electromagnetic waves maxwell`s equations lecture 23 thursday: 8 april 2004

Electromagnetic WavesMaxwell`s Equations

Lecture 23

Thursday: 8 April 2004

ELECTROMAGNETIC WAVES

• E = Em sin(kx - t)

• B = Bm sin(kx - t)

k fT

ck

f

22

2

wave speed

k fT

ck

f

22

2

wave speed

POYNTING VECTOR

•This is a measure of power per area. Units are watts per meter2.

•Direction is the direction in which the wave is moving.

S E B 1

0S E B

1

0

POYNTING VECTOR

•However, since E and B are perpendicular,

S EB

E

Bc

Sc

Ec

B

1

1

0

0

2

0

2

and since

S EB

E

Bc

Sc

Ec

B

1

1

0

0

2

0

2

and since

INTENSITYS

cE kx t

S Ic

E

S Ic

E

EE

m

m

rms

rmsm

1

1

2

1

2

0

2 2

0

2

0

2

sin ( )

Intensity

Sc

E kx t

S Ic

E

S Ic

E

EE

m

m

rms

rmsm

1

1

2

1

2

0

2 2

0

2

0

2

sin ( )

Intensity

RADIATION PRESSUREF

p

t c

U

t cIA

FIA

c

FIA

c

1 1

2

(complete absorption)

(complete reflection)

Fp

t c

U

t cIA

FIA

c

FIA

c

1 1

2

(complete absorption)

(complete reflection)

RADIATION PRESSURE

Pressure =

(complete absorption

(complete reflection)

F

A

pI

c

pI

c

r

r

)

2

Pressure =

(complete absorption

(complete reflection)

F

A

pI

c

pI

c

r

r

)

2

RADIATION PRESSUREMomentum:

(complete absorption)

(complete reflection)

pU

c

pU

c

2

Momentum:

(complete absorption)

(complete reflection)

pU

c

pU

c

2

Comet: Picture taken WEDNESDAY 26 March

1997,at 2000 CST,at White Bear Lake,Minnesota.

MAXWELL’S EQUATIONSMAXWELL’S EQUATIONSMAXWELL’S EQUATIONSMAXWELL’S EQUATIONS

E A E s

B A B s

dq

dd

dt

d d id

dt

B

E

0

0 00

E A E s

B A B s

dq

dd

dt

d d id

dt

B

E

0

0 00

AMPERE’S LAWOriginal:

As modified by Maxwell:

B s

B s

d i

d id

dtE

0

0 0

Original:

As modified by Maxwell:

B s

B s

d i

d id

dtE

0

0 0

Reasons for the Extra Term SYMMETRY

CONTINUITY

SYMMETRY A time varying magnetic field produces an

electric field.

A time varying electric field produces a magnetic field.

SYMMETRYMaxwell' s Equations in Free Space

q i

d dd

dt

d dd

dt

B

E

0 0

0

0 0 0

E A E s

B A B s

Maxwell' s Equations in Free Space

q i

d dd

dt

d dd

dt

B

E

0 0

0

0 0 0

E A E s

B A B s

CONTINUITY

CONTINUITYWithout Maxwell' s modification:

At

At

At

P d i

P d

P d i

1 0

2

3 0

0

:

:

:

B s

B s

B s

Without Maxwell' s modification:

At

At

At

P d i

P d

P d i

1 0

2

3 0

0

:

:

:

B s

B s

B s

CONTINUITYWith Maxwell's modification:

At

At

At

"Displacement current"

P d i

P dd

dti

P d i

d

dti

Ed

Ed

1 0

2 0 0 0

3 0

0

:

:

:

B s

B s

B s

With Maxwell's modification:

At

At

At

"Displacement current"

P d i

P dd

dti

P d i

d

dti

Ed

Ed

1 0

2 0 0 0

3 0

0

:

:

:

B s

B s

B s

CONTINUITYi i

q CV

CA

dV Ed

qA

dEd AE

idq

dt

d

dti

d

E

Ed

0

0 0 0

0

i i

q CV

CA

dV Ed

qA

dEd AE

idq

dt

d

dti

d

E

Ed

0

0 0 0

0

AMPERE’S LAW

AMPERE’S LAWFor path 1:

For path 2:

For path 3:

since

B s

B s

B s

d i

d i i

d i i

i id

d

0

0 0

0 0

( )

( )

For path 1:

For path 2:

For path 3:

since

B s

B s

B s

d i

d i i

d i i

i id

d

0

0 0

0 0

( )

( )

ELECTROMAGNETIC WAVES

Maxwell' s Equations in Free Space

q i

d dd

dt

d dd

dt

B

E

0 0

0

0 0 0

E A E s

B A B s

Maxwell' s Equations in Free Space

q i

d dd

dt

d dd

dt

B

E

0 0

0

0 0 0

E A E s

B A B s

ELECTROMAGNETIC WAVES

E

x

B

t

B

x

E

t

E

B

E

Bc

c

m

m

0 0

0 0

8130 10. m/s

E

x

B

t

B

x

E

t

E

B

E

Bc

c

m

m

0 0

0 0

8130 10. m/s

MAXWELL’S EQUATIONSMAXWELL’S EQUATIONSMAXWELL’S EQUATIONSMAXWELL’S EQUATIONS

E A E s

B A B s

dq

dd

dt

d d id

dti i

B

E

d

0

0 0

0

0

( )

E A E s

B A B s

dq

dd

dt

d d id

dti i

B

E

d

0

0 0

0

0

( )