Download Chapter 32: Maxwell`s Equation and EM Waves

Chapter 32: Maxwell’s Equation
and EM Waves
Slide 29-1
Equations of electromagnetism: a review
•  We’ve now seen the four fundamental equations of electromagnetism,
here listed together for the first time.
•  But one is incomplete: Ampère’s law needs refining
Slide 29-2
Maxwell’s
Adjustment to Ampere’s Law
I
~ = µ0 Ienc
~ · dr
B
•  For the situation on right,
Ampere’s law predicts that the B
field depends on which
Amperian loop is used.
•  Can’t have contradictory
results – either there is a B
field or there isn’t!
•  Maxwell postulated that a
changing electric flux acts as
a source of magnetic fields
(in addition to currents)
Slide 29-3
Maxwell’s Adjustment to Ampere’s Law
•  Need to add a term to the right side of
Ampere’s law to account for the changing
electric flux. This is the displacement
current.
•  Notice, for a parallel plate capacitor, the
electricZflux is
E
=
~ = EA =
~ · dA
E
Q
A=
✏0
✏0
d E
I
=
dt
✏0
•  So Maxwell added a displacement current
(Ampere-Maxwell
term to Ampere’s law:
•  The the rate of change of the flux is
I
✓
⇤ · d⇤⇥ = µ0 (I + Id )enc = µ0 I +
B
•  Corresponding displacement current
density:
d E
0
dt
◆
Law)
enc
~
d
E
J~d = ✏0
dt
Slide 29-4
•  Changing B field induces E Field.
•  Changing E field induces B field
I
d E
~
~
B · dl = µ0 ✏0
dt
I
~ =
~ · dl
E
d
B
dt
Slide 29-5
Consider a large parallel plate capacitor as shown, charging so
that Q = Q0+βt on the positively charged plate. Assuming the
edges of the capacitor and the wire connections to the plates can be
ignored,
what is the magnitude of the magnetic field B halfway between the
plates, at a radius r?
s
z
a
r
I
Q
I
-Q
d
Slide 29-6
I
⇤ · d⇤⇥ = µ0
B
d E
0
dt
r<R:
I
2⇡rB = µ0 ✏0 ⇡r
✏0 ⇡R2
r
B = µ0 I
(r
<
R)
2⇡R2
2
Slide 29-7
I
⇤ · d⇤⇥ = µ0
B
d E
0
dt
r>R:
I
2⇡rB = µ0 ✏0 ⇡R
✏0 ⇡R2
2
µ0 I
B=
(r > R)
2⇡r
Slide 29-8
Maxwell’s equations
•  These equations form a complete description of electric and
magnetic fields
•  Combined with the Lorentz force, F = q(E + vxB), they form the
complete theory of electromagnetism (classically).
Slide 29-9
Maxwell’s equations in vacuum
•  In a vacuum there’s no electric charge density and also no
current density.
Maxwell’s equations in vacuum
•  A changing electric field is a source for a magnetic field,
and a changing magnetic field is a source for an electric
field.
•  These equations infer the possibility of electromagnetic
waves!
Slide 29-10
The Wave Equation
•  Maxwell’s equations can be manipulated to show that, for
situations where E and B only vary in the x direction:
2⇤
⇥ E
= µ0
2
⇥x
⇤
⇥2B
= µ0
2
⇥x
2⇤
⇥ E
0
⇥t2
⇤
⇥2B
0
⇥t2
•  more generally,
2⇤
E
⇥
2⇤
r E = µ0 0 2
⇥t
r2 =
2
x2
x̂ +
2
y2
2⇤
B
⇥
2⇤
r B = µ0 0 2
⇥t
ŷ +
2
z2
ẑ
Slide 29-11
Waves on a String
•  Let’s consider first mechanical waves
(Ch. 15):
•  Consider a string with tension T and
linear mass density µ
•  Application of Newton’s 2nd law gives the
wave equation for the string:
@ 2 y(x, t)
µ @ 2 y(x, t)
=
2
@x
T
@t2
•  Sinusoidal Solutions:

y(x, t) = A cos 2⇡
✓
x
t
P
◆
Slide 29-12
Waves on a String
@ 2 y(x, t)
µ @ 2 y(x, t)
=
2
@x
T
@t2

