PETER PAZMANY CATHOLIC UNIVERSITYConsortiummembersSEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER

**Molekuláris bionika és Infobionika Szakok tananyagának komplex fejlesztése konzorciumi keretben

***A projekt az Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával valósul meg.

Development of Complex Curricula for Molecular Bionics and Infobionics Programs within a consortial* framework**

Consortium leader

PETER PAZMANY CATHOLIC UNIVERSITY

Consortium members

SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER

The Project has been realised with the support of the European Union and has been co-financed by the European Social Fund ***

PHYSICS FOR NANOBIO-TECHNOLOGY

Principles of Physics for Bionic Engineering

Chapter 3. Electrodynamics – I

(A nanobio-technológia fizikai alapjai )

(Elektrodinamika)

Árpád I. CSURGAY,

Ádám FEKETE, Kristóf TAHY, Ildikó CSURGAYNÉ

Table of Contents

3. Electrodynamics - I

1. Experimental Foundation

2. Maxwell’s Equations – Boundary Conditions 3. Electromagnetic Wave Equation

4. Time-harmonic Fields and their Complex Amplitude (Phasor) Representation

5. Plane Wave Propagation

1. Experimental Foundation

Electromagnetic interactions can be described by four vector fields

 The sources of are electric charges

Gauss’s Law for electric field ).

, ( ), , ( ),

, ( ), ,

(r t D r t B r t H r t E

) , (r t D

 



i V i A

V

Q d

dA



D

A

V

 Currents generate curls (whirls) of

Ampere’s Law

 The vector field

has no sources (neither sinks)

Gauss’s Law for magnetic field ),

, (r t H

d _i d

L i A

 l 







) , (r t B

. 0

d 



A

A B

H J

J

 Time-varying magnetic field generates curls (whirls) of

Faraday’s Law of induction

 Time-varying electric field generates curls (whirls) of

Generalized Ampere’s Law ) , (r t E

d d .

L t A

    





) , (r t H

d dt



d dt



J J

σ σ

d d d .

L A t A

     







 In vacuum

 In ‘simple’ materials (linear, time-invariant)

Conduction Polarization Magnetization

0E,

D  ₀ ¹² 1 ⁹

8.854 10 As/Vm 10 As/Vm

   ^  36π  ^

0H,

B   ₀ 1.256 10 ^6 Vs/Am  4π 10 Vs/Am ^7

2 8

0 0 1/ c , c 3 10 m/s

    





J   P  _{0 e}E M  _mH

 

D₀ 1 _e   B  ₀





2. Maxwell’s Equations – Boundary Conditions

Integral form Local (differential) form

I. Ampere’s Law II. Faraday’s Law III. Gauss’s Law for electric field IV. Gauss’s Law for magnetic field V. Materials

d d d

L A t A

     







d d

L t A

    





d d

A V

 V

  





d 0

A

 









 

J E E D   E, B   H. t 

    



H J D E

 B 0



 D

t

 





 B

E

Electric charges generate electric field. Moving charges generate magnetic field as well. Accelerating charges generate radiation.

Electromagnetic field carries energy and momentum

VI. Connection to other fields of physics Energy density

Force acting on a moving charge

1 1

2 2 ,

w   ED BH . m Ws

3 

 ,

B v

E

F  q q 

 

Maxwell equations are ‘complete’ for correct initial and boundary conditions, the solution exists and is unique.

If at t = t₀, we know the fields inside a volume covered by a closed surface, and at each point of the covering surface we know either the tangential component of the electric or the magnetic field from the initial t₀ up to time t then we can calculate the fields inside the volume up to time t (existence and uniqueness theorem).

Conservation of electromagnetic energy in vacuum

The energy in a closed volume changes if and only if radiating power flows in or out through its surface.

Electromagnetic energy is stored in insulators, it always flows in the insulators, never in the wires.

Power flow

through a surface:

Poynting vector:

Momentum:

. d 2 d

1 2

1 ₂

0 2

0







 

A V

t





d

A



^{ }

  ,

S E H





/ .c

 p S

Conservation of electromagnetic energy







Total power generated by the sources (batteries, generators,

etc.) inside the volume

Power dissipated inside the volume

Rate of increase of electric and magnetic

stored energy

Power density of radiation through the

surface

2 gen

1 1

d d d d .

