ELECTROMAGNETICS AND APPLICATIONS

Lecture 5Normal Incidence at Media Interfaces

Luca Daniel

L3-2

• Review of Fundamental Electromagnetic Laws

• Electromagnetic Waves in Media and Interfaceso The EM waves in homogenous Mediao Electromagnetic Power and Energyo EM Fields at Interfaces between Different Media

Fields at boundaries: normal components Fields at boundaries: tangential components Fields inside perfect conductors Fields at boundaries of perfect conductors

o EM Waves Incident “Normally” to a Different Medium Normal incidence to a perfect conductor Standing waves (time domain view and energy) Normal incidence to a dielectric Power balance

o EM Waves Incident at General Angle to a Different Medium

Today’s Outline

TodayToday

General Boundary Conditions:

1 2 s

1 2

n D D

n B B 0

1 2n E E 0

1 2 sn H H J

1 1E ,D

2 2E ,D

1 1H ,B

2 2H ,B

s coulombs/m2

Js

Amperes/m

2 2 2 2E D H B 0

1 s

1

1

1 s

n D

n B 0

n E 0

n H JH is parallel to perfect conductors (and is terminated by surface current)

E is perpendicular to perfect conductors (and is terminated by surface charge)

Inside Perfect Conductors:

Summary of Boundary Conditions

Perfect conductor

sJ

Surface currentAmperes/m

Line current over a perfect conductor

perpendicularH 0

Point charge over a perfect conductor

Perfect conductor

Surface chargeCoulombs/m2

s

tangentialE 0n

n

s

s

n H J

J H

L3-7

Two Examples: quasi-static fields from charge and current near perfect conductors

o s

s

o

n E

E

EM Waves with “Normal” Incidence to Perfect Conductors

Solution Method for Boundary Value Problems:

1) Assume fields on both sides of the boundary in terms of unknown coefficients; typically a sum of terms

2) Write equations for fields that satisfy boundary conditions3) Solve for unknowns and check answer with Maxwell Equations

Example—Normal incidence on perfect conductor:

1) Incident:

2) Match B.C.:

3) Solve:

jkzi iˆE (z) yE e

oE z,tc =

y

z0

jkzr r

jkzt t

ˆReflected: E (z) yE e

ˆTransmitted: E (z) yE e 0

//

r ti

E (0) 0 :

E (z 0) E (z 0) E (z 0) 0

jk0 jk0+ i r r iˆ ˆyE e yE e 0 E E

(given)

Standing Waves – Time Domain View and Energy

PerfectConductor

We [J/m3] = (1/2) E(t,z)|2 = 2 Ei2 sin2kz sin2t

(Where does the energy go?) t = +/2

t =

t = -/2

z

z = 0

E

Standing waves oscillate without moving

Never any We here

Incident:

Reflected: jkzr iˆE y( E )e

jkzi iˆE yE e

Total: jkz jkzi iˆ ˆE yE e e y 2j E sinkz

j te iˆE(t,z) R Ee 2yE sinkz sin tTime Domain:

E = 0 every half cycle (t = 0, , etc.) and every half wavelength for any t

Electric EnergyDensity:

y

2iE

z sin2kz sin2 t

S E H

1

S Re E H 02

= 0 when t = /2, 3, etc.

t = t = 0

SH

t =

23 2 2 2m i

1W [J/m ] = μ H(t,z) = 2ε E cos kz cos ωt2

Standing Waves Magnetic Field

z = 0

2iE

z 4 sinkz coskz sin t cos t

Incident:

Reflected:

Total:

jkzi iˆE yE e

jkzr iˆE y( E )e

jkz jkzi

i

ˆE yE e e

y2jE sinkz

jkz jkzi

i

ˆH x E ( e e )

x2 E coskz

jkzr iˆH +x E e

jkzi iˆH x E e

Time Domain: iˆE(t,z) 2yE sinkz sin t iˆH(t,z) 2x E coskz cos t

s

i

ˆJ n H(z 0 )

ˆ2y E cos t

Magnetic EnergyDensity:

x

z

Normal Incidence to Dielectrics

Normal Incidence

iE

iH rH

rE

x tE

tH

y

z

1

1

i

t

tr

i i

EE and

E E

2

t

t i

i t

0 00 0 00 tjk jk jki r t i r tˆ ˆ ˆAt z : yE e yE e yE e E E E

0 00 0 00

tjk jk jkti r

i i t

EE Eˆ ˆ ˆAt z : x e x e x e

ti r

i i t

EE E

Boundary Conditions for the Electric Field

Boundary Conditions for the Magnetic Field (no surface currents)

t i

t i

1

1

t i

t i

( / )

( / )

xx

Normal Incidence to Dielectrics – Power Balance

Normal Incidence

iE

iH rH

rE

x tE

tH

y

z

0

2

t

t

0 t

0

0

t

t

xx

2

0

i

i i iE

ˆS E H z2

0

r

rE

ˆS z2

tt

t

EˆS z

22

2 r r

i i

S E

S E

220

20

t t t

ti i

S E /T

S E /

21

Example: 04 t

0 0

04 2

t

1

3 2

3

2223

11 899

%

111

9 %

1 or

or