Electromagnetic Fields Ii-summer

Electromagnetic Fields First Semester 2008/2009 Electronic and Comm. Eng. Dept.

Course Outlines Chapter (1):Quasi Stationary Magnetic Fields 1.1 Faraday’s law. 1.2 Induced e.m.f due to motion.

Chapter (2): Maxwell’s Equations and Plane waves 2.1 Displacement currents. 2.2 Differential and integral forms (time domain). 2.3 Sinusoidal time varying fields. 2.4 Derivation and solution of wave equation in unbounded media. 2.5 Plane waves in different media. 2.6 Power, energy and poynting theorem.

Course Outlines (Continued) Chapter (2): Maxwell’s Equations and Plane waves 2.7 Wave polarization and propagation modes. 2.8 Reflection and refraction of plane waves.

Chapter (3): Guided Waves 3.1 Solution of wave Eqn. in bounded media (W.G.). 3.2 Rectangular waveguides (RWG). 3.3 Transverse Electric (TE) modes. 3.4 Transverse Magnetic (TM) modes. 3.5 Power transmission and attenuation inside RWG. 3.6 RWG currents and excitation techniques.

References •

William H. Hayt, “Engineering Electromagnetic,” McGraw-Hill, 1989.

•

Plonsey & Collin, “ Principles and Applications of Electromagnetic Fields,” McGraw-Hill, 1962.

•

F.T. Ulaby, “Fundamentals of Applied Electro-magnetic,” Prentice-Hall, 1997.

Introduction: Classification of fields w.r.t. time:  Static fields: source at rest w.r.t time (static charges) .  Stationary fields: source with uniform motion w.r.t. time, (i.e. D.C. current or ∂⁄∂t =0).  Quasi-Stationary fields: by quasi-stationary field we mean field that is slowly varying with time (∂⁄∂t ≠0) in such a way that all radiation effects on the system can be neglected (for circuit of maximum dimension D; the wave length λ of the operating frequency must satisfy the condition λ≥D ).  General time-varying fields: the variation of frequency is not limited by the dimensions of the circuit.

(1.1) Faraday’s law of induction:  In 1820 C.H. Oersted demonstrated that an electric current affected a compass needle.  After this, Faraday professed his belief that if a current could produce a magnetic effect, then the magnetic effect should be able to produce a current (magnetism).  In 1931, the electric induction phenomenon was discovered as a results of Faraday’s experiments.

Faraday’s first experiment: If two separate coils are wound on an iron ring. One of them is connected through a switch to D.C. battery. It was observed that whenever the current was changed, an induced current would flow in the other coil.

Faraday’s second experiment: If a magnet moves near a coil, an induced current will be produced in the galvanometer.

ψ (t)

 Generally, for any closed path C in space which is linked by a changing magnetic field, the induced voltage (e.m.f) around this path is equal to the negative time rate of change of the total magnetic flux through the closed path. e .m . f = Vind = −

∂ψ ( t ) ∂t

⇒

C S

Faraday’s law of induction (Basic form)

 The minus sign is according to Lenz’s law which states that: “The induced voltage is in such direction that it resists the original change”

 For N-turns loop:

N-turns

Vind= -∂ψ / ∂t

N-turns

∑

ψ

∑

....

ψ (t)

Depend on the distribution

of the flux in each turn for Nturns loop.

ψ ∑ =ψ 1 +ψ 2 +…..

ψ ∑ = Nψ

(different ψ in each turn)

(same ψ in each turn)

Faraday’s law in integral form: form Vind = −

∂ψ ∂t

  ∂   ∫C E.d  = − ∂t ∫S B.dS

Faraday’s law in differential form: form     ∂   ∫ E.d  = ∫ (∇ × E ).dS = − ∫ B.dS ∂t S C S

  ∂B (∇ × E ) = − ∂t

Notes: The electric field has two sources (charges and time varying magnetic field). If there is no time variation (∂ / ∂t =0), gives

(Static case).

Faraday’s law in circuit form: form Vind = −

∂ψ ∂t

L=

;

ψ I

⇒

ψ = LI

∴

Vind = − L

dI dt

(1.2) Induced e.m.f. due to motion: (1.2.1) Moving conductor in static magnetic field: When a conductor is moving through a static magnetic field, an induced voltage is produced in the conductor. The magnitude of this voltage is found from:

Lorentz force law;

 “A particle of charge ( q ) moving with velocity ( v ) in magnetic    field (

B

) experience a force given by, (

 F = qv × B

)”

 B

Example: Consider a conducting wire of  through a length L moving with velocity    v magnetic field B ( v ⊥ B ). Each electron of charge (–e) in the conductor experience a Lorentz force:    Fm = − e v × B (Fm = − evB)

