EMT02

Electromagnetism

2

Contents

• Review of Maxwell’s equations and Lorentz Force Law

• Motion of a charged particle under constant Electromagnetic fields • Relativistic transformations of fields

• Electromagnetic energy conservation • Electromagnetic waves

– Waves in vacuo

– Waves in conducting medium

• Waves in a uniform conducting guide – Simple example TE01 mode

– Propagation constant, cut-off frequency – Group velocity, phase velocity

3

Basic Equations from Vector Calculus





   

 

     

    

      

          



y F x

F x

F z

F z

F y

F F

z F y

F x

F F

F F F F

1 2

3 1

2 3

3 2

1

3 2 1

, ,

: curl

: divergence

, , , vector

a For



 





   

 

  

 

  

z φ y

φ x

φ φ

x,y,z,t φ

, ,

: gradient

, function

scalar a

For _{Gradient is normal to surfaces;}

4

Basic Vector Calculus

F

F

F

F

G

F

F

G

G

F





















2

)

(

)

(

0

,

0

)

(













































































S C

r

d

F

S

d

F









dS

n

S

d







Oriented

boundary C

n



Stokes’ Theorem Divergence or Gauss’ Theorem













S V

S

d

F

dV

F







Closed surface S, volume V, outward pointing normal

What is Electromagnetism?

• The study of Maxwell’s equations, devised in 1863 to represent the relationships between electric and magnetic fields in the presence of electric charges and currents, whether steady or rapidly fluctuating, in a vacuum or in matter.

• The equations represent one of the most elegant and concise way to describe the fundamentals of electricity and magnetism. They pull together in a consistent way earlier results known from the work of Gauss, Faraday, Ampère, Biot, Savart and others.

• Remarkably, Maxwell’s equations are perfectly consistent with the transformations of special relativity.

Why Electromagnetics?

• As devices get smaller and smaller, and frequencies get higher and higher, circuit theory is less able to adequately describe the

performance or to predict the operation of circuits.

• At very high frequencies, transmission line and guided wave theory must be used - high speed electronics, micro/nano electronics,

integrated circuits.

• _{Other applications of Electromagnetics}

-Fiber Optics

Microwave Communication Systems Antennas and wave propagation

Optical Computing

Electromagnetic Interference, Electromagnetic Compatibility Biology and Medicine/Medical Imaging

Why Electromagnetics? ………..

• _{As use of the electromagnetic frequency spectrum increases, the}

demand for engineers who have practical working knowledge in the area of electromagnetic continues to grow.

•_{Electromagnetic engineers design high frequency or optoelectronic}

circuits, antennas and waveguides; design electrical circuits that will function properly in the presence of external interference while not interfering with other equipment.

• _{The electromagnetics technical specialty prepares future engineers}

for employment in industry in the areas of radar, antennas, fiber optics, high frequency circuits, electromagnetic compatibility and microwave communication.

Maxwell’s Equations

Relate Electric and Magnetic fields generated by charge and current distributions.

t

D

j

H

t

B

E

B

D



















































0



1 ,

,

In vacuum 2

0 0 0

0  

 E B H c

D       

E = electric field

D = electric displacement

H = magnetic field

B = magnetic flux density

= charge density

j = current density

₀ (permeability of free space) = 4 10-7

₀ (permittivity of free space) = 8.854 10-12 c (speed of light) = 2.99792458 108 _m/s

9

Equivalent to Gauss’ Flux Theorem:

The flux of electric field out of a closed region is proportional to the total electric charge Q enclosed within the surface.

A point charge q generates an electric field

Maxwell’s 1

st

Equation

0 0

0

1











Q

dV S

d E dV

E E

V S

V

 

 

  

 

 







 



0 2

0 3 0

4 4







q r

dS q

S d E

r r q E

sphere sphere

 









 

 

0











E



Area integral gives a measure of the net charge enclosed;

Gauss’ law for magnetism:

The net magnetic flux out of any closed surface is zero. Surround a magnetic dipole with a

closed surface. The magnetic flux directed inward towards the south pole will equal the flux outward from the north pole.

If there were a magnetic monopole source, this would give a non-zero integral.

