A Drop of the Hard Stuff: How
Maxwell Created His Equations,
What They Mean, and How
They Predicted the Discovery
of Radio
Peter Excell
Professor of Communications
Outline
• Mathematical Physics delivering real
benefits
• Using computers to make it ‘easier’
• Implications
“Pre-requisites”
• Differentiation
• Electric & Magnetic fields
• Electric circuits
“Maxwell’s Equations”
div D  
div B  0
B
curl E  
t
D
curl H 
J
t
“Maxwell’s Equations”
 B 
So: E  curl  
?
 t 
-1
No!
Electromagnetic Waves
• Theories of electric and magnetic fields
evolved independently of the main theories
of optics, but JAMES CLERK MAXWELL
showed how they were related.
• This was one of the greatest achievements
of Physics, particularly as it started as an
abstruse mathematical theory but went on to
become a major applied technology which
has had a major effect in shaping the
modern world: Radio Waves
Electromagnetic Waves
This is equivalent to today’s modern physics
when/if it produces results which are taken
forward to everyday practical applications,
e.g.:
• Superstring Theory
• The Large Hadron Collider
• Stephen Hawking’s stuff
Electromagnetic Fields
• Oersted and Ampere showed how an electric
current could create a magnetic field, causing
‘action at a distance’.
• Faraday showed how a magnetic field could
create a current, but only if it was varying in
time.
• Maxwell generalised Faraday’s and Ampere’s
Laws, combined them, and discovered an
equation for travelling electromagnetic waves.
An approximate explanation of
how Maxwell discovered the
Electromagnetic Wave Equation
(Developed for first-year Elec.
Eng. Students in the 1990s…)
Maxwell’s generalisation of Faraday’s
Law of electromagnetic induction
d
Induced voltage: V  
dt
 is the total magnetic flux, in Webers (Wb), at rightangles to the wire loop. But  = BA, where B is
magnetic flux density (Wb/m2 or T) and A is coil area.
Generalising Faraday
Let A = xy, then:
d
V
dt
dB
V   xy
dt
Now suppose the ‘voltmeter’ side is opened up so
that we just see an electric field, E, across a gap. The
electric field in the gap is approximately given by:
V
dB
E  , So : E   x
y
dt
Generalising Faraday
dB
E  x
dt
Differentiate both sides with respect to x: this has to
be a partial derivative because E and B are functions
of both time and position:
E
B

x
t
This is a simple statement of Maxwell’s
generalisation of Faraday’s Law.
Generalising Faraday
E
B

x
t
The complete form is:
B
curl E  
t
Park that.
…………………..
Now we look at Ampère’s Law:
Generalising Ampère
Ampère’s Law describes the magnetic flux set up by
a current-carrying wire:
The current I sets up a magnetic flux density B:
I
I
B
H 
2 r
2 r
Generalising Ampère
I
I
B
H 
2 r
2 r
Why?
1. Because Ampère found that current-carrying
circuits exerted a force on each other, due to their
magnetic fields, inversely proportional to distance
and proportional to the current.
Generalising Ampère
I
I
B
H 
2 r
2 r
Why?
2. Because B = H by definition:
•B is magnetic flux density
•H is magnetic field strength
• is permeability (a constant of ‘space’, linking the
force produced by magnetic fields to the current
creating them)
Generalising Ampère
Now suppose that there is a capacitor in the circuit:
There is no current in the gap in the capacitor (the
dielectric), but experiment (and reasoning) shows
there must still be a magnetic flux and field.
Generalising Ampère
What is the equivalent of ordinary conduction current
in the capacitor gap?
Start with the basic equation for a capacitor:
dV
IC
dt
For a parallel-plate capacitor (ignoring ‘fringing’):
C
A
g
V
and E  ,  V  Eg
g
Where g is the gap width
Generalising Ampère
A
C
g
 is permittivity (another constant of ‘space’, linking
the force produced by electric fields to the voltage
creating them)
Generalising Ampère
Now combine:
dV
A
IC
,C
dt
g
and V  Eg
A
 dE 
 dE 
 I
 g   A 
g  dt 
 dt 
This is the so-called ‘Displacement Current’ – a
key Maxwell innovation.
Generalising Ampère
 dE 
 I  A 
 dt 
The equivalent current density, J, (A/m2) is:
I
dE dD
J  

A
dt
dt
because D (electric flux density) = E
Generalising Ampère
Now go back to the circuit and take a magnified view
of part of the wire:
Now consider the wire to be a thin-walled tube so
that all current is forced to flow on its surface:
Generalising Ampère
Take ‘r’ to be equal to the ‘wire’ radius, so we are
observing the magnetic fields right on the metal
surface:
I
J
, but
2 rx
I
H
2 r
hence: H = Jx
Differentiating with respect to x:
dH
J
dx
Generalising Ampère
Now suppose there is a gap in the tube, forming a
narrow capacitor:
In the gap, J is replaced by dD/dt, but this should now
be written as a partial derivative, to allow for spatial
variations.
Generalising Ampère
In a general case,
both conduction
and displacement
currents may be
present, hence:
H D

J
x
t
This is a simplified form of the Maxwell-Ampère
Equation, Maxwell’s generalisation of Ampère’s Law.
Generalising Ampère
H D

J
x
t
The full vector version is:
D
curl H 
J
t
The Wave Equation
In vacuum (or air) there can be no conduction current
and the Maxwell-Ampère equation reduces to:
H D
E


x
t
t
But Faraday’s Law (as generalised by Maxwell) gave
us another such equation:
E
B

x
t
Notice the symmetry…
H
 
t
The Wave Equation
Differentiate the Faraday-Maxwell equation with
respect to x:
E
H
 
