The Laws of Electromagnetism

1
Chapter 34

• The Laws of Electromagnetism
• Maxwells Equations
• Displacement Current
• Electromagnetic Radiation

2
The Electromagnetic Spectrum
infra -red
ultra -violet
Radio waves
g-rays
m-wave
x-rays
3
The Equations of Electromagnetism (at this point
)
Gauss Law for Electrostatics
Gauss Law for Magnetism
Faradays Law of Induction
Amperes Law
4
The Equations of Electromagnetism
..monopole..
Gausss Laws
1
?
2
...theres no magnetic monopole....!!
5
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..?
6
...lets take a look at charge flowing into a
capacitor...
...when we derived Amperes Law we assumed
constant current...
7
...lets take a look at charge flowing into a
capacitor...
...when we derived Amperes Law we assumed
constant current...
E
B
.. if the loop encloses one plate of the
capacitor..there is a problem I 0
Side view (Surface is now like a bag)
8
Maxwell solved this problem by realizing that....
Inside the capacitor there must be an induced
magnetic field...
How?.
9
Maxwell solved this problem by realizing that....
Inside the capacitor there must be an induced
magnetic field...
How?. Inside the capacitor there is a changing E
?
B
A changing electric field induces a magnetic
field
E
10
Maxwell solved this problem by realizing that....
Inside the capacitor there must be an induced
magnetic field...
How?. Inside the capacitor there is a changing E
?
B
A changing electric field induces a magnetic
field
E
where Id is called the displacement
current
11
Maxwell solved this problem by realizing that....
Inside the capacitor there must be an induced
magnetic field...
How?. Inside the capacitor there is a changing E
?
B
A changing electric field induces a magnetic
field
E
where Id is called the displacement
current
Therefore, Maxwells revision of Amperes Law
becomes....
12
Derivation of Displacement Current
For a capacitor, and .
Now, the electric flux is given by EA, so
, where this current , not being associated with
charges, is called the Displacement current,
Id. Hence and
13
Derivation of Displacement Current
For a capacitor, and .
Now, the electric flux is given by EA, so
, where this current, not being associated with
charges, is called the Displacement Current,
Id. Hence and
14
Maxwells Equations of Electromagnetism
Gauss Law for Electrostatics
Gauss Law for Magnetism
Faradays Law of Induction
Amperes Law
15
Maxwells Equations of Electromagnetismin Vacuum
(no charges, no masses)
Consider these equations in a vacuum.....
......no mass, no charges. no currents.....
16
Maxwells Equations of Electromagnetismin Vacuum
(no charges, no masses)
17
Electromagnetic Waves
These two equations can be solved simultaneously.
The result is
18
Electromagnetic Waves
B
E
19
Electromagnetic Waves
B
E
Special case..PLANE WAVES...
satisfy the wave equation
Maxwells Solution
20
Plane Electromagnetic Waves
Ey
Bz
c
x
21
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
22
F
v
Moving wave F(x, t) FP sin (kx - ?t )
x
What happens at x 0 as a function of time?
F(0, t) FP sin (-?t)
For x 0 and t 0 ? F(0, 0) FP sin (0) For x
0 and t t ? F (0, t) FP sin (0 ?t) FP
sin ( ?t) This is equivalent to kx - ?t ? x
- (?/k) t F(x0) at time t is the same as
Fx-(?/k)t at time 0 The wave moves to the
right with speed ?/k
23
Plane Electromagnetic Waves
Ey
Bz
Notes Waves are in Phase, but fields
oriented at 900. k2p/l. Speed of wave is
cw/k ( fl) At all times EcB.
c
x