Maxwell’s 2

nd

Equation





B





0















B



0

B



d

S



0

Gauss’ law for magnetism is then a statement that There There are no magnetic monopoles

Equivalent to Faraday’s Law of Induction:

(for a fixed circuit C)

The electromotive force round a circuit          is proportional to the rate of change of flux of magnetic field,

       through the circuit.

Maxwell’s 3

rd

Equation

t

B

E

















dt d S

d B dt

d l

d E

S d t B S

d E

C S

S S

   

  



   

 

 









 



 

 







E



d

l





N S

Faraday’s Law is the basis for

electric generators. It also forms the basis for inductors and

transformers.







Maxwell’s 4

th

Equation

t

E

c

j

B



















2 0

1



Originates from Ampère’s (Circuital) Law :

Satisfied by the field for a steady line current (Biot-Savart Law, 1820):

I

S

d

j

S

d

B

l

d

B

C S S



























₀













₀

r I B

r r l

d I

B











2 4

0 3

0

 





current line

straight a

For

  

j

B







₀







Ampère

13

Need for Displacement

Current

• Faraday: vary B-field, generate E-field

• Maxwell: varying E-field should then produce a B-field, but not covered by Ampère’s Law.

Surface 1 Surface 2

Closed loop Current I

 Apply Ampère to surface 1 (flat disk): line

integral of B = ₀I

 Applied to surface 2, line integral is zero

since no current penetrates the deformed surface.

 In capacitor, , so

 Displacement current density is

t E j_d

  

 

0



dt dE A dt

dQ

I   ₀

A ε

Q E

0







t

E

j

j

j

B

_d



























0 0 0

0







14

Consistency with

Charge Conservation

Charge conservation: Total current flowing out of a region equals the rate of

decrease of charge within the volume.

From Maxwell’s equations:

Take divergence of (modified) Ampère’s equation

0

    

 

  

 

 

  









t j

dV t

dV j

dV dt

d S

d j



 

 





 

t j

t j

E t

c j B

      

   

  

 

  



  

 

  

   



  

 



 

 

0 0

1

0 0

0 0

2 0

Charge conservation is implicit in Maxwell’s Equations

15

Maxwell’s

Equations in Vacuum

In vacuum

Source-free equations:

Source equations

Equivalent integral forms

(useful for simple

geometries)

2 0 0 0 0 1 , , c H B E

D         

0 0          t B E B    j t E c B E     0 2 0 1 _           

















































S

d

E

dt

d

c

S

d

j

l

d

B

dt

d

S

d

B

dt

d

l

d

E

S

d

B

dV

S

d

E





























2 0 0

1

0

1







Example: Calculate E from B

      0 0 0 0 sin r r r r t B

B_z 











B



dS

dt

d

l

d

E







t r B E t B r t B r dt d rE r r           cos 2 1 cos sin 2 0 0 2 0 2 0         t r B r E t B r t B r dt d rE r r           cos 2 cos sin 2 0 2 0 0 2 0 0 2 0 0         Also from t B E         dt E c j B    _{ }  

 ₀ 1₂ then gives current density necessary to sustain the fields

r z

17

Lorentz Force Law

• Supplement to Maxwell’s equations, gives force on a charged particle moving in an electromagnetic field: • For continuous distributions, have a force density • Relativistic equation of motion

– 4-vector form:

– 3-vector component:



E v B



q

f     

B j

E

f_d       

   

      

   



dt p d dt dE c f

c f v d

dP

F  

 

, 1

, 

 



m v



f q



E v B



dt

d _   _ 

  





18

Motion of charged particles

in constant magnetic fields

1. Dot product with v:    





  constant is constant is 0 So 1 But 0 2 2 2 0 v dt d dt d v dt d v c v γ B v v m q v dt d v                              











m v



q

 

v B

dt d B v E q f v m dt

d _{  }  _  _{ }  _{    } 

0 0

No acceleration with a magnetic field

 

 

0, constant

0 // 0          v v B dt d B v B m q v dt d B        

2. Dot product with B:

constant also constant and constant _//   v v

Motion in constant magnetic field









 

 

0 0

0 2

0

frequency angular

at

radius with

motion circular

m m

m qB v

ω

qB v m ρ

B v m

q v

B v m

q dt

v d

 



 

 

 



 



  

Constant magnetic field gives uniform spiral about

B with constant energy.

