Maxwell s Equations and Electromagnetic
Waves
(Chapter 35)
James Clerk Maxwell, Scottish Physicist (1831-1879)
Everyday Experience: Cars on the Highway
•  Car 2 has velocity v2=100 mph
•  Car 1 is a police car. It has velocity v1=65mph
Magnetic Force: Frame of Reference Determines
Velocity!
•  When we say an object is moving
with velocity v we always mean a
velocity v relative to a particular
reference frame .
•  The velocity of the particle is
different with respect to different
reference frames.
The Lorentz force:
!
! ! !
F = q E+v!B
!
!
From F = ma, how do you
(
)
distinguish electric and magnetic forces?
Galilean Transformations
!
An object has a velocity v measured relative to
frame S. Measured in the frame S', the same object
has the velocity:
! ! !
v ' = v !V
The acceleration of the object measures in frame S
!
is dv / dt. The acceleration of the object measured
in frame S' is:
!
!
!
!
dv ' dv dV dv
=
!
=
dt dt dt dt
Acceleration is the same in both reference frames!!
!
!
!
F = ma = mdv / dt (Frame S)
!
!
!
F ' = ma ' = mdv '/ dt (Frame S')
Inertial reference frames move at a constant velocity relative to each other.
The laws of physics must be the same in all inertial reference frames!
What happened to the force ?
In S’ the particle is stationary. S is a frame moving with
velocity v relative to S.
In S’, what is the force on the particle?
Electric and Magnetic Fields are the Same Force!
The total force is independent of the inertial reference frame!
!
! !
F = qv ! B (Frame S)
!
! !
F ' = qv ! B (Frame S')
In frame S', the velocity is zero.
There can only be an electric force in S' :
!
!
F ' = qE ' (Frame S')
! ! !
E ' = v ! B (Relationship between B field in S and E field in S')
E or B? It Depends on Your Perspective
Whether a field is seen as electric or magnetic
depends on the motion of the reference frame relative
to the sources of the field.
The Galilean field transformation equations are
where V is the velocity of frame S' relative to frame S and
where the fields are measured at the same point in space by
experimenters at rest in each reference frame.
NOTE: These equations are only valid if V << c.
Ampère s law Revisited: Maxwell s Addition
Whenever total current Ithrough
passes through an area bounded
by a closed curve, the line
integral of the magnetic field
around the curve is
The figure illustrates the
geometry of Ampère s law. In
this case, Ithrough = I1 − I2 .
Ampere s Law: B Field Independent of Surface
Shape!
•  Ampere s law makes no reference to the shape of the surface
through which the current flows.
•  It only makes reference to the boundary of that surface at which
the magnetic field is evaluated.
•  Two different surfaces that have the same boundary must
therefore have the same current flowing through them.
Ampere s Law: Something is Not Correct!
•  Consider a capacitor that is being charged/discharged.
•  Through the surface S1, there is a current I: No problem using
Ampere s law.
•  S1 and S2 have the same boundary: Magnetic field is the same
for both surfaces.
There is a Current Flowing Between the
Capacitor Plates
! e = EA = Electric flux between capacitor plates
Q
!e =
"0
d! e 1 dQ
=
= I / "0
dt
" 0 dt
d! e
I = "0
dt
Change in the electric flux is the same
as the current in the wire. Therefore Ampere's
law gives correct answer for S2 if we modify it
to be:
! !
d! e &
#
Bid
s
=
µ
I
+
"
(
0%
0
")
$
dt '
The Displacement Current
The electric flux due to an electric field E across a surface
area A is
! !
! !
! e = "" EidA # Ei A (for constant E field)
The displacement current is defined as
Maxwell modified Ampère s law to read
Maxwell s Equations
!
! ! !
PLUS: F = q E + v ! B
(
)
Maxwell Predicted the Electromagnetic
Spectrum We Use Today....
•  Before Maxwell, everything outside visible light was
unknown. He predicted that e.m. waves like light should exist
at higher and lower frequencies than light.
•  In 1886 Heinrich Hertz in Germany built the first primitive
radio receiver and antenna and proved Maxwell s prediction
by producing and measuring radio waves.
•  Nikola Tesla filed the first patent for radio communication in
1897. Beginning of wireless age
First: A Review of Waves....
