# 24.4-7, 24.11x

Urban and Civil

Nov 16, 2013 (4 years and 5 months ago)

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Four equations (integral form) :

Gauss’s law

Gauss’s law for magnetism

Ampere
-
Maxwell law

0
ˆ

inside
q
dA
n
E

dA
n
B
dt
d
l
d
E
ˆ

dt
d
I
l
d
B
elec
path
inside
0
_
0

+ Lorentz force

B
v
q
E
q
F

Maxwell’s Equations

0
ˆ

A
n
B

Time varying magnetic field makes electric field

Time varying electric field makes magnetic field

Fields Without Charges

Key idea: Fields travel in space at certain speed

Disturbance moving in space

a wave?

1. Simplest case: a pulse (moving slab
)

A Simple Configuration of Traveling Fields

vE
B
0
0

E=Bv

vBv
B
0
0

2
0
0
1
v

m/s

8
0
0
10
3
1

v
Based on Maxwell’s equations, pulse must propagate at speed of light

E=cB

A Pulse: Speed of Propagation

Electromagnetic pulse can propagate in space

How can we initiate such a pulse?

Short pulse of transverse

electric field

Accelerated Charges

1.
Transverse pulse
propagates at speed of
light

2.
Since
E(t)

there must
be
B

3.
Direction of
v

is given
by
:

B
E

E

B

v

Accelerated Charges

We can qualitatively predict the direction.

What is the magnitude?

Magnitude can be derived
from Gauss’s law

Field ~
-
qa

r
c
a
q
E
2
0
4
1


1.

The direction of the field is opposite to

qa

2. The electric field falls off at a rate

1/
r

Magnitude of the Transverse Electric Field

Field of an accelerated charge

1

2

3

4

vT

𝑎

A

B

Φ
𝑆

𝛼

Φ


Φ

Accelerates for
t
, then coasts for
T

at
v=at

to reach B.

cT

ct

𝜃

r>
cT

o
bserve

Φ

; outer shell

Φ


inner shell of acceleration zone

𝐸


>
𝐸

since B is closer, but
Φ

=

Φ

since areas compensate

Φ

+
Φ

=
0

No charge

Φ
𝑆
=
0

𝐸
𝑆

𝐸
𝑟𝑎

𝐸
𝑡𝑎𝑛

tan

(
𝛼
)
=
𝐸
𝑡𝑎𝑛
𝐸
𝑟𝑎

=
𝑣𝑖𝑛
(
𝜃
)
𝑐

𝐸
𝑡𝑎𝑛
=
𝐸
𝑟𝑎
𝑣𝑖𝑛
(
𝜃
)
𝑐

Field of an accelerated charge

1

2

3

4

vT

𝑎

A

B

Φ
𝑆

𝛼

Φ


Φ

cT

ct

𝜃

𝐸
𝑆

𝐸
𝑟𝑎

𝐸
𝑡𝑎𝑛

𝐸
𝑡𝑎𝑛
=
𝐸
𝑟𝑎
𝑣𝑖𝑛
(
𝜃
)
𝑐

𝐸
𝑟𝑎
=
1
4𝜋
𝜀
0


2

𝐸
𝑡𝑎𝑛
=
1
4𝜋
𝜀
0


2
𝑣𝑖𝑛
(
𝜃
)
𝑐

𝑎
=
𝑣
/


𝐸
𝑡𝑎𝑛
=

4𝜋
𝜀
0
𝑎𝑖𝑛
(
𝜃
)
𝑐
2



=

/
c

𝑎𝑖𝑛
𝜃
=
𝑎

𝐸
𝑟𝑎𝑖𝑎𝑡𝑖𝑣
=

4𝜋
𝜀
0

𝑎

𝑐
2


Plane Electromagnetic Waves

A
plane
wave consists of electric and magnetic fields that vary in
space only in the direction of the wave propagation.

The fields are perpendicular
to each other and to the
direction of propagation.

p
p
ˆ
,sin
ˆ
,sin
E x t E kx t j
B x t B kx t k

 
 
𝐸

Positive Charge in EM wave

A particle will experience electric
force during a short time

d/c:

qE
F
elec

p

p

0

F
e
l
e
c

t

q
E

d
c

What will happen to the ball?

It will oscillate

Energy was transferred from E/M field to the ball

m
c
qEd
m
p
K
K
2
1
2
0
2
2
Amount of energy in
the pulse is ~
E
2

m
c
qEd
K
2
1
2
Ball gained energy:

Pulse energy must decrease

)
(J/m

3
2
0
2
0
1
2
1
2
1
B
E
Volume
Energy

E=cB

2
0
0
2
0
2
0
2
0
1
1
2
1
1
2
1
2
1
c
E
c
E
E
Volume
Energy

2
0
E

There is E/M energy stored in the pulse:

)
(J/m

3
2
0
E
Volume
Energy

Pulse moves in space:

there is energy flux

Units:

J/(m
2
s) = W/m
2

During

t
:

E
n
e
r
g
y

0
E
2

A

c

t

2
0
E
c
t
A
Energy
flux

EB
flux
0
1

used:

E=cB,

0

0
=1/c
2

Energy Flux

EB
flux
0
1

The direction of the E/M radiation was given by

B
E

Energy flux, the “Poynting vector”:

)
1
0
2
W/m

(in

B
E
S

S

is the rate of energy flux in E/M radiation

It points in the direction of the E/M radiation

John Henry

Poynting

(1852
-
1914)

Energy Flux: The Poynting Vector

Intensity,

𝐼
=

In the vicinity of the Earth, the energy density of radiation
emitted by the sun is ~1400 W/m
2
. What is the approximate
magnitude of the electric field in the sunlight?