rigidity

Magnetic

0

q

p

q

v

m

20

Motion in constant Electric Field

Solution of

 

E

m q v

dt

d _ 

0





Constant E-field gives uniform acceleration in straight line











m v



q E

dt d B

v E q f

v m dt

d _{  }  _  _{ }  _{   } 

0 0

2

0 2

2 0

1

1 _

  

   

 

  

    

 t

m qE c

v t

m qE

v   



is

    

  

    

   



 1 1

2

0 2

0

c m qEt qE

c m x

v dt

dx

 

c m qE

t m qE

0 2

0

for 2

1 _



qEx

• According to observer O in frame F, particle has velocity v, fields are E and B and Lorentz force is

• In Frame F, particle is at rest and force is

• Assume measurements give same charge and force, so

• Point charge q at rest in F:

• See a current in F, giving a field • Suggests

Relativistic Transformations of E and B



E v B



q

f      

E q f  

B v

E E

q

q   and     

0 ,

4 ₀ 3 

 B

r r q

E  



E v

c r

r v

q

B   0  ₃    1₂  

4



E v

c B

B    1₂  

Ro

ug

h i

de

a





// //

2

// //

,

,

B

B

c

E

v

B

B

E

E

B

v

E

E





































_



















 

 





22

Potentials

• Magnetic vector potential:

• Electric scalar potential:

• Lorentz Gauge:

A B

A

B        



 0 such that

0

1

2

_











A

t

c















f(t)

,

A





A











t A E

t A E

t A E

t A A

t t

B E

    

 

   



    

 

   

 

    

  

       



 

 

 

 



 

 with , so

0

23

Electromagnetic 4-Vectors

Α ,

1 ,

1 0

1

4

2    

  

      



 _

  

 

   

A c

t c A

t c

 

 

Lorentz Gauge

4-gradient

4

4-potential A

Current

4-vector

_

_

_

_

_

_

_



0 0

0  ( , )  ( , ) where 

 



j c v

c V

J

v j

 

 

Continuity

equation ,



,



0

1

4   

   

   



 _

  



 j

t j

c t

c

J









Charge-current

transformations















 











,

₂

c

j

v

v

j

j

x

x

24                                              z y x z y x A A Ac c v c v A A Ac       1 0 0 0 0 1 0 0 0 0 0 0 ' ' ' '

Relativistic Transformations

• 4-potential vector:

• Lorentz transformation

• Fields:        A c

A 1 , 



E v B



E E E c E v B B B

B      

                  _    

 _{// }



_{ 2 // // }



_

//



x vt



x y y y A x A B A

B y x

z _{ }    

             

  and , ' 

                                 

 and , ₂

c vx t t z z t A z E t A E z z     

Example: Electromagnetic Field

of a Single Particle

• Charged particle moving along x-axis of Frame F

• P has

• In F, fields are only electrostatic (B=0), given by

t c

vx t

t t

v b

r x

b vt

x _{P p}  p _  

  



 _

 

   

 _{( ,'0, ), so '} 2 2 _'2_{, ' 2}

' 



Origins coincide at t=t=0

Observer P

z

b

charge q

x

Frame F _v Frame F’

z’

x’

3 3

3 _' , ' 0, ' _'

' '

' ' '

r qb E

E r

qvt E

x r

q

E   _P  _x   _y  _z 

t v x

t v x

x_P  _P   _P   

 ( ) so

Transform to laboratory frame F:







2 2 2 2



32 2 3 2 2 2 2 ' 0 ' t v b b q E E E t v b vt q E E z z y x x             

0

'

2













z x z z y

B

B

E

c

E

c

v

B





3

3

,

'

0

,

'

_'

'

'

'

r

qb

E

E

r

qvt

E

_x





_y



_z





E v B



E E E c E v B B B B                         _         





// // 2 // // 3

r

r

v

q

B











At non-relativistic energies,  ≈ 1, restoring the Biot-Savart law:

27

Electromagnetic Energy

• Rate of doing work on unit volume of a system is

• Substitute for j from Maxwell’s equations and re-arrange into the form



E

j

B



v

E

j

E

v

f

v







_d















































E D B H



t S

H E

S t

D E

t B H

S

t D E

E H

H E

E t

D H

E j

  



 

 

 



 

 

 

 

 



  

  

  

 

       

  

    

  

 

  

   

 

   

   



2 1

where

28



B H E D





E H



t E

j       

 _{  }

   



 _{  }

   



2 1





_



     

 E D B H dV E H dS

dt d dt

dW       

2 1

electric + magnetic energy

densities of the fields

Poynting vector gives flux of e/m

energy across boundaries

Integrated over a volume, have energy conservation law: rate of doing work on system equals rate of increase of

stored electromagnetic energy+ rate of energy flow across boundary.

