Maxwell’s Equations and
Electromagnetic Waves
Setting the Stage - The
Displacement Current
• Maxwell had a crucial “leap of insight”...
Will there still be a magnetic
field around the capacitor?
A Beautiful Symmetry...
• A changing magnetic
flux produces an Electric
field
• A changing electric
flux produces a Magnetic
field
An extension of Ampere’s
Law...
• Maxwell reasoned that Ampere’s Law
would also apply to the displacement
current.
Bdl


(
I

I
)
o
displacement

Clever application of
Gauss’ Law here!
d E dQ
o

 I displacement
dt
dt
Maxwell’s Equations (first
glimpse)
• Faraday’s Law:
d
Edl


B
dA
n


dt
• Ampere’s Law:
d
Bdl


I



E
dA
o
o
o
n


dt
Maxwell’s Equations – Integral
Form
•
Faraday’s Law:
•
Ampere’s Law:
•
Gauss’ Law
d
 Edl   dt  Bn dA
d
 Bdl  o I  o o dt  En dA
Q
 E  dA 
o
 B  dA  0
Maxwell’s Equations –
Differential Form
 E  dA  
V
Q    dV
V

 E 
o
(  E )dV 
Q
o
Gauss’ Theorem – integral
of a flux equals volume
integral of divergence
Maxwell’s Equations –
Differential Form
d
 Edl   dt  Bn dA
d
 E  dl   (  E )dA   dt  Bn dA
B
 E  
Stoke’s Theorem: “Integral
t
around the path equals flux
of the curl”
Maxwell’s Equations –
Differential Form
d
 Bdl   (  B)dA  o o dt  En dA
E
  B   o o
t
Maxwell’s Equations –
Differential Form

E 
o
B
 E  
t
B  0
E
  B   o o
t
The Wave Equation
• How fast will a wave
travel along a string of
density ?
Two Ways to M’Es…
• Abstract:
 ( A)  ( A)   A
2
• Physical:
– Imagine a plane wave
of electric field in zdirection
 0 
E ( x, t )   0 
 Ez ( x, t ) 
Go to Rob Salgado’s sim of this
Moving Fields…
• Moving E-Field
leads to…
• Moving B-Field
leads to…
It’s Alive!
• Well, at least it’s a wave! Combining
the last two equations leads us to:
 Ey
2
x
2
 Ey
2
  o o
t
2
• example - consider the electric field part of an
electromagnetic wave described by:
E ( x, t )  Eo sin(kx   t ) j  Eo cos(kx   t )k
The Poynting Vector
• Light waves (and all electromagnetic waves)
carry energy
EB
u  uE  uB 
o c
• A wave has an intensity
I  uaverage c
Poynting Vector
1 Eo Bo
I
S
av
2 o
S
1
o
EB
Radiation Pressure
• Light waves (and all electromagnetic waves)
exert pressure
I
P
c
Accelerating Charges Radiate
Power!
• We can show (by dimensional argument) that
an accelerating charge should radiate energy
at a rate given by:
2
kq a
P a 3
c
2
• More detailed argumentation reveals that a = 2/3