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Lecture 32Energy and momentum.
Standing waves.
Energy in a EM wave
Energy density due to an electric field: 20
12
u Eε=
Energy density for an EM wave: 2 20
0
1 12 2
u E Bεµ
= +
but0 0
BE cBε µ
= =
2 20 0 0
0
1 12 2
u E Eε ε µµ
= +
20u Eε=
Energy density equallysplit between E, B fields
Energy density due to a magnetic field: 2
0
12
u Bµ
=
Energy transport
How much energy goes through a surface of area A in time dt ?
Energy in this “box”:
y
x
z
propagation
cdt
20dU udV E Acdtε= =
Energy flow per unit time and per unit area: 20
1 dUS cEA dt
ε= =0
EBµ
=
Definition: Poynting vector0
1S E Bµ
= ×r rr Energy flow per unit
time and per unit area
I S=Intensity:
ACT: Plane harmonic wave
z
x
y
Propagation• P
x
z
y
At the time shown, the magnetic field point P (on the y axis) is:
A. Bmax i
B. Bmax j
C. 0
E/B are the same at all points in each yz plane!
Propagation direction is ,so is in the direction and is in the direction
E BE x
B y
×r r
r
r
Energy in the harmonic wave
( ) ( )2max max
0 0
1 1 ˆˆcosS E B E B kx t j kωµ µ
= × = − ×rr
( )maxˆcosE E kx t jω= −
r
( )maxˆcosB B kx t kω= −
r
( )2max max
0
1 ˆcosE B kx t iωµ
= −
( )2max max max max
0 0
1 1cos2
I S E B kx t E Bωµ µ
= = − =
Direction +x (as expected…)
max max0
12
I S E Bµ
= =
ACT: Emax and distance
An isotropic radio transmitter emits power in all directions. What is the ratio of the amplitudes of the E field at distances of 100 m and 200 m from the source E max(100)/E max(200) ?
A. 1B. 2C. 4
Energy is uniformly distributed in a sphere of radius r (r = distance to source):
2
power powerarea 4
Irπ
= =
max max
02E BI S
µ= =Intensity is
max1Er
µ
max
max
E cB
=
2max
02Ecµ
=
Emission of EM waves
How does an EM wave begin? When a charge is accelerated.
Moving charged infinite sheet
Oscillating dipole
Whenever a charge is accelerated, it loses energy due to radiation.
→ Bad thing when you’re trying to accelerate a particle
→ Good thing when you can use the radiation!
• synchrotron radiation produces X-rays
• detection of black holes
• any emission antenna
Momentum
EM waves carry energy… and momentum. (And mechanical waves, too, btw.)
Basic idea of momentum: p = mv→ Mass m moving with speed v (say to the right)
→ (Kinetic) energy flows (to the right)
21 12 2
KE mv pv= =
A very hand-waiving trick to get momentum without the mass:
KEpv
:
Using the proper mathematical tools (special relativity), one obtains
lightKE pc=
Radiation pressure
If EM waves carry radiation, they can exert a force (and thus a pressure) when they hit a surface:
1pressure pFA A t
∆= =
∆Power
cA=
Ic
=
if radiation is completely absorbedlightKEp
c∆ =
1 KEcA t
=∆
Sc
=0
EBcµ
=
0
2 2If radiation is completely reflected, , so pressurelightKE EBpc cµ
∆ = =
Light pressure, though “light”, has noticeable effects comet’s tail →pushed away from the sun*.
*Note: The dust tail is pushed away by radiation; the ion tail is pushed away by the solar wind!
Standing electromagnetic waves
EM wave propagating between two plates of a perfect conductor:
Conducting wall ⇒ E-field must be zero there
⇒ original wave and reflected waves produce a standing wave with condition E = 0 on both ends:
( ) ( )max maxcos cosyE E kx t E kx tω ω= − − +
( ) ( )max2 sin sinE kx tω= −
E-field nodes:
nodes30, , , ...
2 2Ex λ λλ− =
( )( )
max
max
Original wave: cos
Reflected wave: cosy
y
E E kx tE E kx t
ω
ω
= −
= − + x
y E
λ
And the B field?
( )( )
max
max
Original wave: cos
Reflected wave: cosy
y
E E kx tE E kx t
ω
ω
= −
= − +
( )( )
max
max
cos cos
z
z
B B kx tB B kx t
ω
ω
⇒ = −
⇒ = +
Note: No minus sign for !We need propagation
BE B× =r r
( ) ( )( ) ( )
max max
max
cos cos 2 cos cos
zB B kx t B kx tB kx t
ω ω
ω
= − + +
= −
B-field nodes:
nodes3 5, , ...
4 4 4Bx λ λ λ− =
x
zB
λ
DEMO: Marshmallows and microwave
Doppler effect
Just like for mechanical waves, if the source or the observer of an EM are moving, the received frequency can be different from the emitted frequency.
The equations are different, though, because…
… nothing can go faster than light!
We can’t simply add velocities à la Galilean. We need relativity.
Spaceship moves with speed v Light from star travels at
c +v relative to spaceship???
Spaceship moves with speed v
The equations are different, though, because…
Spaceship moves with speed v
The equations are different, though, because…
Spaceship moves with speed v Light from star travels at
c +v relative to spaceship???
The equations are different, though, because…
Spaceship moves with speed v
No relativity in 222, so let’s forget about the math.
But here’s some examples of Doppler’s effect in EM radiation anyway:
• police speed radars
• weather radars (detect motion of rain droplets)
• in astronomy: red shift/blue shift
Most stars are made of H, so their spectrum must be the same
Spectrum of the sun (optical wavelengths)
Spectrum of object X
Lines are at λ larger than expected (red shift)
Object must be moving away from us
It turns out that all distant galaxies are moving away from us
The universe must be expanding!
