Download Maxwell`s Equations

Maxwell’s Equations
• We have been examining a variety of
electrical and magnetic phenomena
• James Clerk Maxwell summarized all of
electricity and magnetism in just four
equations
• Remarkably, the equations predict the
existence of electromagnetic waves
Gauss
q

 ( E )   E A 
n
0
• The electric field is
due to electric charges.
– Related to electric flux
• Electric field lines
start or stop on a
charge or make a
closed loop.
No Monopoles
( B)   Bn A  0
• There are no magnetic
charges.
• Magnetic field lines
can only make a
closed loops.
• An emf is
induced by a
varying magnetic
field within a
closed path.
– Magnetic forces
will move
charges.
• This implies that
a changing
magnetic field
creates an
electric field.
Faraday
d( B)
fem  Circ ( E )   E||l  
dt
Ampere
• Ampere’s Law:
Circ( B)   B||l  0 I
Earlier, we just went on a
closed path enclosing
surface 1. But according to
Ampere’s Law, we could
have considered surface 2.
The current enclosed is the
same as for surface 1. We
can say that the current
flowing into any volume
must equal that coming out.
IV Maxwell’s Equation
• Suppose we have a charged capacitor and it
begins to discharge
Surface 1 works
but surface 2 has
no current
passing through
the surface yet
there is a
magnetic field
inside the
surface.
Problems with Ampere’s Law
Circ( B)   B||l  0 I
B 2r   o I
o I
B
2r
But if we use Surface 2 ...
Circ( B)   B||l  0 I
B 2r   o  0 
B 0
?????
Maxwell’s correction to Ampere’s Law
Q  CV
o A
C
d
V  Ed
 o A 
Q
  Ed   o AE  o  E
 d 
dQ
d E
I
 o
Called “displacement current”, Id
dt
dt
Ampere’s Extended lawIV Maxwell equation:
• A magnetic field is
induced by an electric
current.
• There is an electric flux as
well.
– Changes create magnetic
fields
• This implies that a
changing magnetic field
creates an electric field.
 E 

Circ ( B)     I  

t 

B field surrounds electric
field, although there is no
“current” flowing here
• Maxwell noted the
symmetry between
electric and magnetic
fields.
( B)
• Changing magnetic
Circuitazi one( E )  E||l  
t
fields create electric
fields
 E 

Circuitazi one( B)  B||l    I  

– Current and
t 

changing electric
fields create
q
Flusso ( E )  EnS 
magnetic fields
0
– Electric field lines
Flusso ( B)  0
originate from
charges or form
closed loops
– Magnetic field lines
form closed loops
Maxwell’s Equations:




Electromagnetic Waves
• So, a magnetic field will be
produced in space if there is a
changing electric field
• But, this magnetic field is
changing since the electric field
is changing
• A changing magnetic field
produces an electric field that is
also changing
• We have a self-perpetuating
system
A capacitor being charged by a current ic has a
displacement current equal to iC between the plates,
with displacement current iD = e A dE/dt. This
changing E field can be regarded as the source of the
magnetic field between the plates.
Electromagnetic Waves
Notice that the electric and magnetic fields are at right angles
to one another! They are also perpendicular to the direction
of motion of the wave.
This picture defines the coordinate system we will use in our
discussion. Wave propagates along the x-axis. The electric
field varies in the y-direction and the magnetic field in the zdirection.
Wave Motion
• Changing electric and magnetic fields
create a wave.
– Electric field creates a magnetic field
– Magnetic field creates an electric field
Electromagnetic Waves
Close switch and current
flows briefly. Sets up
electric field. Current flow
sets up magnetic field as
little circles around the
wires. Fields not
instantaneous, but form in
time. Energy is stored in
fields and cannot move
infinitely fast.
Speed of EM Waves
We are going to apply Faraday’s Law to the imaginary moving
rectangle abcd. Compute the magnetic flux change
 B BA By 0vt
1) fem 


 By 0v
t
t
t
Speed of EM Waves
• We can say the emf around the loop is the
sum of the individual emfs going along each
straight line segment in the loop
• We look at the work done in moving a test
charge around the loop
• 2) fem = W/q = Fd/q = Ed = Ey0 = By0v
3) E = Bv
Speed of EM Waves
Now we are going to look at the change in electric flux. Set a
new imaginary rectangle and play the same game as before.

Ez
vt
E
0
Ampere’s Law:  B||l  00
 00
 00 Ez0v
t
t
4) Bz 0  0 0 Ez0v
Speed of EM Waves:
4) Bz 0  0 0 Ez0v  B  0 0 Ev
3) E  Bv
4) B  0 0 ( Bv )v
 1  0 0v 2
v
v
1
0 0
1
8.85 1012  4 107
v  3 108 m / s
Fields are functions of both
position (x) and time (t)
dE
dB

dx
dt
Partial derivatives
are appropriate
B
E
  o  o
x
t
dB
dE
 o o
dx
dt
E
 B

2
x
x t
2
E
B

x
t
 B
2E
 o o 2
t x
t
2E
2E
 o o 2
2
x
t
This is a wave
equation!
The equation’s solution
E  E y  Eo sin  kx  t 
2E
2E
 o o 2
2
x
t
2E
2
 k E o sin  kx  t 
2
x
2E
2
  E o sin  kx  t 
2
t
k 2Eo sin  kx  t   oo2Eo sin  kx  t 
2
1

2
k
o o
The speed of light
(or any other electromagnetic radiation)

k
 f
f  v

1
vc 
k
o  o
SPETTRO ELETTROMAGNETICO
generazione di un’onda e.m.
E
B
ESPERIMENTO DI HERTZ
Hertz nel 1886 riuscì per la prima volta a produrre e a rivelare le onde elettromagnetiche di cui Maxwell
aveva previsto l’esistenza.
Le onde elettromagnetiche furono generate da oscillazioni di cariche elettriche lungo un circuito.
Generatore di onde em
Rivelatore di onde em
La trasmissione delle onde era rilevata
da un cerchio di grosso filo di rame
interrotto da uno spazio di lunghezza
regolabile tra due sferette.
Generatore
di differenza di
potenziale
Il passaggio di una corrente
oscillante nel cerchio di rame si
manifestava attraverso una
scintilla che illuminava le due
sferette
Le onde generate con questo apparato avevano una frequenza di 10 9 Hz