Traveling Waves
f (x) = y = y0 sin(kx)
Consider the substitution: x ! x ± vt
gL (x,t) = y0 sin(kx + kvt) = y0 sin(kx + " t) (Wave moves to the left at speed v)
gR (x,t) = y0 sin(kx # kvt) = y0 sin(kx # " t) (Wave moves to the right at speed v)
Traveling Waves: Frequency and Period
gR (x,t) = y0 sin(kx ! " t)
Each point on the wave moves to right at velocity v = " / k.
If we look at a single position x0 , it is only a function of time.
Take x0 = 0 for simplicity:
gR (0,t) = !y0 sin(" t)
Properties of Traveling Waves
Mathematical Function for traveling wave: y = y0 sin ( kx ± ! t ) OR y0 cos ( kx ± ! t )
1. Wavelength: " (Distance in x between peaks of the wave at a single time)
2. Wave Number: k = 2# / "
3. Period: T (Time between peaks of the wave at single point)
4. Frequency: f = ! / 2# = 1 / T
5. Speed of propagation: v = ! k = " f
6. Direction of propagation: + sign= towards the left. - sign=towards the right.
Traveling waves obey the differential equation:
2
!2 y
!
y
2
=v
2
!t
!x 2
(You must know this equation!)
“Derivation” of Electromamagnetic Waves
Maxwell's Equations (Starting Point of Derivation):
! !
! !
"! EidA = 0
"! BidA = 0
! !
! !
d ! !
d ! !
"! Eids = " dt ! BidA "! Bids = !0µ0 dt ! EidA
Faraday’s Law
Ampere’s Law
Only One Wave- Not Two!
Ampere's Law and Faraday's Law yield:
!E
!E
!Bz
!Bz
=" y
AND
= "µ 0! 0 y
!t
!x
!x
!t
Differentiating each with respect to x and t they can be combined as:
!2 E y
!2 E y
= µ 0! 0 2
!x 2
!t
AND
!2 Bz
!2 B z
= µ 0! 0 2
!x 2
!t
Hence, E y and Bz represent the same wave traveling in the x-direction.
Let E y = E0 f (x " vt) AND Bz = B0 f (x " vt)
Velocity:
!E y
!Bz
="
# vB0 f '(x " vt) = E 0 f (x " vt)
!t
!x
# vB0 = E0
The magnitude of the E field determines that of the B field and vice versa.
Electromagnetic waves can
exist at any frequency, not
just at the frequencies of
visible light. This
prediction was the harbinger
of radio waves.
Properties of Electromagnetic Waves:
!
!
1. E and B are perpendicular to direction of
propagation. ("transverse waves")
!
!
2. E is perpendicular to B.
! !
3. E ! B is in the direction of propagation.
4.The waves travel at the speed c=1/ " 0 µ0
!
!
5. E = c B everywhere.
Electromagnetic Spectrum in Perspective
Energy and Power of Electromagnetic Waves
•  Electromagnetic waves transmit energy.
James Bond in “Goldfinger”
Energy Flow of Electromagnetic Waves
The energy flow of an electromagnetic wave is described
by the Poynting vector defined as
The magnitude of the Poynting
vector is
S has units W/m2 or power per area.
!0 ! 2
1 !2
U=
E +
B = energy density of electromagnetic field.
2
2 µ0
! !
"U
### "t dV + "
## SidA = 0 Conservation of Energy
Intensity is a More Useful Measure for Waves.
•  Since E and B are oscillating in time, S is not
constant but also oscillates.
•  What is useful is the average energy transfer over
many oscillations.
•  The intensity is the Poynting vector averaged over
the oscillations.
An Example….
The electric field of an EM wave is
!
8 "1
#
E = (20.0V / m) ĵ cos$(6.28 !10 m )x " ! t %&
a)What is the wavelength?
b)What is the frequency?
c) What is the magnetic field amplitude?
d) In what direction is the magnetic field?
e) What would we call this type of wave ?
Radiation
Pressure
•  Electromagnetic waves
carry energy
•  They also carry
mechanical momentum
that can exert a force on
objects, p=E/c
Momentum transfered by light
absorption:
energy absorbed
!p =
c
Crook s
Radiometer
Radiation Pressure is Different for Reflected,
Absorbed, and Transmitted Light
Radiation Pressure (absorption):
Prad = Frad / A =
1 " !pabs % 1 " 1 !Erad %
$
'= $
' = I /c
A # !t & A # c !t &
I=light intensity
A=area of surface
In general:
Prad = QPR I / c
Force
At the earth s surface
the radiation pressure
due to sun light is
approximately 10-6N/m2. It
is very weak.