Solution:

EB
flux
0
1

2
0
E
c

N/C

725
0

c
flux
E
Note: this is an average (rms) value

Exercise

A laser pointer emits ~5 mW of light power. What is the
approximate magnitude of the electric field?

Solution:

1.
Spot size: ~2 mm

2.
flux = (5
.
10
-
3
W)/(3.14
.
0.001
2

m
2
)=1592 W/m
2

3.
Electric field:

N/C

773
0

c
flux
E
(rms value)

What if we focus it into 2 a micron spot?

Flux will increase 10
6

times,
E

will increase 10
3

times:

N/C

000
,
773

E
Exercise

E

field starts motion

Moving charge in magnetic field:

B
v
q
F
mag

F
mag

What if there is negative charge?

B
v
q
F
mag

F
mag

What is its magnitude?

Average speed:

v/
2

c
E
v
q
B
v
q
F
mag
2
2

1
2
)
/(
2


c
v
qE
c
E
v
q
F
F
elec
mag

Net momentum:

in transverse direction: 0

in longitudinal direction: >0

Relativistic energy:

E
2

p
c

2

m
c
2

2
Quantum view: light consists of photons with zero mass:

2
2
pc
E

Classical (Maxwell): it is also valid, i.e. momentum = energy/speed

B
E
S

0
1

)
N/m

(in

2
B
E
c
c
S

0
1

Momentum flux:

Momentum Flux

Units of Pressure

What is the magnitude of the electric field due to sunlight
near
the Earth
(

~
𝐼
=
1400
W/m
2
)? What is the force
due to
sunlight
on a sail with the area 1 km
2

at this location?

Solution
:

If
reflective surface?

2
N/m

6
10
3
.
9

Total force on the sail:

N

3
.
9

F
Exercise: Solar Sail

Atmospheric pressure is ~ 10
5

N/m
2

=
1
𝜇
0
𝐸𝐵
=
1
𝜇
0
𝐸
𝐸
𝑐
=
𝐸
2
𝜇
0
𝑐

𝐸
=
𝜇
0
𝑐

=
725 N/C

If absorbed
, pressure =

𝑆

=
1400

W
/
m
2
3
×
1
0
8
m
/
s
=
4
.
66
×
1
0

6

N/m
2

Electric fields are not blocked by matter: how can
E

decrease?

Re
-

Positive charge

Electromagnetic Spectrum

Need to create oscillating motion of electrons

LC circuit: can produce oscillating motion of charges

To increase effect: connect to antenna

Visible light

Heat up atoms, atomic vibration can reach visible frequency range

Transitions of electrons between different quantized levels

How can we produce electromagnetic radiation of a desired frequency?

AC voltage

(~300 MHz)

What will happen if distance is increased twice?

no

light

polarized

along one axis…

…and it can be
unpolarized
:

𝐸
𝑟𝑎𝑖𝑎𝑡𝑖𝑣
=

4𝜋
𝜀
0

𝑎

𝑐
2


Making polarized light

Turning polarization

Polaroid sunglasses and camera filters:

reflected light is highly polarized:

can block it

Considered:

using polarized car lights and polarizers
-
windshields

Polarized Light

In which of these situations will the
bulb light?

A)
A

B)
B

C)
C

D)
None

E)
B and C

Why there is light coming from the sky?

Why is it polarized?

Why is it blue?

x

x
0
s
i
n

t

E
~
a

d
2
x
d
t
2

y
0

2
s
i
n

t

Energy flux:

4
2
~
~

E
Ratio of blue/red frequency is ~2

scattering intensity ratio is 16

Why is sun red at sunset? Is its light polarized?

Why the Sky is Blue

Why there is no light going through a cardboard?

Electric fields are not blocked by matter

Electrons and nucleus in cardboard reradiate light

Behind the cardboard reradiated E/M field cancels original field

Cardboard

1.

too small to be observed in most cases

2.
E/M fields can affect charged particles: nucleus and electrons

Both fields (E and M) are always present

they ‘feed’ each other

But usually only electric field is considered (
B=E/c
)

Effect of E/M Radiation on Matter

Effect of Radiation on a Neutral Atom

Main effect:

brief electric kick sideways

Neutral atom:

polarizes

Electron is much lighter than nucleus:

can model atom as outer electron
connected to the rest of the atom by a
spring:

F=eE

Resonance

See 15.P.47

t
E
E
y

sin
0

t
F
eE
F
y
y

sin
0

Amplitude of oscillation will depend
on how close we are to the natural
free
-
oscillation frequency of the ball
-
spring system

Resonance

E/M radiation waves with frequency ~10
6

Hz has big effect on
mobile electrons in the metal of radio antenna:

can tune radio to a single frequency

E/M radiation with frequency ~ 10
15

Hz has big effect on organic
molecules:

retina in your eye responds to visible light but not radio waves

Very high frequency (X
-
rays) has little effect on atoms and can