P1 L

 Fm

 E ind P2

 v

 Fe

Which force the electron to move toward P1,

leaving positive charges at P2. The displaced charges setup an induced electric field which opposes the displacement of the charges due to Lorentz force. Then,

   Fe = qE ind = − eE ind

(Fe = − eE ind )

When sufficient charges have been built up equilibrium is established,

Fm = Fe

   E in = V × B

E ind = vB

The induced voltage between the ends of the conductor is given by: Vind

P2

     = ∫ E ind . d  = ∫c ( V× B) ⋅ dL P1

Vind

P2

P2   = ∫ E ind .d  = vB ∫ d P1

P1

Vind = vBL

(1.2.2) Moving conductor in time varying magnetic field:   B( t ) B( t ) C  v

After dt

dS

 d

 vdt S

Vind

 ∂B     = −∫ .dS + ∫ ( v × B).d  S ∂t C Contribution due to time variation

Contribution due to motion

Example: Within a certain, ε=10-11 F/m and µ=10-5 H/m. If Bx=2*10-4cos 105t sin 10-3y T: (a) Use Δ x H = ε ∂E/ ∂t to prove that: E=-20000 cos10-3y sin105t v/m. solution

  B x 2 ∗ 10−4 cos105 t sin 10−3 y B = µH ⇒ H x = = µ 10−5 aˆ x

 ∂ ∆× H = ∂x Hx

aˆ y

aˆ z

∂ ∂y

∂ ∂z

Hy 0

ε

=

20 cos105 t sin 10−3 y

=

∂Hx ∂Hx aˆ y − aˆ z ∂z ∂y

= − 10−3 ∗ 20 cos105 t cos10−3 y aˆ z

0

Hz 0

∂E = −10−3 ∗ 20 cos105 t cos10−3 y aˆ z ∂t − 10 −3 ∗ 20 cos 10 −3 y ∫ cos 105 t dt Ez = ε

 E = −20000 cos 10 −3 y sin 10 5 t aˆ z

v/m

=

− 10−3 ∗ 20 10−11

sin 105 t − 3 cos10 y 5 10

Introduction: In this chapter we will concern with time varying electromagnetic field, and we shall then find that the electric and magnetic field are related to each other. I.e. a time varying magnetic field producing an electric field and a time varying electric field producing magnetic field, results in a phenomena of wave propagation.

(2.1) Displacement current: The British physicist Maxwell’s was the first one who postulated the electromagnetic wave propagation, his first study starts from the basic equations of the electric and magnetic fields including the time variation.   ∂B ………………. Faraday’s law of induction. (1) ∇ × E = − ∂t   ………………. Ampere’s circuital law. ( 2) ∇ × H = J  ………………. Gauss’ flux theorem. ( 3) ∇ .D = ρ  ………………. Law of continuity of B-lines (magnetic Gauss’ law). (4) ∇ .B = 0  ∂ρ (5) ∇ .J = − ∂t

……………….

Continuity equation (law of conservation of charges).

Maxwell’s pointed out that the above equations form a set which is inconsistent (He shows its inadequacy for time varying conditions). How ? Taking the divergence of both sides of eqn.(1):

  ∂B (1) ∇ × E = − ∂t

………………. Faraday’s law of induction.

  ∂ ∇.(∇ × E ) = − (∇.B ) ∂t 0

 ∇.B = 0

which gives the same result as eqn.(4) (i.e dependant equation).

Taking the divergence of both sides of eqn.(2):   ( 2) ∇ × H = J

………………. Ampere’s circuital law.

  ∇.(∇ × H) = ∇.J

 ∇.J = 0

 which is in contradiction with eqn.(5). For this reason, Maxwell add term ∂D to ∂t 0

eqn.(2) which gives:

  ∂  ∇.(∇ × H) = ∇.J + (∇.D) ∂t 0

 ∂ρ ∇.J = − ∂t

which gives the same result as eqn.(5) (i.e dependant equation).

 ( ∇ .D = ρ )

 ∂D   = J d ….. Maxwell’s current density (Displacement current density) ∂t



….. Its unit is [A/m2].    ∂D  I d = ∫ Jd .dS = ∫ .dS ….. Displacement Current [A]. s S ∂t

To discuss the displacement current density:

Differential form

   ∂D 2) ∇ × H = J + ∂t

Integral form

Remarks

    ∂   ∫ H.d  = ∫ J.dS + ∫ D.dS ..... Ampere’s circuital law. ∂t S C S I

    ∂   ∫ H .d  = I + I d = ∫ J .dS + ∫ D .dS C S ∂t S

Example: to illustrate the physical nature of the displacement current. d.c

a.c ε

 The –ve charges accumulate in one plate and

c

a.c

C

I

changed which change the direction of dipoles that represent a displacement for the electrons and a current Id will flow. C S S ε

I

From Ampere’s circuital law:

    ∫ H.d  = ∫ J.dS = I

c

c

a.c

ε

 The polarity of the capacitor plates is

the +ve charges accumulate at the other plate, so the dielectric material will polarized and there is no movements of dipoles (no current). S

∼

S

do

  From Ampere’s circuital law: ∫ H.d  = ??

dv  εS  d  I c = C =   (Ed o )  dt  d o  dt 

c

Ic ∂D = Jd = S ∂t

(2.2) Maxwell’s equations differential and integral form (time domain):

Differential form   ∂B 1) ∇ × E = − ∂t    ∂D 2) ∇ × H = J + ∂t  3) ∇.D = ρ

Remarks Integral form   ∂   ∫ E.d  = − ∫ B.dS ………… Faraday’s law of ∂t S C induction.     ∂   ∫ H.d  = ∫ J.dS + ∫ D.dS ..... Ampere’s circuital law. ∂t S C S   ∫ D.dS = ∫ ρdv ……………. Gauss’ flux theorem. S

V

  ∫ B.dS = 0 …………………........ Law of continuity of B-lines

 4) ∇.B = 0

S

 ∂ρ & ∇.J = − ∂t

  ∂ ∫ J.dS = − ∫ ρdv …………........ Continuity equation ∂t V S

Constitutive Relations:   D = εE ;

  J = σE ;

(magnetic Gauss’ law)

  B = µH

Where σ, ε and µ are the medium parameters.

(law of conservation of charges)

Note: Jconv = ρv J

Jcond = σ E

Conduction current density (Jcond): Motion of charges usually electrons in a region of zero net charge density. Convection current density (Jconv): Motion of volume charge density (ρ) Displacement current density (Jd): Third type of current density.

  ∂D =J d ∂t

Electromagnetic quantities (review):  E ...........Electric field intensity [Volts/meter ; V/m].

 H ...........Mgnetic field intensity [Amperes/m ; A/m].  B ...........Magnetic flux density [Webers/m2 ; wb/m 2 or Tesla ; T].  D ...........Electric flux (Displacement current) density [Coulombs /m 2 ; C/m 2 ]  J ............Electric current density [Amperes/m2 ; A/m2 ]. ρ ............Electric charge density [Coulombs/m3 ; C/m3 ].

[Moh/m ; /m]. [Farad/m ; F/m]. Ω

σ ...........Electric conductivity ε ............Dielectric permitivity

µ ...........Magnetic permeability [Henery/m ; H/m].

In free-space:*

ε = ε o = 8.854 × 10 −12

[F/m ].

µ = µ o = 4π × 10 − 7

[H/m].

How Maxwell’s equations used to show wave equation? Consider the electric and magnetic fields in a region does not include any sources, which called:

Source-free

Source-free wave equation : Source-free: the solution region does not include any sources.  Jimp = 0 & ρ = 0

(2.4) Derivation and solution of wave equations in unbounded media: 2.4.1 Types of media ( according to the values of ε , µ andσ ). 2.4.2 Source-free wave equation. 2.4.2.1 Time form. a. In free-apace ( ε = ε o , µ = µo and σ = 0 ). b. Lossless dielectric ( ε = ε o ε r , µ = µoµr and σ = 0 ). c. Lossy dielectric ( ε =ε o ε r , µ = µoµr and σ ≠ 0 ).

2.4.2.2 Complex form.

2.4.3 Properties of plane wave.

There are many media to derive and solve the wave equation. Let’s start by free space.

a) In free-apace: ( ε = ε o , µ = µo and σ = 0 )

(2.4.2.1)Time form:   ∂H ∇ × E = −µo ..... (1) ∂t

 ∂E ∇ × H = εo ..... (2) ∂t  ∇ .D =0 ..... (3)  ∇ .B =0 ..... (4)  & ∇ .J =0 ..... (5)

To derive the wave equation for the electric field: Take the curl of both sides of eqn. (1) :   ∂ ∇ × ∇ × E = −µo (∇ × H ) ∂t

Using vector relationship, we get    ∂ ∂E 2 ∇ (∇.E ) − ∇ E = − µo (εo ) ∂t ∂t 0

Similarly;

  ∂ E 2 ∇ E − µoεo 2 = 0 ∂t 2

Generally:

 2  ∂ ∇ − µo εo  2   ∂ t   2

 2  ∂ H 2 ∇ H − µoεo 2 = 0 ∂t

 E     =0 H   

Source-free wave equation in free-space (2nd order P.D.E.)

Vector relationship:

 ∇.D = 0

c) Lossy dielectric: (dielectric with finite conductivity) ( ε =ε o ε r , µ = µo µr and σ ≠ 0 )

Sinusoidal time varying fields:

Types of media ( according to the values of ε , µ andσ ):

Complex form (general medium):

Solution of source-free wave equation (complex form):

Properties of plane wave :

Classification of media

(3.1) Solution of wave equation in bounded media