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The derivation of
Maxwell’s Equations
∇.E
∇×B
∇×E
∇.B
by Brandon Speedie
Our recent feature on the history of electronics
covered many prominent contributors to the field.
Two names stand out above others; their work
is commonly referred to as the ‘second great
unification in physics’.
D
avid Maddison’s History of Electronics series was published in the
October, November and December 2023 issues (siliconchip.au/
Series/404). It mentioned hundreds
of people who laid the foundations
for modern electronics. Englishman
Michael Faraday was one of the standouts in that list, with significant contributions to the understanding of electromagnetics.
Faraday was born in 1791 to a
poor family. He had an early interest
in chemistry, but his family lacked
the means to formally educate him.
Instead, he became selftaught through
books and an unbounded curiosity
for experimentation. This practical
approach continued throughout his
career and set the blueprint for his
breakthroughs in electromagnetics,
despite having no formal training.
Faraday was responsible for many
notable discoveries, including the
concept of shielding (the Faraday
Cage), the effect of a magnetic field
on the polarisation of light (the Faraday Effect), the electric motor (an
early homopolar type, see Fig.1), the
Faraday’s coil and ring experiment
demonstrated electromagnetic induction.
Source: Ri – siliconchip.au/link/abv3
electric generator (an early dynamo,
see Fig.2), and the fact that electricity
is a force rather than a ‘fluid’ (as was
the understanding at the time).
He also theorised that this electromagnetic force extended into the
space around currentcarrying wires,
although his colleagues considered
that idea too farfetched. Faraday
didn’t live long enough to see his concept accepted by the scientific community.
It was an experiment with an iron
ring and two coils of wire in 1831 that
proved a defining moment for the vocation we now call electrical engineering. By passing a current through one
coil, Faraday observed a temporary
current flowing in the second coil,
despite the lack of a galvanic connection between them.
We now refer to this phenomenon as
electromagnetic induction, the property behind many common products
such as transformers, electric motors,
speakers, dynamic microphones, guitar pickups, RFID cards etc. Most notably, this principle is involved in generating the bulk of our electricity. It was
a remarkable achievement, later earning Faraday the moniker, “the father
of electricity”.
James Clerk Maxwell
Maxwell was born in 1831 in Scotland. His comfortable upbringing and
access to education contrasted with
Faraday. Recognising his academic
potential, his family sent him to technical academies and University to
foster his curiosity about the world
around him.
Maxwell had long admired Faraday’s
work but understood that he was fundamentally a tinkerer with only a basic
understanding of mathematics. Maxwell recognised that his own strengths
in mathematics were needed to unify
Faraday’s experimental results, along
with the work of other notable contributors such as Carl Friedrich Gauss and
Hans Christian Ørsted.
In 1860, Maxwell’s employment
moved to King’s College, where he
came into regular contact with Faraday. During this period, he published
a fourpart paper, “On Physical Lines
of Force”, using concepts Faraday had
Figs.1 & 2: Faraday’s homopolar
motor (left) and Faraday’s disc
generator (right).
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Silicon Chip
Australia's electronics magazine
siliconchip.com.au
Fig.3: an application of the cross
product. The torque of an axle
can be calculated from the
cross product of the radius
and force vectors.
the two vectors together, then multiplying that by the cosine of the angle
between them. The cosine is at a maximum if the two vectors point in the
same direction and zero if they are
orthogonal.
If the vectors this is applied to are
unit vectors (vectors of length one),
the result is simply the cosine of the
angle between them.
Divergence (∇●)
introduced many decades earlier.
It contained the four expressions
we now know as Maxwell’s equations
that tie together electricity, magnetism
and light as a single phenomenon: the
electromagnetic force. This is called
the ‘second great unification in physics’ because Sir Isaac Newton’s trailblazing work with motion and gravity
is considered the first.
Vector calculus
To understand the notation of Maxwell’s equations, a quick primer on
vector calculus is in order. Electromagnetism works in threedimensional
space, which can make mathematical
representations confusing. We will
cover the basics here, using figures to
help visualise the equations. The formulas will follow the differential form
derived by Oliver Heaviside from Maxwell’s original paper.
Combining Del and the dot product is commonly referred to as the
divergence operator. When used on
a vector field, it returns a scalar field
representing its source at any particular point. For example, calculating
the divergence on atmospheric wind
speed would give a view of pressure
differences.
Cross product (×)
The cross product is a vector operation to calculate the ‘normal’ of two
vectors, resulting in a new vector perpendicular to the two input vectors.
Curl (∇×)
Combining Del and the cross product yields the curl operator. When
applied to a vector field, its result is
a vector field that shows the rotation
or circulation.
Returning to the Meteorology example, calculating the curl of wind speeds
in the atmosphere will return vorticity, a measure of cyclone or anticyclone rotation. Negative vorticity usually correlates with low pressure and
unstable weather (cyclonic rotation),
and positive vorticity with high pressure and fine weather (see Figs.4 & 5).