F-M:
x
 E
  H 



 
2
x  t 
x
2
Diff:
t
  H 
   
t  x 
The exchange in the order of the differentiations is
allowable if there is no mechanical movement.
Now we can combine this equation with the MaxwellAmpère equation above to give an equation in E as
the only field component.
The Wave Equation
Differentiated
F-M:
 E
  H 
  

2
x
t  x 
2
H
E

M-A:
x
t
2
2
 E
 E
Eliminate H:
  2
2
x
t
Note: the electric fields in the two equations are not quite the
same, and we have to change the sign of one when combining
them, by dropping the minus sign. This fudge doesn’t arise
when the full vector treatment is used: a weakness of the very
simplified treatment used here.
The Wave Equation
2
2
 E
 E


2
2
x
t
This is a WAVE EQUATION for electric fields: a
momentous result.
This is a double simple harmonic motion type of
equation with respect to x and t simultaneously. It is
satisfied by solutions which oscillate sinusoidally with
respect to x and t simultaneously, e.g.
E  E0 cos([t  x  ])
The Wave Equation
E  E0 cos([t  x  ])
x  has the dimensions of time,
i.e. distance/speed, so 1 /  is a speed.
The term
( is a frequency)
 and  can be found from laboratory experiments
with electromagnets and capacitors:
= 4 x 10-7 H/m, and  = 8.85 x 10-12 F/m, so:
8 m/s
=
2.998x10
1 / 
THE SPEED OF LIGHT!
The Wave Equation
This truly remarkable result, first derived by Maxwell,
showed that electromagnetic waves could exist,
which travelled at the speed of light.
It strongly implied that light was such an
electromagnetic wave (later proved to be so),
- but that light-like waves could exist over a much
wider spectrum than was then known (infraredultraviolet).
The Wave Equation
It also implied that such waves could exist at low
frequencies, in which case they:
A. Could bend round corners and over the horizon
B. Could go through clouds and fog (and walls)
C. Could be created by ordinary electric circuits
Exploitation
James Clerk Maxwell died in 1879, before he could
test his theory.
The existence of such ‘radio’ waves was first
demonstrated by Heinrich Hertz, working in
Germany, in 1888. Hertz’ experiments were only on a
laboratory scale, and he also died young.
Radio was commercialised by Guglielmo Marconi
who first transmitted information in a radio signal in
1895 and got a signal across the Atlantic in 1901.
Exploitation
Marconi found influential backers….
The invention of the radio valve helped the technology:
the imperative of working with very weak signals
initiated electronics.
The sinking of the Titanic showed the importance of
radio in ships.
The First World War gave a massive boost to radio.
After the war, entertainment applications drove the
technology.
TV – Radar – Satellites – Mobile phones…
Einstein
The wave has a velocity, but with respect to what?
It was thought that space was filled with aether, which
carried light waves, even in a vacuum.
This would mean that tests of the speed of light would
give different results as the Earth moved through
space: this was found not to be the case (MichelsonMorley Experiment).
Einstein
Lorentz proposed that everything, including the
measuring equipment, ‘shrank’ in the direction of
movement through the aether, so that speed change
would be undetectable (‘Lorentz Contraction‘).
Albert Einstein showed that the aether was not
necessary…
Einstein
Albert Einstein showed that the aether was not
necessary if it was accepted that the speed of light
was a fundamental constant of the universe.
Since the speed is based on the constants  and 
this seems reasonable.
But since velocity = distance/time it means that
distance and time are no longer absolute
quantities.
Einstein
This leads to the very surprising effect of Time Dilation
which means that clocks appear to run slower in
systems moving relative to the observer.
This is the basis of Einstein’s Special Theory of
Relativity: a severe upset to ‘common sense’
understanding!
Einstein
Note that the theory really grew from consideration of
Maxwell’s work on the electromagnetic nature of light.
The well-known equation E = mc2 is a secondary
consequence.
Using computers to make it ‘easier’
Differentiation can be replaced by a ‘finite-difference
approximation’, e.g.:
E E( x1 )  E( x 2 )

x
x1  x 2
E E( t1 )  E( t 2 )

t
t1  t 2
These ‘discretised’ versions can easily be
programmed into a computer, but space and time
have to be chopped up into tiny segments, so the
sums are easy(ish), but there can be billions of
them…
Using computers to make it
‘easier’
Another approach:
Convert Maxwell’s Equations to integral forms using
Stokes’ and Gauss’ theorems, then form an ‘Integral
Equation’ approach.
- Equivalent to the ‘action at a distance’ view of field
phenomena.
- The finite difference approach is equivalent to
Faraday’s view of a pervasive field.
- This gets down to the epistemology of science…
Computational examples
Finite-difference
model of a row of
biological cells
with a 900MHz
wave incident
from the left
Integral-equation model of detailed current distribution on self-resonant coils
Finite-difference model of a sphere simulating a human head, with a 900MHz
wave incident from the left.
Finite-difference model of a human head, with mobile phone adjacent at bottom
Finite-difference grid structure for a human head, with mobile phone and human
hand models
Structure for a hybrid model, using finite-difference for the sphere (simulating a
head) and integral-equation method for the phone and antenna
Field distribution predicted by a hybrid model, using finite-difference for the human
head and integral-equation method for the phone and antenna (invisible in the
rectangular box)
Thermal finite-difference model for human head, with added electrical warming
predicted by electrical finite-difference model (thermal model by Sapienza
University of Rome)
Does it Matter?
•
•
•
•
•
•
•
•
•
•
Unified field theory
Radio systems
Radar
Military applications
Medical applications
Difficult maths
Exotic physics
Supercomputing
MIMO: IEEE 802.11n etc
Don’t forget the applications
Does it Matter?
• ‘Big ideas’
• ‘Polymathism’
• Breadth vs depth
– To discuss