri idt V t i t I t

dt C

 

If we use phasors in the RLC circuit, Re , Re 1

( ) Re / 1

j t s j t

s

s s

i t s

I

di j I e idt e

dt j

R j L I V

C

i t V R j L e

C

 



 

 

 

 

   

 

  

      

   

       



0 0 0

j s

j s

V V e I I e ^



 (Scalar) phasors that contain amplitude and phase information but are independent of time t.

Time-harmonic Maxwell’s & wave equations

• Vector phasors.

• Time-harmonic (cos 

   

   

, , , Re

, , ,

, Re

, , ,

j t

j t

x y z t e

x y z

x y

x e

z z

t y





 

  

 

  

E H

E H

0 j

j



 



 

  

 

   E

E H

H

H J E





=0

=

t

t



 

   





  

 D

E B

B

H J D





Time-harmonic Maxwell’s & wave equations

Time-harmonic wave equations (nonhomogeneous Helmholtz’s equations)

 

2 2

2 2

2

2 2

2 2

2

2 2

( , ) ( , )

( , )

( ) ( ) ( )

2

( ) ( )

(

wave nu

) ( )

mber

=







 

 

  

 

  





  

    



    

  

  

    

  



   

 

  

A A J A A J

p

V R k V R R

R t V R t

V R t

t V R j R

R

V R u

k t

k

Time-harmonic retarded potential

• Phasor form of retarded scalar potential

• Phasor form of retarded vector potential.

•

  ¹   ^V  

4

jkR

V

V R R e d

R



 





 

 

   

 

4

jkR V

R R e d

R

 







 





A

 

 

cos

j t kx j t kx

t kx

e ^ e ^ e



 



 

   

2 2

When 2 1

1 ... 1

2

, static expressions(Eq. 3-39 & Eq. 5-22)

jkR

kR R

e jkR k R

V R R

 



 

    



A

Time-harmonic retarded potential

• Example: Find the magnetic field intensity H and the value of β when = 9₀

  ^{z t , }

^{5 cos 10} 

^{t   z}  ^{ V/m }

E a

 

 

9

0

10 (1/s)

5 1

j z

z y e

z j













  

E a

E H

 

0

0 0

1

0 5 0

1 5 5 ( )

x y z

j z

j z j z

x x x x

z j x y z

e

e e H z

j z



 





 



 

  

    

 

  

        

a a a

H

a a a

 

 

   

     

2 2

0

0 0 0

10 0

9 0

1 1

z 5

3 3 10 rad/m

5 0.0398

, 5 0.0398 cos 10 10

j z

y x y

j z j z

x x

j z

x x

H e

j j z

c

z e e

z t e t z









    

       











 



 

  

        

    

   

    

E H a a

H a a

H a a

The EM Waves in lossy media

• If a medium is conducting (σ≠0), a current J=σE will flow

• 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, δ_c

 

 complex permitti 

,

vity

c

c

j j j

j j

j

   

 

   

 



 

 

       

 

 

   

H E E E

=

  



The EM Waves in lossy media

• Loss tangent, δ_c

• Good conductor if

• Good insulator if

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

tan 

  , 

: loss angle

 

  



J



E

   

 

  

 

     

 

 

 

H _cE E

c

j j

j j

>>1





<<1





=

t j

   



 

         

H J E E

The electromagnetic spectrum

• Spectrum of electromagnetic waves