2 2

V V S

V V V

t



  

           

   

V

E J E D H B J E H A

Boundary conditions

How does the electromagnetic field change at the boundary of two materials, characterized by and ?

Maxwell equations hold at the boundary as well.

Surface current density (J_n) changes the tangential component of H.

2 2

2,





1



1 1,







d d

L _A t

  

      









d

t t n d n

H H l J A D A

     t 

1 2 _n.

  

n (H H ) J

1 1 1,







H1

H2



A

2 2 2,







l A

The tangential component of E goes through the surface always continuously

d d d

L    dt _A  









d 0

t t d n

E E l B A

t

 

      

 

.

2 0

1  

(E E ) n

1 1 1,







E1

E2



A

2 2

2,







l

, d

d





V A



D

, 0 d 



s

s B

,

d _{n1 abcd n2 efgh abcdefgh n1 n2 S}

s

D D

V S

D S

D  









1 2



  

n (D D )

, 0 0

d  ₁  ₂   ₁  ₂ 



s

B B

S B S

B s

B

.

2 0

1  

(B B ) n

1 1 1,







2 2

2,







c g f

b h

e a

d

 0



(E E )

n _{1 2}

 ₁  ₂  A

n (H H ) J n (D ₁ D )₂ 



 ₁  ₂  0 n (B B )

1 1 1,







2 2

2,







1

Et

2

Et

1

Ht

2

Ht

1

Dn

2

Dn

1

Bn

2

Bn

n

JA



+

3. Electromagnetic Wave Equation

In 1864 Maxwell predicted the existence of EM waves. Hertz proved it 20 years later experimentally, by generating artificial electromagnetic radio wave. Light turned to be electromagnetic wave. Optics has been integrated into electromagnetism.

Wave equation and solution

, ) 0

, ( 1

d

) , ( d

2 2 2

2

2 



 



 





t t x v

x t

x



 ^

 ^ 

 

Wave Equation in Source Free Region

2 2

t t

t 

 













 





 











 

 





 E

H E B E

E

  

0 0

  

  D B

H B

E D













 



 























 t

E E E

E

E



 



 























 t

B B B

B

B



2 2

t t

t 

 













 





 











 

 



 B

E B D B

H

   

Gamma-rays X-rays

Ultraviolet Visible light Infrared IR Microwave Radio

Audio

nm 900

300









3 10

3 ¹⁶   ¹⁹

μm 1mm

1 









1mm

km 1000

10





10km

1m 30kHz300MHz PHz

THz GHz MHz kHz

[m]

m]

[



 [nm]

4. Time-harmonic Fields and their Complex Amplitude (Phasor) Representation

In case Maxwell equations are linear, then the sinusoidal time variations of source functions of a given frequency produce steady-state sinusoidal time variations of the field vectors (E, D, B, H) of the same frequency.

Time-dependent field vector

Complex amplitude ‘phasor’ of the field vector

where

. ) cos(

) (

) cos(

) (

) cos(

) ( )

, (

k r

j r

i r

r F























z z

y y

x x

t F

t F

t F

t













).

( ,

, , ,

,F F t

F_{x y z}







.

x j y z

j j

x y z

F e ^ F e ^ F e ^

  

F(r) i j k

( , )t  Re{ ej^t}, F r F(r)

The complex amplitudes (‘phasors’) satisfy the time-independent Maxwell equations

D(r) J(r)

H(r)   



 j







t) Re ( ) ^j ,

(r  H r  H





t) Re ( ) ^j ,

(r  B r  B





t) Re ( ) ^j ,

(r  E r  E





t) Re ( ) ^j ,

(r  D r  D

t

t e

t e







j  j 



 ^



 







 

  

 



e ^ t ^





 







 Re H(r) e^{j t} Re J(r) ^j Re D(r) ^j

Mutatis mutandis

In this case ε, σ and μ can be frequency dependent tensors, with complex components at ω.