Solar Sail : IKAROS (Interplanetary Kite-craft Accelerated by Radiation
Of the Sun) Probe going to Venus 2010 (Japanese Space Agency)
An Example…
You want to suspend horizontally and 8.5in by 11in sheet
of paper with a vertical light beam. Assume the dimensions
of the light beam matches that of the paper. The paper mass
is 1.0g.
What is the light intensity needed?
What if the paper is instead a mirror ?
Dipole Antennas
•  Static charges or constant currents DO NOT produce EM waves
•  An oscillating electric dipole is the simplest and most common
type of antenna for producing electromagnetic waves.
•  Reversing the charges of an electric dipole does two things:
–  It reverses the direction of the electric field.
–  The charge motion results in a current that produces a B-field
How to Make a Dipole Antenna
Attach two lengths of wire to an
oscillating voltage source
Dipole antenna works best when the
wires of length L/2 each satisfy:
c
L = ! /2=
2f
f = frequency of AC voltage
! =c / f =wavelength of radiation.
λ/4
Why do radio stations have large
antennas?
AM Radio stations have frequencies 535-1605kHz.
Imagine a radio station at 700kHz,
3 " 10 8 m / s
!=c/ f =
= 429m
700 " 10 3 Hz
A quarter wavelength dipole antenna must have total length: ! / 4 = 107m # 351 feet
Oscillating Dipole Moment Radiates an Electric
Field That Oscillates
Simultaneously, the current associated with
the changing dipole creates a magnetic field.
Dipole Antennas Produce Spherical Waves NOT
Plane Waves
! sin "
E!
sin(kr # $ t + % )
r
sin 2 "
I(" ) = I 0 2
r
Polarization
•  There are two possible
orientations of the
electric/magnetic fields
for a wave propagating
in any given direction
•  These orientations are
orthogonal to each
other.
Certain materials (particularly
plastics) only allow light that
is polarized along a specific
direction to pass through them.
Polarized sunglasses
only allow vertically polarized
light to pass through and block
horizontally polarized light.
Malus s Law
Suppose a polarized light wave of intensity I0 approaches a
polarizing filter. θ is the angle between the incident plane of
polarization and the polarizer axis. The transmitted intensity is
given by Malus s Law:
If the light incident on a
polarizing filter is
unpolarized, the transmitted
intensity is
In other words, a polarizing filter
passes 50% of unpolarized light
and blocks 50%.
Last Example…
Unpolarized light of intensity I0 is incident of
three polarizing filters:
-First filter has a vertical polarization
-Second filter has polarization 45o relative to
vertical
-Third filter has horizontal polarization.
What is the intensity of the light that gets
through ?
Derivation of Electromagnetic Waves
(Skip for Lecture!)
We will derive the equation for electromagnetic waves.
Since this is an introductory course,
we must make some assumptions:
1. The electromagnetic waves propagate in the x-direction
2. The waves are plane waves: The E and B fields are
the same everywhere in the yz-plane
3. There are no currents or free charges.
Maxwell's Equations (Starting Point of Derivation):
! !
! !
"! EidA = 0
"! BidA = 0
! !
d ! !
"! Eids = " dt ! BidA
! !
d ! !
"! Bids = # 0 µ0 dt ! EidA
Step 1: Ampere-Maxwell Equation
! !
d ! !
" Bids = µ0! 0 dt " EidA
Apply to red rectangle:
! !
" Bids = Bz (x,t)l # Bz (x + dx,t)l
$Ey (
%
d ! !
µ0! 0 " EidA = µ0! 0 ' ldx
&
dt
$t *)
$Ey
Bz (x + dx,t) # Bz (x,t)
= # µ0! 0
dx
$t
let dx + 0,
$Ey
$Bz
= # µ0! 0
$x
$t
Step 2: Faraday s Law
! !
d ! !
"! Eids = " dt ! BidA
Apply to the red rectangle in xy plane:
! !
"! Eids = E y (x + dx,t)l " Ey (x,t)l
#Bz
d ! !
