Transcript Document 7122630

Maxwell’s Equations
Chapter 32, Sections 9, 10, 11
Maxwell’s Equations
Electromagnetic Waves
Chapter 34, Sections 1,2,3
The Equations of Electromagnetism
(at this point …)
Gauss’ Law for Electrostatics
 E  dA  
 B  dA  0
q
0
Gauss’ Law for Magnetism

Faraday’s Law of Induction

Ampere’s Law
 B  dl   I

dB
E  dl  
dt
0

The Equations of Electromagnetism
Gauss’s Laws
1
2
 E  dA  
..monopole..
q
0
 B  dA  0
...there’s no
magnetic monopole....!!
?
The Equations of Electromagnetism
Faraday’s Law
3

dB
E  dl  
dt
Ampere’s Law
4
 B  dl   I
.. if you change a
magnetic field you
induce an electric
field.........
0
.......is the reverse
true..?
Look at charge flowing into a capacitor
Ampere’s Law
 B  dl  0I
Here I is the current piercing the
flat surface spanning the loop.

B
E
Look at charge flowing into a capacitor
Ampere’s Law
 B  dl  0I
B
E
Here I is the current piercing the
flat surface spanning the loop.
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.
Side view: (Surface
is now like a bag:)
E
B
Look at charge flowing into a capacitor
It must still be the case that B
around the little loop satisfies
B
E
 B  dl   I
0
where I is the current in the
wire. But that current does
not pierce the surface.
What does pierce the
surface? Electric flux - and
that flux is increasing in
time.
E
B
Look at charge flowing into a capacitor
q  0 EA
dq
d(EA)
I
 0
dt
dt
dE
I  0
dt
Thus the steady current in the
wire produces a steadily
increasing electric flux. For the
sac-like surface we can write
Ampere’s law equivalently as
dE
 B  dl  00 dt
B
E
E
B
Look at charge flowing into a capacitor
B
E
The best way to write this result is
dE
 B  dl  0I  00 dt
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.
E
B
Maxwell-Ampere Law
B
This result is Maxwell’s
modification of Ampere’s law:
E
dE
 B  dl  0I  00 dt
Can rewrite this by defining the
displacement current
(not really a current) as
dE
I d  0
dt
Then
 B  dl   (I  I )
0
d
Maxwell-Ampere Law
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:
E
A changing electric field induces a magnetic field
This is easy to see: just apply the new version of Ampere’s 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
dE
 B  dl  00 dt
dE
B2r  00 r 
dt
2
 B
00 r dE
2
dt
Maxwell’s Equations of Electromagnetism
Gauss’s Law for Electrostatics
 E  dA  
q
0
Gauss’s Law for Magnetism
 B  dA  0

Faraday’s Law of Induction


Ampere’s Law
dE
 B  dl  0I  00 dt

dB
E  dl  
dt
Maxwell’s Equations of Electromagnetism
Gauss’s Law for Electrostatics
 E  dA  
q
0
Gauss’s Law for Magnetism
 B  dA  0

Faraday’s Law of Induction


Ampere’s Law
dE
 B  dl  0I  00 dt

dB
E  dl  
dt
These are as symmetric as can be between electric and
magnetic fields – given that there are no magnetic charges.
Maxwell’s Equations in a Vacuum
Consider these equations in a vacuum: no charges or currents
 E  dA  
q
0
 B  dA  0
dB 
 E  dl   dt


dE
 B  dl  0I  00 dt



 E  dA  0
 B  dA  0

dB
E  dl  
dt
dE
 B  dl  00 dt
Maxwell’s Equations in a Vacuum
Consider these equations in a vacuum: no charges or currents
 E  dA  
q
0
 B  dA  0
dB 
 E  dl   dt


dE
 B  dl  0I  00 dt



 E  dA  0
 B  dA  0

dB
E  dl  
dt
dE
 B  dl  00 dt
These integral equations have a remarkable property: a wave solution
Plane Electromagnetic Waves
Ey
Bz
dE
 B  dl  00 dt
dB
 E  dl   dt
This pair of equations is

solved simultaneously by:
c
x
E(x, t) = EP sin (kx-t) ĵ
as long as
B(x, t) = BP sin (kx-t) ẑ
E p Bp   /k  1/ 00
F(x)

Static wave
F(x) = FP sin (kx + )
k = 2  
k = wavenumber
 = wavelength
x
F(x)

Moving wave
v
x
F(x, t) = FP sin (kx - t)
 = 2  f
 = angular frequency
f = frequency
v=/k
F
v
x
At time zero this is F(x,0)=Fpsin(kx).
Moving wave
F(x, t) = FP sin (kx - t )
F
v
x
Moving wave
F(x, t) = FP sin (kx - t )
At time zero this is F(x,0)=Fpsin(kx).
Now consider a “snapshot” of F(x,t) at a later fixed time t.
F
v
x
Moving wave
F(x, t) = FP sin (kx - t )
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 sin{k[x-(/k)t]}
This is the same as the time-zero function, slide to the right a
distance (/k)t.
F
v
x
Moving wave
F(x, t) = FP sin (kx - t )
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 sin{k[x-(/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
Plane Electromagnetic Waves
E(x, t) = EP sin (kx-t) ĵ
B(x, t) = BP sin (kx-t) ẑ
These are both waves, and both have wave speed /k.
Plane Electromagnetic Waves
E(x, t) = EP sin (kx-t) ĵ
B(x, t) = BP sin (kx-t) ẑ
These are both waves, and both have wave speed /k.
But these expressions for E and B solve Maxwell’s equations only if
 /k 1/ 00
Hence the speed of electromagnetic waves is c 1/ 00 .


Plane Electromagnetic Waves
E(x, t) = EP sin (kx-t) ĵ
B(x, t) = BP sin (kx-t) ẑ
These are both waves, and both have wave speed /k.
But these expressions for E and B solve Maxwell’s equations only if
 /k 1/ 00
Hence the speed of electromagnetic waves is c 1/ 00 .
Maxwell plugged in the values of the constants and found

c 1/
00  3 108 m /s  the speedof light

Plane Electromagnetic Waves
E(x, t) = EP sin (kx-t) ĵ
B(x, t) = BP sin (kx-t) ẑ
These are both waves, and both have wave speed /k.
But these expressions for E and B solve Maxwell’s equations only if
 /k 1/ 00
Hence the speed of electromagnetic waves is c 1/ 00 .
Maxwell plugged in the values of the constants and found

c 1/
00  3 108 m /s  the speedof light

Plane Electromagnetic Waves
E(x, t) = EP sin (kx-t) ĵ
B(x, t) = BP sin (kx-t) ẑ
These are both waves, and both have wave speed /k.
But these expressions for E and B solve Maxwell’s equations only if
 /k 1/ 00
Hence the speed of electromagnetic waves is c 1/ 00 .
Maxwell plugged in the values of the constants and found

c 1/
00  3 108 m /s  the speedof light

Thus Maxwell discovered that light is electromagnetic radiation.
Plane Electromagnetic Waves
Ey
E(x, t) = EP sin (kx-t) ĵ
B(x, t) = BP sin (kx-t) ẑ
Bz
• Waves are in phase.
• Fields are oriented at 900 to one
another and to the direction of
propagation (i.e., are transverse).
8
c

1/



3
10
m /s
• Wave speed is c
0 0
• At all times E=cB.


c
x
The Electromagnetic Spectrum