Just about the whole of electromagnetism is contained in the
Maxwell equations:
G G
’ <E
U
Ho
G
G G
wB
’uE = wt
G G
’ <B
0
G
G
G G
wE
’ u B = P o j P oH o
wt
2
3
Gauss’ law of electrostatics
G G
For several discrete charges ³ E<da
S
¦q
i
Ho
The electric flux through a closed surface S is proportional to
the total charge enclosed within S.
G
For a continuous charge distribution U (r )
G G
³ E < da
S
1
G
³ U (r )dV
Ho V
More specifically, the net outward flux of the electric field across
any closed surface S is equal to the total electric charge enclosed
within the region V bounded by S, divided by the permittivity of
free space H o.
4
Gauss’ law of electrostatics ... 2
G G
By the divergence theorem: ³ E<da
S
G
? ³ div EdV
V
1
V
G
³ U (r )dV
Ho V
G
G
§
U (r ) ·
? ³ ¨ div E ¸ dV
Ho ¹
V©
G
? div E
G
³ div EdV
0
must be true for
any volume V
G
U (r )
Ho
Maxwell I
Electric field diverges outwards from positive
charge and inwards towards negative charge.
5
An animation of the motion of a positive charge moving past a massive charge
which is also positive. The smaller charge is deflected away from the larger
charge because of their mutual repulsion. This repulsion is primarily due to a
pressure transmitted by the electric fields surrounding the charges.
6
An animation of the motion of a negative charge moving past a
massive positive charge. The negative charge is deflected toward the
positive charge because of the attraction between them. This attraction
is primarily due to a tension transmitted by the electric fields
surrounding the charges.
7
Gauss law of magnetism
The net outward flux of any magnetic field across any closed
surface S is zero (there can be no magnetic monopoles).
G G
³ B<da 0
S
Then by the divergence theorem:
G
div B
0
Maxwell II
Magnetic fields never diverge or converge.
They always form closed loops (circulate).
8
The animation shows two co-axial wire loops carrying current in the same
sense. The loops attract one another. We show the field configuration
here using the "iron filings" representation. The bottom wire loop carries
three times the current of the top wire loop.
9
Faraday’s law of electromagnetic induction
H induced
w) B
wt
where
)B
G G
³ B<da
S
An emf can induced in a closed circuit by changing magnetic
fields or by the motion of the circuit through a magnetic field.
The circuit referred to could be a closed electric circuit or
imaginary closed loop in space.
G
The potential difference
along a small section d A of a
G
G induced
circuit C is given by E<d A .
G G
w G G
? H induced
vC³ E<d A wt ³S B<da
where the surface S is bounded by the circuit C.
It follows that induced electric fields are non-conservative.
10
Faraday’s law of electromagnetic induction ... 2
G G
v³ E<d A
C
By Stokes’ theorem:
G G
v³ E<d A
C
w G G
³ B < da
wt S
³
G G G
’ u E <da
S
G G G
w G G
? ³ ’ u E <da = ³ B<da
wt S
S
G
§ G G wB · G
must be true for
? ³¨’uE ¸<da = 0
any circuit C
wt ¹
S©
G
G
wB
Maxwell III
? curl E = wt
Changing magnetic fields induce electric fields which curl
around the changing magnetic fields.
11
Ampere’s Law
Magnetic fields may be produced by electric currents:
G G
vC³ B<d A Po I
G
The line integral of a magnetic field B around any closed path
C is equal to the permeability of free space Po multiplied by the
current I through the area enclosed by the path.
When theG current
is distributed continuously as a current
G
G G
density j(r ) , then
I = j<da
³
S
(The currentG IGacross any surface S bounded by the closed path C is
the flux of j(r ) across S.)
G G
G G
Po ³ j<da
Then v³ B<d A
C
S
It follows that magnetic fields produced by electric currents are
also non-conservative. Therefore the magnetic fields cannot be
12
derived by scalar potentials, but only vector potentials.
Ampere’s Law ... 2
G G
v³ B<d A
C
By Stokes’ theorem:
G G
v³ B<d A
C
?
S
G G G
’ u B < da
³
S
³
G G
G G G
’ u B <da = Po ³ j<da
³
G G
G G
’ u B P o j < da = 0
S
?
G G
Po ³ j<da
S
S
G
G
? curl B = Po j
must be true for
any circuit C
Maxwell IV
(almost ...)
Magnetic field lines curl around
electric currents (think of the magnetic
field around a conducting wire).
G
B
I
13
Suppose we have five rings that carry a number of free positive charges that are not moving. Since there
is no current, there is no magnetic field. Now suppose a set of external agents come along (one for each
charge) and simultaneously spin up the charges counterclockwise as seen from above, at the same time
and at the same rate, in a manner that has been pre-arranged. Once the charges on the rings start to
accelerate, there is a magnetic field in the space between the rings, mostly parallel to their common axis,
which is stronger inside the rings than outside. This is the solenoid configuration.
As the magnetic flux through the rings grows, Faraday's Law tells us that there is an electric field induced
by the time-changing magnetic field that is circulating clockwise as seen from above. The force on the
charges due to this electric field is thus opposite the direction the external agents are trying to spin the
rings up in (counterclockwise), and thus the agents have to do additional work to spin up the charges
because of their charge. This is the source of the energy that is appearing in the magnetic field between
the rings-the work done by the agents against the "back emf".
