0 Bv Dt

x

D

jH

tx

BE

Maxwell’s equations

constitutive relations

• D = E

• B = H

• j = E

jE

EE

HH

tt

HExHEEH xx

dsHEx dvxHE

Poynting’s theorem

22

x

E1 Hdv

2 t t

dv

E H ds

E j

Poynting’s theorem

• em power that leaves

• = - (stored em energy) / t

• - lost em powerE x H

Spherical radiation

• E x H • ds = EH 42

E2 42

constant

• E 1/

Cylindrical radiation

• E x H • ds = EH 2r L E2 2r L constant

• E 1/(r)1/2

r L

aLa

I

L

Vx o

2

2 dsHE

22 E1 Hdv 0

2 t t

oV Ix 2 aL

L 2 a

E H ds

2o2

V Idv L a

L a

E j

oV Ix 2 aL

L 2 a

E H ds

2E1dv

2 t

0 dvjE21 H

dv 02 t

2

V1 d

dA2 t

21 A V2 d t

xE

ωsinωtd

ε

2

rH

t

B

dlH

dst

D

r

d

V = Vocos t

= 2rH

t

V

d

επr2

Boundary conditions

E1 E2

B1 B2

Pillbox S Loop L

components

Integral form of Maxwell’s equations

dsHx

dvD

dvB

dsEx dlE

dsB

t

dlH dsD

j

t

dsD dvv

dsB 0

Normal component of B

Bn1s + Bn2 s = 0

Bn1 Bn2

Normal components of B are continuous

No magnetic monopoles!!

dvB dsB 0

Normal component of D

Dn1s + Dn2 s = s s

Dn1 Dn2

Normal components of D differ by surface charge

density

Electric charge

dvD dsD dvv dss

Tangential component of H

Ht1L + Ht2 L = js L

Ht1 Ht2

Tangential components of H differ by surface current

density

surface current

dsHx dlHdsD

j

t dljs

Tangential component of E

Et1L + Et2 L = 0 Et1 Et2

Tangential components of E are continuous

dsEx dlE

dsB

t0

example

2n4 E 1 5 3

5 4 1 n tE u u

?2E1= 1 2= 4

s 2C3

m

2 2 4 n tE u u

example

2y 1yB B0

4 1

5 4 1 x yB u u

?2B

A3 ms yJ u

5 16 2 x yB u u

1= 1 2= 4

xy

example

2y 1yB B3

4 1

5 4 1 x yB u u

?2B

A3 ms zJ u

5 28 2 x yB u u

1= 1 2= 4

xy

Tangential components of the electric field intensity are continuous

Normal components of the displacement flux density differ by a surface charge density

Normal components of the magnetic flux density are continuous

Tangential components of the magnetic field intensity differ by a surface current density

Tangential component of Emetal

Et1L + Et2 L = 0 Et1 Et2

Tangential component of E is zero.

but Et2 = 0

dsEx dlE

dsB

t0

images• Charge +Q• Equipotential

contours• Electric field E

• Image charge -Q

• Equipotential contours

• Electric field E

images

• Charge +Q

• Image charge -Q

2d

• Image charge –Q

• Image charge +Q

• Image charge +Q

Antenna on top of the ground

• Underneath an antenna is an array of conductors

• This creates a ground plane• This effectively makes the antenna

twice as tall

an egg – after

an egg – before

tooth pick

CD

microwave oven experiments -- dangerous

aluminum foil