Maxwell

1
Maxwells Equations

• Chapter 32, Sections 9, 10, 11
• Maxwells Equations

Electromagnetic Waves
Chapter 34, Sections 1,2,3
2
The Equations of Electromagnetism (at this point
)
Gauss Law for Electrostatics
Gauss Law for Magnetism
Faradays Law of Induction
Amperes Law
3
The Equations of Electromagnetism
..monopole..
Gausss Laws
1
?
2
...theres no magnetic monopole....!!
4
The Equations of Electromagnetism
.. if you change a magnetic field you induce an
electric field.........
Faradays Law
3
Amperes Law
4
.......is the reverse true..?
5
Look at charge flowing into a capacitor
E
B
Amperes Law
Here I is the current piercing the flat surface
spanning the loop.
6
Look at charge flowing into a capacitor
E
B
Amperes Law
Here I is the current piercing the flat surface
spanning the loop.
E
For an infinite wire you can deform the surface
and I still pierces it. But something goes wrong
here if the loop encloses one plate of the
capacitor in this case the piercing current is
zero.
B
Side view (Surface is now like a bag)
7
Look at charge flowing into a capacitor
E
It must still be the case that B around the
little loop satisfies
B
E
where I is the current in the wire. But that
current does not pierce the surface.
B
What does pierce the surface? Electric flux - and
that flux is increasing in time.
8
Look at charge flowing into a capacitor
E
B
E
B
Thus the steady current in the wire produces a
steadily increasing electric flux. For the
sac-like surface we can write Amperes law
equivalently as
9
Look at charge flowing into a capacitor
E
B
The best way to write this result is
E
B
Then whether the capping surface is the flat
(pierced by I) or the sac (pierced by electric
flux) you get the same answer for B around the
circular loop.
10
Maxwell-Ampere Law
E
B
This result is Maxwells modification of Amperes
law
Can rewrite this by defining the displacement
current (not really a current) as
Then
11
Maxwell-Ampere Law
E
B
This turns out to be more than a careful way to
take care of a strange choice of capping surface.
It predicts a new result
A changing electric field induces a magnetic field
This is easy to see just apply the new version
of Amperes law to a loop between the capacitor
plates with a flat capping surface
B
x
x x x x
x x x x x
x x
12
Maxwells Equations of Electromagnetism
Gausss Law for Electrostatics
Gausss Law for Magnetism
Faradays Law of Induction
Amperes Law
13
Maxwells Equations of Electromagnetism
Gausss Law for Electrostatics
Gausss Law for Magnetism
Faradays Law of Induction
Amperes Law
These are as symmetric as can be between electric
and magnetic fields given that there are no
magnetic charges.
14
Maxwells Equations in a Vacuum
Consider these equations in a vacuum no charges
or currents
15
Maxwells Equations in a Vacuum
Consider these equations in a vacuum no charges
or currents
These integral equations have a remarkable
property a wave solution
16
Plane Electromagnetic Waves
Ey
Bz
This pair of equations is solved simultaneously
by
c
x
as long as
17
Static wave F(x) FP sin (kx ?) k 2?
? ? k wavenumber ? wavelength
Moving wave F(x, t) FP sin (kx - ?t) ?
2? ? f ? angular frequency f frequency v ?
/ k
18
F
v
Moving wave F(x, t) FP sin (kx - ?t )
x
At time zero this is F(x,0)Fpsin(kx).
19
F
v
Moving wave F(x, t) FP sin (kx - ?t )
x
At time zero this is F(x,0)Fpsin(kx). Now
consider a snapshot of F(x,t) at a later fixed
time t.
20
F
v
Moving wave F(x, t) FP sin (kx - ?t )
x
At time zero this is F(x,0)Fpsin(kx). Now
consider a snapshot of F(x,t) at a later fixed
time t. Then
F(x, t) FP sinkx-(?/k)t
This is the same as the time-zero function, slide
to the right a distance (?/k)t.
21
F
v
Moving wave F(x, t) FP sin (kx - ?t )
x
At time zero this is F(x,0)Fpsin(kx). Now
consider a snapshot of F(x,t) at a later fixed
time t. Then
F(x, t) FP sinkx-(?/k)t
This is the same as the time-zero function, slide
to the right a distance (?/k)t. The distance it
slides to the right changes linearly with time
that is, it moves with a speed v ?/k. The wave
moves to the right with speed ?/k
22
Plane Electromagnetic Waves
These are both waves, and both have wave speed
?/k.
23
Plane Electromagnetic Waves
These are both waves, and both have wave speed
?/k. But these expressions for E and B solve
Maxwells equations only if
Hence the speed of electromagnetic
waves is
24
Plane Electromagnetic Waves
These are both waves, and both have wave speed
?/k. But these expressions for E and B solve
Maxwells equations only if
Hence the speed of electromagnetic
waves is Maxwell plugged in the values of the
constants and found
25
Plane Electromagnetic Waves
These are both waves, and both have wave speed
?/k. But these expressions for E and B solve
Maxwells equations only if
Hence the speed of electromagnetic
waves is Maxwell plugged in the values of the
constants and found
26
Plane Electromagnetic Waves
These are both waves, and both have wave speed
?/k. But these expressions for E and B solve
Maxwells equations only if
Hence the speed of electromagnetic
waves is Maxwell plugged in the values of the
constants and found
Thus Maxwell discovered that light is
electromagnetic radiation.
27
Plane Electromagnetic Waves
Ey
Bz
c

• Waves are in phase.
• Fields are oriented at 900 to one another and to
  the direction of propagation (i.e., are
  transverse).
• Wave speed is c
• At all times EcB.

x
28
The Electromagnetic Spectrum
infra -red
ultra -violet
Radio waves
g-rays
m-wave
x-rays