Over the time when the magnetic field is increasing in the animation, the agents moving the charges to a
higher speed against the induced electric field are continually doing work. The electromagnetic energy
that they are creating at the place where they are doing work (the path along which the charges move)
flows both inward and outward. The direction of the flow of this energy is shown by the animated texture 14
patterns. This is the electromagnetic energy flow that increases the strength of the magnetic field in the
space between the rings as each positive charge is accelerated to a higher and higher velocity.
The animation shows the magnetic field configuration around a
permanent magnet as it falls under gravity through a
conducting non-magnet ring. The current in the ring is indicated
by the small moving spheres. In this case, the magnet is light,
the ring has zero resistance, and the magnet levitates above
the ring. The motions of the field lines are in the direction of the
local Poynting flux vector.
15
The animation shows the magnetic field configuration around a
conducting non-magnetic ring as it falls under gravity in the magnetic
field of a fixed permanent magnet. The current in the ring is indicated by
the small moving spheres. In this case, the ring is heavy and has zero
resistance, and falls past the magnet. The motions of the field lines are
in the direction of the local Poynting flux vector.
16
The 4 “Maxwell” equations above express laws that came to
Maxwell from other sources. But Maxwell’s genius was to see
that law IV was incomplete.
Consider a capacitor being charged. As the charge flows to
the capacitor plates, a magnetic field rings the wire.
But what about between the plates?
G
B
I
No magnetic field here?
Does the magnetic field stop abruptly between the plates where
there is no current?
Maxwell though not ...
17
Maxwell reasoned that if changing magnetic fields induce
electric fields (Faraday), then, symmetrically, changing
electric fields might induce magnetic fields.
Although there was no experimental evidence for this at the time,
Maxwell added an extra term to his fourth equation, saying that
magnetic fields also curl around changing electric fields:
G
G
G
wE
curl B = Po j PoH o
Maxwell IV
wt
This term generates a
magnetic field between the
capacitor plates as the
electric fields builds up.
Some years later, this magnetic field was detected ..!
18
Maxwell’s equations:
G
U (r )
I
G
div E
III
G
G
wB
curl E = wt
Ho
II
G
div B
0
G
G
G
wE
IV curl B = Po j PoH o
wt
For time independent electric and magnetic fields:
G
G
G
U (r )
div B
0
div E
I
II
Ho
III
G
curl E = 0
IV
G
G
curl B = Po j
19
G
wE
The displacement current density term PoH o wt in Maxwell’s fourth
equation led to a very large unexpected physics jackpot ...
Imagine a single electric charge
being vibrated. In the space near the
vibrating charge, the charge’s
electric field is changing, so it
induces a magnetic field curling
around it.
But the magnetic field is also
changing, so it induces more
electric field, which induces more
magnetic field, etc, etc ...
The result is an electromagnetic wave of fields rippling
out from the vibrating charge at the speed of light.
20
Electromagnetic radiation continued ...
In empty space Maxwell III and IV become
G
G
G G
G G
wE
wB
and
’ u B = P oH o
’uE = wt
wt
G G G
?’ u ’ u E
w G G
’uB
= wt
G
G G G
w§
wE ·
?’ u ’ u E = ¨ PoH o
¸
wt ©
wt ¹
G
2
G G G
G
wE
2
? ’ ’ <E ’ E = P o H o 2
wt
G
2
G
wE
2
? ’ E = P oH o 2
wt
21
G
G
wE
2
’ E = P oH o 2
wt
2
This partial differential has electric field solutions that represent
waves propagating in empty space with speed
1
c
= 2.998 ×108 m s-1
P oH o
These waves together with associated magnetic field waves
G
2
G
wB
2
’ B = P oH o 2
wt
together describe constitute an electromagnetic wave.
22
23
24
The electrostatic scalar potential
G G
For time independent fields ’ u E = 0
G
showing that electrostatic field E is a conservative field.
G G
Since ’ u ’M = 0 for any scalar field M .
G
G
we can write E = ’V
where the scalar field V is called the electrostatic potential.
When the electric charge distribution is described
by a charge
G G
density U then we can use Gauss’ Law ’<E
U H o to obtain:
G G
Poisson’s
2
’ < ’V
U Ho
or
’V
U Ho
equation
If U = 0 everywhere, then
2
’V
Laplace’s
0
equation
25
The magnetostatic vector potential
The sources of magnetostatic fields are steady current density
distributions:
G
G G
Amperes’ law: ’ u B = Po j
G
i.e. B is not conservative.
G G
We also know that ’<B
(no magnetic monopoles)
0
G
G G G
Since ’< ’ u F = 0 for any vector field F ,
G
G
G G
we can write B = ’ u A for some A
G
where A is the magnetic vector potential.
26
The magnetostatic vector potential ...2
G
In order to define A uniquely, we need to specify its divergence.
G G
For time independent fields ’< A
0 .
G G
G G G
Therefore ’ u B = ’ u ’ u A
G
G G G
2
’ ’< A ’ A
G
G G
2
’ A
since ’< A
G
? ’ A
2
G G
’uB
0
G
= Po j
27
Electrostatic scalar potential
Knowing U and the
boundary conditions for V
2
’V
U Ho
V
Magnetostatic vector potential
G
Knowing j and the G
boundary conditions for A
G
’ A
2
G
A
G
G
E = ’V
G
E
G
Po j
G
G G
B = ’u A
G
B
28