BidA = ldx
!
dt
#t
Combine the two sides:
E y (x + dx,t) " Ey (x,t)
dx
let dx $ 0
#Ey
#B
=" z
#x
#t
#Bz
="
#t
Step 3: Combine Equations.
!Ey
!Ey
!Bz
!Bz
="
= "# 0 µ0
!x
!t
!x
!t
Take x-derivative of 1st eq. and substitute in 2nd eq.:
!2 Ey
!x 2
!2 Ey
2
!2 Ey
! $ !Ey '
! !Bz
! !Bz
= &
="
="
= # 0 µ0 2
)
!x % !x (
!x !t
!t !x
!t
= # 0 µ0
!2 Ey
2
WAVE EQUATION!!
!x
!t
For B-field: Take t-derivative of 1st eq. and subsitute in 2nd eq.
! 2 Bz ! $ !Bz ' ! $ !Ey '
! $ !Ey '
1 ! 2 Bz
= &
= &"
=" &
=
)
2
)
)
!t
!t % !t ( !t % !x (
!x % !t ( # 0 µ0 !x 2
! 2 Bz
! 2 Bz
= # 0 µ0 2
2
!x
!t
AGAIN, WAVE EQUATION!!
Only One Wave- Not Two!
We derived separate wave equations for E and B-fields:
!2 Ey
!2 E y
! 2 Bz
!2 B z
= µ0" 0
AND
= µ0" 0
2
2
2
!x
!t
!x
!t 2
But these are not independent waves since they come from
the same coupled equations:
!Ey
!Ey
!Bz
!Bz
=#
AND
= # µ0" 0
!t
!x
!x
!t
Hence, E y and Bz represent the same wave traveling in the x-direction.
Let Ey = E0 f (x # vt) AND Bz = B0 f (x # vt)
Velocity:
!Ey
!Bz
=#
$ vB0 f '(x # vt) = E 0 f (x # vt)
!t
!x
$ vB0 = E0
The magnitude of the E field determines that of the B field and vice versa.
How to Create Perfect Plane Waves
Shake a flat uniformly charged sheet up and down. This
results in a oscillating current in the yz plane.
The Electric Field is Deformed by Shaking
at Velocity v
•  Information can only
propagate at c
(Einstein s Relativity).
•  The electric field from
the sheet is unchanged
at distances x>cT.
•  x<cT, the electric field
is deformed by the
shaking.
cT
tan ! = E1 / E0 = vT / cT = v / c
! ! ! $ " '$
v '
ˆ
E = E1 + E0 = &
&% i + ĵ )(
)
c
% 2# 0 c (
There is a Resistance to the Shaking...
•  The displacement creates a
field in the y direction that
pushes the charge sheet in
the opposite direction of the
motion.
•  We can calculate how much
must be done by an external
force to overcome this
resistance
!
!
# v! &
dFe = dqE1 = ! dA %
ĵ = Upward force on area element dA of sheet
$ 2" 0 c ('
!
!
# v! &
dFext = )dFe = )! dA %
ĵ
$ 2" 0 c ('
!
!
# v 2! 2 &
dFext ds
Power that must be delivered to sheet: P = *
i dA = A %
dA dt
$ 2" 0 c ('
What About B Field?
•  For x>ct, B-field must
be zero.
•  For x<ct, there is a B
field since the moving
charge sheet is a
current
! #% +( µ0! v / 2)k̂
B1 = $
%& "( µ0! v / 2)k̂
x>0
x<0
Notice:
E1
1
=
= c which is a good sign!
B1 c' 0 µ0
Rate at Which Energy is Carried Away by Fields
! 1 ! !
1 $ v"
S=
E 1 ! B1 =
µ0
µ0 &% 2# 0 c
' $ µ 0" v ' $ v 2" 2 ' ˆ
ĵ ) ! &
k̂ ) = &
i
( % 4 # 0 c )(
( % 2
This is only for the right side! Adding the power per unit area
transmitted by the fields on both sides, one finds that it exactly equals
the mechanical power per unit area delivered by the external force:
P v 2" 2
=
A 2# 0 c
Mechanical Energy of Shaking is Transferred to
the Electromagnetic Radiation
The work you do to shake the sheet is converted to
electromagnetic energy carried away by the waves with 100%
efficiency.