D(r) J(r)

H(r)   



 ^j



B(r) E(r)   



 j



B(r)  0



D(r)  r

  ^

 

D E J 



  

5. Plane waves, Reflection and Refraction

Uniform Plane Wave Propagation in Free Space Propagation in z direction

x y

z

.

 0



 



 



 





x y

x x

B E

B E

 

0 0

0 0 j .

j ^{x y z}

x y z

B B B

z

E E E

 



        

     

i j k

E i j k

E(r) B(r)

,

j _x

y B

z

E  





  ^x j B_y,

z

E  





  0  j



Transverse electric–magnetic propagation (TEM) Two independent solutions and

 

0 0 0

0

0

0 0 j .

j ^{x y z}

x y z

E E E

z

B B B

  



   

    

   

 ^{x y z}

i j k

B(r)

a a a

B E

0 ,

0 x

y E

z

B  j 



  ^x _{0 0} E_y,

z

B  j 



  0  j₀₀ E_z.

TEM

 0

 _z

z B

E

y x

B

E , B

, E

.

x x y

z B E z

B

0 0 2 0

2 0 2

j

  

  



 

 

 ₂ ² _{0 0} 0

2  





x

x B

z

B

  

x

x y E

z B z

E

0 0 2 2

2

j



  



 

 

 ₂ ² _{0 0} 0

2  





x

x E

z

E

  

z x

z x

x E e E e

E





^B

^{ }

 ^H

^e

^{ H}

^e



c

 









 c  1 / 



 3  10

m/s

z x

z x

x E e E e

E  ^{ j0}  ^{ j0}

z x

z x

x H e H e

H  ^{ j0}  ^{ j0}

y

x B

z  j





y

x E

z H

0 0

0 j

 









 



 

 _x^{  z} _x^{  z}

y E e E e

H ^{0 j 0}

0 j 0

0

0  













z x

y H e H e

E ₀ ^{j 0} ₀ ^{j 0}

0

j j j

1













 





0 0 0

0

0 0 0 0

120 π 377

   

 ^    ^  ^{   }

x z x z

y E e

E e

H ^{j0 j0}





 

 



 E^y 





The solution is the sum of the two independent waves

We return back to the time-domain

  



Eˆ  E_x^e^j0z  E_x^e^j0z  E_y^e^j0z  E_y^e^j0z

j i

H 









 

 



 



 

 ^







 

 

y z y z

x z x z

E e E e

E e E e

0 0

0

0 j

0 j

0 j

0 j

0

ˆ ^{   }









c

 







0

120π 377

 









t

z, ) Re ˆ ^j

(  E

E ^{H(z,t)  Re}





Note that the coefficients are complex numbers, with amplitude and phase

 In general, elliptically polarized, TEM wave

 Linear and circular polarizations are special cases,



 





_,^,_y  _x,^,_y ^{j x,,y}

x E e

E ^

 

   

   

j

0 0

0 0

( , ) Re ˆ

cos cos

cos cos

t

x x x x

y y y y

z t e

E t z E t z

E t z E t z



     

     

   

   

  

 

        

 

       

E E

i j

 

   

   

j H

H











     



 











     

























y y

y y

x x

x x

t

z E t

z E t

z E t

z E t

e t

z





 





 





 





 



0 0

0 0

0 0

0 0

j

cos cos

cos cos

Re ˆ )

, (

Free space propagation Conductive medium

Analogy

Uniform Plane Wave Propagation in Conductive Media

There is an analogy between the free space propagation and propagation in linear media



 

 









 











0 j

0

0 r

H E   



 j





 j



H E   



 j



E E

E

H 



 

 

















 







  

0 0

0 0

0

j

j j

j



















 



 

 







 







j j

j

j j

ˆ j



 



 

 



 



 

 



0 0

0









 

 





 

 

j

Linear Polarization

Phasor vectors in space

Solution in space-time



 

 





 









 

 





 

 

j

 





E i

 

_

_

 

H j

    



E z,t  Re e^jt E_m^e^{ z}  E_m^e^{ z}

 



 



 



 

 ^{  } j

H ^{t m z} E^m e ^z

E e e

t

z ^{  }





,

 

H z,t  ^z

y

 

E z,t 

0 k