Transcript 24.4-7, 24.11x

Maxwell’s Equations
Four equations (integral form) :
q
Gauss’s law

 E  nˆdA 
Gauss’s law for magnetism

 B  nˆA  0
Faraday’s law
Ampere-Maxwell law
+ Lorentz force
inside
0


 
d 
 E  dl   dt  B  nˆdA
 
d elec 

 B  dl  0  I inside_ path   0 dt 


 
F  qE  qv  B
Fields Without Charges
Time varying magnetic field makes electric field
Time varying electric field makes magnetic field
A Simple Configuration of Traveling Fields
Key idea: Fields travel in space at certain speed
Disturbance moving in space – a wave?
1. Simplest case: a pulse (moving slab)
A Pulse: Speed of Propagation
B  0 0vE
E=Bv
B  0 0vBv
1  0 0v 2
v
1
0 0
 3  108 m/s
E=cB
Based on Maxwell’s equations, pulse must propagate at speed of light
Accelerated Charges
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: E  B
E
v
B
Magnitude of the Transverse Electric Field
We can qualitatively predict the direction.
What is the magnitude?
Magnitude can be derived
from Gauss’s law
Field ~ -qa

Eradiative 

1  qa
40 c 2 r
1. The direction of the field is opposite to qa
2. The electric field falls off at a rate 1/r
Field of an accelerated charge
Φ𝑆
𝐸𝑆
1
cT
4
Φ𝐵
3
vT
ct
𝐸𝑡𝑎𝑛
𝐸𝑟𝑎𝑑
r>cT observe Φ𝐴 ; outer shell
𝛼
Φ𝐴
𝜃
A 𝑎B
Accelerates for t, then coasts for T
at v=at to reach B.
2
Φ𝐵 inner shell of acceleration zone
𝐸𝐵 > 𝐸𝐴 since B is closer, but
Φ𝐵 =−Φ𝐴 since areas compensate
Φ𝐵 + Φ𝐴 = 0 No charge Φ𝑆 = 0
𝐸𝑡𝑎𝑛
𝑣𝑇𝑠𝑖𝑛(𝜃)
tan(𝛼) =
=
𝐸𝑟𝑎𝑑
𝑐𝑡
𝐸𝑡𝑎𝑛
𝑣𝑇𝑠𝑖𝑛(𝜃)
= 𝐸𝑟𝑎𝑑
𝑐𝑡
Field of an accelerated charge
Φ𝑆
𝐸𝑆
1
cT
4
Φ𝐵
3
𝐸𝑟𝑎𝑑
𝐸𝑟𝑎𝑑
𝛼
𝐸𝑡𝑎𝑛
2
1 𝑞
=
4𝜋𝜀0 𝑟 2
1 𝑞 𝑣𝑇𝑠𝑖𝑛(𝜃)
=
4𝜋𝜀0 𝑟 2
𝑐𝑡
𝑎 = 𝑣/𝑡
𝐸𝑡𝑎𝑛
vT
ct
𝐸𝑡𝑎𝑛
Φ𝐴
𝜃
A 𝑎B
𝐸𝑡𝑎𝑛
𝑣𝑇𝑠𝑖𝑛(𝜃)
= 𝐸𝑟𝑎𝑑
𝑐𝑡
𝑇 = 𝑟/c
𝑞 𝑎𝑠𝑖𝑛(𝜃)
=
4𝜋𝜀0 𝑐 2 𝑟
𝑎𝑠𝑖𝑛 𝜃 = 𝑎⊥
𝑞 −𝑎⊥
𝐸𝑟𝑎𝑑𝑖𝑎𝑡𝑖𝑣𝑒 =
4𝜋𝜀0 𝑐 2 𝑟
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.
𝐸
E  x, t   Ep sin  kx  t  ˆj
B  x, t   Bp sin  kx  t  kˆ
Positive Charge in EM wave
Energy of E/M Radiation
A particle will experience electric
force during a short time d/c:
Felec  qE
 d
p  p  0  Felec t  qE  
 c
What will happen to the ball?
It will oscillate
Energy was transferred from E/M field to the ball
2
p 2  qEd   1 
K  K  0 

 

2m  c   2m 
Amount of energy in
the pulse is ~ E2
Energy of E/M Radiation
Ball gained energy:
2
 qEd   1 
K  
 

 c   2m 
Pulse energy must decrease
Energy 1
1 1 2
 0E 2 
B (J/m 3 )
Volume 2
2 0
E/M radiation: E=cB
Energy 1
1 1 E
1
1 
2
2
2

