Modern Physics
Unit 1: Classical Models and the Birth of Modern Physics
Lecture 1.3: Energy in a Wave, Radiation Pressure,
and Constructive/Deconstructive Interference
Ron Reifenberger
Professor of Physics
Purdue University
1
Maxwell’s Equations - Fundamental Properties
of E&M (1864)
+
S
E diverges from point charge
I
N
Changing B induces current
B
I
B
B is continuous
Current produces B
2
Maxwell’s Equations - Modern Notation
two constants of proportionality : ε 0 and µo
3
Using Equations for E and B fields,
Maxwell Predicts an Electromagnetic
Wave!
The wave is continuous
λ
c = f λ
Units: [c] in m/s;
[λ] in m; [f] in s-1 = Hz
E
and
c
important predictions from Maxwell :=
ε 0 µo c 2 1=
B
1
c is velocity of EM wave=
=
constant =3 × 108 m/s
ε 0 µo
4
Prediction: the Electromagnetic Spectrum
c = f λ
Violet
~ 4.3 x 10-7 m
~ 698 THz
visible light – wavelengths
to
to
to
Red
~7.5 x 10-7 m
~ 400 THz
5
Subsequent work from 1864-1890s
predicted many properties of EM waves
Focus on energy transport
 Intensity
 Power
 Energy density
 Momentum transfer
 Radiation pressure
6
Energy is transported by an EM wave (1880s)
The Poynting vector S specifies the instantaneous power per
unit area transported by an EM wave at a point in space at
an instant of time.
[E(t) and B(t) represent instantaneous values]
E(t)
 1   E(t) B(t)
S = E ×B =
μo
μo
c=
B(t)
Notation: E(t)=Eo sin (ωt)
E (t)
B (t)
=
1
εo μo
 E2 ( t )
S=
= cεo E2 ( t )
cμo

Eo2
S
= cεo
AVERAGE
2
2
= cεo Erms
Units: [W/m2]
7
The time-averaged value of S is called the
intensity I of the wave

I≡ S
1
2
= cεo Eo2 = cεo Erms
AVERAGE
2
Eo
1
since c =
=
Bo
εo μo

I≡ S
AVERAGE
2


B
1
1
2 2
o
= cεo ( c Bo )=
cεo 

2
2
ε
μ
 o o
1 c 2 c 2
=
Bo = Brms
2 μo
μo
The intensity I specifies the power (in
Watts per m2) carried by an EM wave in
free space, averaged over time.
8
Be able to distinguish between
closely related concepts
 Intensity (in W/m2) of an EM wave: I
 Power (in W or J/s) carried (or transmitted) by EM wave through
an area A: P=I×A
 Energy (in J) carried by an EM wave in a time ∆t: U=P∆t=(I×A)∆t
 Energy density (in J/m3) of an EM wave: utot=I/c
These quantities (I, P, U and utot) are used to
define the momentum and the radiation pressure
exerted by an EM wave.
9
The time averaged energy density of an EM wave
KEY IDEA: Energy is stored in
both E and B fields
Total absorber
Hole in mask,
area A
time aver. power thru hole : P = IA [W ]
time aver. energy thru hole : ΔU = PΔt [J ]
 J 
Define time aver. energy density utot in 3  :
 m 
c∆t
ΔU
PΔt
IAΔt
utot =
=
=
(AcΔt ) (AcΔt ) (AcΔt )
I 11
1 c 2
2
=  cε=
E
Bo 

o o 
c c2
 c  2μo 
1
1 2
= εoEo2 =
Bo
2
2μo
utot =
utot
2
utot = εoErms
=
1 2
Brms
μo
Units: J/m3
utot
Energy U
passing thru
hole in time ∆t
Time aver.
radiation
intensity I
over time
∆t
Important to distinguish between what we
can measure and what we can calculate.
10
An EM wave exerts a net force on absorber
absorber
Because of mutual effect of both E
and B, a net force is exerted on
charged particles at the surface of
an absorber. A net force means a
change in momentum over time.
Recall:
Δv Δ ( mv ) Δp
In general, F = ma = m
=
=
Δt
Δt
Δt
where Δp is change in momentum.
Electric Force
(F=qE) gives charge
q a velocity v.
Lorentz Force:
F=qv×B
11
Consequence of net force on absorber
1. For EM radaition*, U = pc, so
ΔU = cΔp (valid for total adsorption)
If EM wave has time aver. power P, intensity I, we have
ΔU PΔt IAΔt
= =
c
c
c
time aver. net momentum delivered by EM wave when totally adsorbed
=
Δp
2. If EM wave with time aver. power P, intensity I, we also have a
time aver. radiation force
Fradiation
Pradiation
PΔt
Δp
P IA
=
= c = =
Δt
Δt c c
Fradiation I time aver. radiation pressure exerted by EM wave 
=
=


c when totally adsorbed (units : Nm-2 or Pa)
A

* We use U for energy because E is already used for electric field. Using this notation in a
familiar context, for example, of particles – we would write U=½ mv2 =½ pv
12
Interference
- A Phenomenon Unique to Waves -
Q: What happens when two waves “collide”?
A: Waves combine according to the Principle of Superposition – the
resulting wave is the sum of the displacements of the two waves.
Water Waves
from Two
Sources
13
Huygens Principle (1629-1695)
How to predict how a wave will propagate? Huygens claims each point on a wave
contains a future version of the wave itself.
At some time t, every point on
a wavefront can be considered
as the source of a secondary
wave (a spherical wavelet). If
ALL these secondary wavelets
move outward at the same
speed as the original wave,
then their tangent line at some
later time t’ will define the
location of the new wavefront.
Wavefront: an
“imaginary” line in space
that defines where all
points on the wave have
same phase.
ray
New wavefront
tangent line
r=vt’
Wave
front
wavefront
14
Young’s Double Slit (1803)
Key Idea: Two coherent light beams interfere
after following different paths
bright when dsinΘ =nλ
n = integer
Θ
(n=1)
d
Out of phase
by 180o (or π)
(n=1)
What do the wavy lines represent?
15
SUMMARY
1
εoμo
Speed of EM Waves (m/s)
c=
Ratio of Peak Fields
Eo = cBo
Definition of Wavenumber (m-1)
k≡
Frequency-Wavelength
relationship
λf = c
 1  
S = E ×B
μo
Definition of Poynting Vector S
(W/m2)
Energy density
(J/m3)
Intensity of EM Wave (W/m2)
(time average of Poynting vector
for sinusoidal, linearly polarized ,
plane EM wave)
Momentum Transfer
(complete absorption)
2π
λ
(kg m/s)
Radiation Pressure ( N/m2 or
Pa) (complete absorption)
utot
Eo2 1 Bo2
1  2 1 2  1  Eo2 1 Bo2 
=  εo E + B  =  εo
+
=
 =εo
2
μo  2  2 μo 2 
2 μo 2
EoBo
Eo2
cBo2 εo c 2
I= S =
=
=
=
Eo
2μo 2μo c 2μo
2
p=
U
; U = utot × volume
c
I
Pradiation =
c
16
Conclusion
By the late 1800’s virtually every
aspect of an EM wave was
understood
17