Review of Waves

• 1D wave equation is with

general solution

• Simple plane wave:

2 2 2 2

2 ₁

t u v

x u

  

 

)

(

)

(

)

,

(

x

t

f

vt

x

g

vt

x

u













t



k

x



3D

:

sin





t



k





x





sin

:

1D

k

   2

Wavelength is Frequency is

  

2

Superposition of plane waves. While shape is relatively undistorted, pulse travels with the group velocity

Phase and group velocities

k t

x v

x k t

p

 

    

  

 0

 



  

 _dk

e k

A( ) i (k)t kx

dk d

v_g  

Plane wave has

constant phase at peaks



t kx



sin

2

 t kx 

31

Wave packet structure

• Phase velocities of individual plane waves

making up the wave packet are different,

Electromagnetic

waves

• Maxwell’s equations predict the existence of electromagnetic waves, later discovered by Hertz.

• No charges, no currents:

0

0  

              B D t B E t D H      









2 2 2 2

t

E

t

D

B

t

t

B

E































































 



E E E E     2 2             2 2 2 2 2 2 2 2 2 : equation wave 3D t E z E y E x E E                   

Nature of Electromagnetic

Waves

• A general plane wave with angular frequency  travelling in the direction of the wave vector k has the form

• Phase = 2  number of waves and so is a Lorentz invariant.

• Apply Maxwell’s equations

)]

(

exp[

)]

(

exp[

₀

0

i

t

k

x

B

B

i

t

k

x

E

E



































i t

k i

 



 

 

B E

k B

E

B k

E k

B E

 

 



  

 





 

 

 



  

 

   



 0 0

Waves are transverse to the direction of propagation, and and are mutually perpendicular_E_{, B}

k



x k t   

34

Plane Electromagnetic Waves

E c B

k t

E c

B   

 

2 2

1 _{   } 

  

 

 



2

that deduce

with Combined

kc k

B E

B E

k

 

 

 

 



c k 

 speed of wavein vacuumis _

  

 

2 Frequency

k 2 Wavelength

  _

Reminder: The fact that is an invariant tells us that

is a Lorentz 4-vector, the 4-Frequency vector. Deduce frequency transforms as

x k t   



   

  

 k

c



,







v c

v c k

v

  

  

   

Waves in a Conducting Medium

• (Ohm’s Law) For a medium of conductivity ,

• Modified Maxwell:

• Put

E

j









t

E

E

t

E

j

H





































E

i

E

H

k

i























conduction current

displacement current

)]

(

exp[

)]

(

exp[

₀

0

i

t

k

x

B

B

i

t

k

x

E

E







































D

4 0

8

-12 0

7

10 57

. 2 1

. 2 ,

10 3

: Teflon

10 ,

10 8

. 5 :

Copper

  

 

 

 

 



D D 

 

  

Dissipation factor

Attenuation in a Good Conductor

 





 









since 0

with Combine 2 2                                E k i k E i E k k E k E i H k E k k H E k t B E                                     

E

i

E

H

k

i























For a good conductor D >> 1,  k  i  k 



1 i



2

, 2  

    depth -skin the is 2 where 1 1 , exp exp is form Wave                           

 _ x x _{k i}

t

Charge Density in a Conducting Material

• Inside a conductor (Ohm’s law)

• Continuity equation is

• Solution is

E

j





_



















































t

E

t

j

t

0

0





 





_

_e

 t 0

So charge density decays exponentially with time. For a very good conductor, charges flow instantly to the surface to form a surface charge density and (for time varying fields) a surface current. Inside a perfect conductor ()

Maxwell’s Equations in a Uniform

Perfectly Conducting Guide

z

y x

Hollow metallic cylinder with perfectly conducting boundary surfaces

Maxwell’s equations with time dependence exp(iwt) are:

 





                                      0 2 2 2 2                                                      H E E H i E E E E i t D H H i t B E           Assume ) ( ) ( ) , ( ) , , , ( ) , ( ) , , , ( z t i z t i e y x H t z y x H e y x E t z y x E            

Then



2 ₍ 2 2₎



 ₀

         H E t     

g is the propagation constant

Can solve for the fields completely in terms of E_z and H_z

40

Special cases

• Transverse magnetic (TM modes):

– Hz=0 everywhere, Ez=0 on cylindrical boundary

• Transverse electric (TE modes):

– Ez=0 everywhere, on cylindrical boundary

• Transverse electromagnetic (TEM modes):

– Ez=Hz=0 everywhere

– requires

0

 



n H_z











41

A simple model: “Parallel Plate

Waveguide”

Transport between two infinite conducting plates (TE₀₁ mode):

2 2

2 2

2 2 2

t

) (

,

satisfies )

( where )

( ) 0 , 1 , 0 (

  

 

 

  



 

K E

K dx

E d E

x E e

x E

E i t z

Kx A

E

   

  

cos sin i.e.

To satisfy boundary conditions, E=0 on x=0 and x=a, so integer ,

,

sin n

a n K

K Kx

A

E   _n  

Propagation constant is

 

  

 

 n

c c

n

K a

n

K _ 

      



 1 where

2 2

2

z x y

x=₀

42

Cut-off frequency,



 w<wc gives real solution for g, so attenuation

only. No wave propagates: cut-off modes.

 w>wc gives purely imaginary solution for g,

and a wave propagates without attenuation.

 _{For a given frequency w only a finite number of}

modes can propagate.

















 

a n e

a x n A

E a

n

c z

t i c

 

   

  

 _{1 , sin}  _,

2

 

 

 

 n a

a n

c   

 For given frequency, convenient to

choose a s.t. only n=1 mode occurs.





12

2 2 2

1 2

2 ₁

, _

  

 

 

 



  

 

 

 c

c

k ik

43

Propagated Electromagnetic Fields

From









      

 

  

  





 

  

  

 

  

    



kz t

a x n a

n A H

H

kz t

a x n Ak

H E

i H

A t

B E

z

y x

 

  

 

  



sin cos

0

cos sin

real, is

assuming ,

 

 

z x

44

Phase and group velocities in the simple

wave guide





















2 2 12

c

k

Wave number:

wavelength space

free the

, 2

2 _{ }



 

 



k

Wavelength:

velocity

space

-free

than

larger

,

1





_



k

v

_p

Phase velocity:





velocity

space

-free

than

smaller

1

2 2

2

























k

dk

d

v

k

_{c g}

45

Calculation of Wave Properties

• If a=3 cm, cut-off frequency of lowest order mode is

• At 7 GHz, only the n=1 mode propagates and

GHz

5

03

.

0

2

10

3

2

1

2

8



















a

f

c

c









c k

v

c k

v

k k

g p

c

 

 

 

 

 

 

 

 



 



1 8

1 8

1 8

9 2

1 2 2

2 1 2 2

ms 10

1 . 2

ms 10

3 . 4

cm 6 2

m 103 10

3 / 10 5

7 2

 

 

 

 







a

n

46           2 2 2 2 2 2 2 2 0 2 2 0 2 since 8 1 4 1 energy Magnetic 8 1 4 1 energy Electric                            





a n k W k a n a A dx H W a A dx E W e a m a e  

Flow of EM energy

along the simple guide







t kz



a x n A a n H H E k H kz t a x n A E E E z y y x y z x                  sin cos , 0 , cos sin , 0

Fields (w>w_c) are:

Time-averaged energy: Total e/m energy density

a A W 2 4 1  

47

Poynting Vector

Poynting vector is

S





E





H







E

_y

H

_z

,

0

,



E

_y

H

_x



Time-averaged:





a x n kA

S 



2 2

sin 1

, 0 , 0 2 1





Integrate over x:



2

4

1 akA S_z 

So energy is transported at a rate: _g

m e

z k _v

W W

S

 

 

Electromagnetic energy is transported down the waveguide with the group velocity

Total e/m energy density

a A

W 2

4 1_