 0E 
    0 E 1 


E
2 
0
Volume 2
2 0  c 
2


c
0 0


2
Energy Flux
There is E/M energy stored in the pulse:
Energy
  0 E 2 (J/m 3 )
Volume
Pulse moves in space:
there is energy flux
Units: J/(m2s) = W/m2
During t:


Energy   0 E 2 A  ct 
Energy
flux 
 c 0 E 2
 A  t 
flux 
1
0
EB
used: E=cB, 00=1/c2
Energy Flux: The Poynting Vector
flux 
1
0
EB
 
The direction of the E/M radiation was given by E  B
Energy flux, the “Poynting vector”:
 1  
S
EB
0
(in W/m 2 )
• S is the rate of energy flux in E/M radiation
• It points in the direction of the E/M radiation
Intensity, 𝐼 = 𝑆
John Henry
Poynting
(1852-1914)
Exercise
In the vicinity of the Earth, the energy density of radiation
emitted by the sun is ~1400 W/m2. What is the approximate
magnitude of the electric field in the sunlight?
Solution:
flux 
E
1
0
EB  c 0 E 2
flux
 725 N/C
c 0
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)/(3.14.0.0012 m2)=1592 W/m2
3. Electric field:
E
flux
 773 N/C
c 0
(rms value)
What if we focus it into 2 a micron spot?
Flux will increase 106 times, E will increase 103 times:
E  773,000 N/C
Momentum of E/M Radiation
• E field starts motion
• Moving charge in magnetic field:

 
Fmag  qv  B
Fmag
What if there is negative charge?

 
Fmag   q v  B
‘Radiation pressure’:
What is its magnitude?
Average speed: v/2
v
vE
Fmag  q B  q
2
2 c
Fmag
vE
v
q
/( qE ) 
 1
Felec
2c
2c
Fmag
Momentum Flux
Net momentum:
in transverse direction: 0
in longitudinal direction: >0
Relativistic energy:
 
E   pc  mc
2
2
2 2
Quantum view: light consists of photons with zero mass: E 2   pc 2
Classical (Maxwell): it is also valid, i.e. momentum = energy/speed
 1  
S
EB
0
Momentum flux:

S
1  

E  B (in N/m 2 )
c 0 c
Units of Pressure
Exercise: Solar Sail
What is the magnitude of the electric field due to sunlight near
the Earth ( 𝑆 ~𝐼 =1400 W/m2)? What is the force due to
sunlight on a sail with the area 1 km2 at this location?
Solution:
1
1 𝐸
𝐸2
𝑆 = 𝐸𝐵 = 𝐸 =
𝜇0
𝜇0 𝑐 𝜇0 𝑐
If absorbed, pressure =
𝑆
𝑐
=
1400 W/m2
3×108 m/s
𝐸=
= 4.66 × 10−6 N/m2
If reflective surface?
Total force on the sail: F  9.3 N
Atmospheric pressure is ~ 105 N/m2
𝜇0 𝑐 𝑆 =725 N/C
 9.3  106 N/m 2
Re-radiation: Scattering
Positive charge
Electric fields are not blocked by matter: how can E decrease?
Electromagnetic Spectrum
E/M Radiation Transmitters
How can we produce electromagnetic radiation of a desired frequency?
Need to create oscillating motion of electrons
Radio frequency
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
Polarized E/M Radiation
AC voltage
(~300 MHz)
𝐸𝑟𝑎𝑑𝑖𝑎𝑡𝑖𝑣𝑒
𝑞 −𝑎⊥
=
4𝜋𝜀0 𝑐 2 𝑟
What will happen if distance is increased twice?
E/M radiation can be polarized
along one axis…
no
light
…and it can be unpolarized:
Polarized Light
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
In which of these situations will the
bulb light?
A)
B)
C)
D)
E)
A
B
C
None
B and C
Why the Sky is Blue
Why there is light coming from the sky?
Why is it polarized?
Why is it blue?
x  x0 sin  t 
d2x
E ~ a  2  y0 2 sin  t 
dt
Energy flux: ~ E 2 ~  4
Ratio of blue/red frequency is ~2  scattering intensity ratio is 16
Why is sun red at sunset? Is its light polarized?
Cardboard
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
Effect of E/M Radiation on Matter
1. Radiative pressure – 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 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
Radiation and Neutral Atom: Resonance
E y  E0 sin t 
Fy  eE y  F0 sin t 
Amplitude of oscillation will depend
on how close we are to the natural
free-oscillation frequency of the ballspring system
Resonance
Importance of Resonance
E/M radiation waves with frequency ~106 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 ~ 1015 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
pass through matter (your body):
X-ray imaging