✓
y(x, t) = A cos 2⇡
(2⇡)2
2
µ (2⇡)2
=
T P2
phase velocity:
P
x
=
t
P
s
◆
T
µ
Alternate Expression:
y(x, t) = A cos (kx
k = 2⇡/
!t)
! = 2⇡f = 2⇡/P
Slide 29-13
Clicker Question
Two traveling waves 1 and 2 are described by the equations.
y1 (x, t) = 2sin(2x − t)
y2 (x, t) = 4sin(x − 2t)
All the numbers are in the appropriate SI (mks) units.
Which wave has the higher speed?
A) Wave 1
B) Wave 2
C) Both have the same speed.
Slide 29-14
Clicker Question
@2E
@x2
@2E
✏ 0 µ0 2 = 0
@t
•  What is the velocity of an electromagnetic wave?
1. 
✏ 0 µ0
1
2.  ✏ µ
0 0
3. 
4. 
r
p
1
✏ 0 µ0
✏ 0 µ0
Slide 29-15
Electromagnetic Waves
•  Maxwell postulated (1865) that light is an electromagnetic wave.
•  Electromagnetic waves travel in a vacuum with the speed of
r
light,
1
c=
✏ 0 µ0
= 3 ⇥ 108 m/s
•  Electromagnetic waves can exist having any frequency, not just
at the frequencies of light.
•  Foreshadow of radio waves!
•  Confirmed by Heinrich Hertz (1887)
Slide 29-16
Plane Waves as Solutions
2
@ E
@x2
@ 2 y(x, t)
1 @y(x, t)
= 2
2
@x
v
@t2
2
@ E
✏ 0 µ0 2 = 0
@t
E(x, t) = E0 sin(kx
!t)
(in y direction)
B(x, t) = B0 sin(kx
!t)
(in z direction)
k 2 E0 sin(kx
⇥t) =
0 µ0 (
)⇥ 2 E0 sin(kx
⇥t)
This is a solution if
⇥
v= =
k
1
0 µ0
Also, since kEp =
=c=3
108 m/s
Bp
1
Bp = Ep
c
Slide 29-17
Electromagnetic Plane Waves
•  Plane wave are waves propagating in one direction with one
wavelength. (E and B do not vary with respect to the other two
dimensions)
•  E and B are transverse to
each-other and to direction
of propagation
(E ⇥ B gives direction of propagation)
•  No medium required for
wave to propagate!
•  Example (shown in figure):
~
E(x,
t) = Ep sin(kx !t) ĵ
~
B(x,
t) = Bp sin(kx !t) k̂
Slide 29-18
Clicker question
• 
At a particular point, the electric field of an
electromagnetic wave points in the + y direction, while
the magnetic field points in the − z direction. Which of
the following describes the propagation direction?
A.  + x
B.  − x
C.  either + x or − x but you can’t tell which
D.  − y
Slide 29-19
Clicker question
•  A planar electromagnetic wave is propagating through
space. Its electric field vector is given by
⇥ = Ep cos(kz
E
t)î
Its magnetic field vector is
⇥ = Bp cos(kz
1) B
⇥ = Bp cos(ky
2) B
t)ĵ
t)k̂
⇥ = Bp cos(ky
3) B
⇥ = Bp cos(kz
4) B
t)k̂
⇥ = Bp sin(kz
5) B
t)î
t)ĵ
Slide 29-20
The Electromagnetic Spectrum
Slide 29-21
Clicker question
•  Which type of radiation travels with the highest speed?
1.  visible light
2.  X-rays
3.  Gamma-rays
4.  radio waves
5.  they all have the same speed
Slide 29-22
An Active Galaxy seen in Multiple
Wavelengths
Slide 29-23
Antennae
An electric field parallel to an antenna
(electric dipole) will “shake” electrons
and produce an AC current.
A magnetic dipole antenna (for AM
radios) should be oriented so that the
B-field passes into and out of the
plane of a loop (as the wave passes
by), inducing a current in the loop.
Slide 29-24
Producing electromagnetic waves
•  Electromagnetic waves are generated ultimately by accelerating
electric charge.
•  Details of emitting systems depend on wavelength, with most efficient
emitters being roughly a wavelength in size.
•  Radio waves are generated by alternating currents in metal antennas.
•  Molecular vibration and rotation produce infrared waves.
•  Visible light arises largely from atomic-scale processes.
•  X rays are produced in the rapid deceleration of electric charge.
•  Gamma rays result from nuclear processes.
A radio transmitter and antenna
Electric fields of an oscillating electric dipole
Slide 29-25
Energy in EM Waves
•  Recall that the energy density due to electric and magnetic fields
(in a vacuum) is given by
1
1 2
2
u = uE + uB =
2
✏0 E +
2µ0
B
•  The Poynting vector describes the rate of energy flow per unit
area (W/m2 in SI). It’s magnitude is given by
(B 2 = E 2 /c2 = µ0 ✏0 E 2 )
S = uc = (uE + uB )c = E 2 c
1
~
•  A more general equation is S~ = E~ ⇥ B
µ0
•  For plane waves (traveling in x direction with E oriented in z
direction):
1
~
S=
Ep Bp cos2 (kx
µ0
!t)î = ✏0 Ep2 c cos2 (kx
!t) î
Slide 29-26
Energy in EM waves
•  Electromagnetic waves transport energy
•  Averaging over the time variations of the
oscillating fields gives the average value, also
called the intensity:
1
I =< S >= ✏0 Ep2 c
2
•  For an isotropic source of radiation having a
luminosity (output power) L:
L
I=
4⇡r2
•  Notice this implies that the electric field goes
as 1/r (rather than 1/r2 for stationary charges)
Slide 29-27
Example: The luminosity of the Sun is L=3.8 x 1026 W. Earth is a
distance r=1.5 x 1011 m away. What is the intensity of sunlight here
on Earth?
Slide 29-28
Flux and Solar Heating
Flux is the rate of energy incident
on a surface per unit area
Slide 29-29
•  The US uses about 4 million GWh of energy per year.
Roughly what surface area would be needed to supply
this energy solely using solar farms?
Slide 29-30
Seasons
Why is it hotter in summer than in winter?
1.  The Earth is closer to the Sun.
2.  The Earth is tilted towards the Sun, causing half the
Earth to be closer to the Sun
3.  The Earth is tilted towards the Sun, causing the
sunlight to be more direct (more intense).
4.  The Sun becomes more luminous.
Slide 29-31