#1) Gauss’ law of magnetism
∇●B=0
Maxwell’s first equation is named
after German physicist Henrich Gauss.
Fig.4 (top): a
wind speed
plot showing
rotational winds
off the east coast
of Australia and
in the southern
ocean. Source:
BoM, siliconchip.
au/link/abv4
Derivative (d/dt)
Fig.5 (bottom):
calculating the
curl of the wind
speed yields
the vorticity,
which more
clearly shows the
cyclonic rotation
off the east coast
(blue) and the
anticyclone in the
southern ocean
(red). Negative
vorticity (blue) is
associated with
atmospheric
instability,
positive (red)
usually means
fine weather. The
same operation
can be used on
a 3D electric or
magnetic field to
derive its source.
Source: BoM,
siliconchip.au/
link/abv5
The derivative operator, d, is shorthand for the Greek letter delta (Δ),
which in mathematics refers to a
change or difference. ‘t’ refers to time,
so d/dt therefore means the change in
a parameter over time or more commonly, ‘rate of change’. The symbol ‘∂’
instead of ‘d’ indicates a partial derivative, which is used when differentiating a function of two or more variables.
Nabla / Del (∇)
Del is the vector differential operator. It is equivalent to the derivative
operator above but can be applied
to more than one dimension. In our
examples, it will be applied to a 3D
field.
Dot product (●)
A dot product is an operation
between two vectors that gives a scalar
(numeric) result. The result is equivalent to multiplying the magnitudes of
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A common example is to derive an
axle’s torque from its radius and force
vectors. The resulting torque vector is
orthogonal to both vectors and points
in the direction of its angular force
(see Fig.3).
Australia's electronics magazine
November 2024 91
Fig.6:
Gauss’ law
of magnetism
with reference to a permanent
magnet. Any field lines exiting
‘north’ wrap around the magnet
and enter at the ‘south’ end. The
net magnetic field source is zero
for any surface cutting through
this field (eg, the square), or for
the whole magnet in total.
Fig.7: Gauss’ law in an airgapped
capacitor (eg, a tuning gang). A
voltage source forces a positive
charge to build up on the top plate
& a negative charge on the bottom
plate. An electric field forms
between the charged regions.
Fig.8: similar to Fig.7 but with
a plastic film dielectric, which
has a higher permittivity than
air. Electric dipoles in the
dielectric orientate themselves to
cancel some of the electric field
strength, increasing the effective
capacitance.
Here, B is the magnetic field. Simply
stated, the sum of all magnetic fields
emanating from an interface will
always add to zero.
This is most obvious when looking
at the magnetic field lines surrounding a bar magnet (see Fig.6). Any field
lines exiting ‘north’ wrap around the
magnet and enter at the ‘south’ end.
Considering any isolated area, or the
entire magnet as a whole, there is no
magnetic field source.
#2) Gauss’ law
∇●E=ρ÷ε
Also called Gauss’ flux theorem.
Here, E is the electric field, ρ is the
charge density (the amount of electric
charge per volume) and ε is the permittivity of the material or medium (calculated as ε0εr, where ε0 is the vacuum
permittivity and εr is the relative permittivity; in a vacuum εr = 1).
This law states that electric charge
is the source of an electric field. The
strength of that field is proportional
to the amount of charge and inversely
proportional to the permittivity of the
supporting material.
This phenomenon is most apparent
in a capacitor, where an accumulation
of negative charge (electrons) builds
up on one plate, and a positive charge
(protons or holes) on the other (Fig.7). A
dielectric between the plates supports
the electric field. Its electric dipoles
will be orientated opposite to the direction of the electric field and therefore
store some of that electric field strength.
Film capacitors use a plastic dielectric such as polypropylene or polystyrene, materials which have a relatively low permittivity, meaning they
have few electric dipoles to orientate
themselves against the field, leaving
it mostly intact (Fig.8).
In contrast, ceramic capacitors typically use a much higher permittivity
dielectric, such as barium titanate,
which will orientate many dipoles in
response to the applied field and cancel much of the electric field strength
(Fig.9). These dipoles provide a higher
capacitance per unit area for ceramic
capacitors compared to film caps.
#3) Faraday’s law of induction
∇ × E = ∂B/∂t
Fig.9: this is like Figs.7 & 8
but with a ceramic dielectric.
The high permittivity allows
many dipoles to cancel a large
proportion of the electric field.
This arrangement has very high
capacitance per area.
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Silicon Chip
Here, E is the electric field and B
is the magnetic field, so ∂B/∂t is the
change in magnetic field over time.
This equation mathematically
formalises Faraday’s coil and ring
Australia's electronics magazine
experiment. It is the notable law of
electromagnetic induction, where a
timevarying magnetic field induces
an orthogonal electric field. The stronger the magnetic field, or the faster its
rate of change, the stronger the resulting electric field.
This law is most familiar in rotating generators such as hydroelectric,
gas, coal and windpowered electricity production. As the alternator spins,
its rotor produces a changing magnetic
field for the stator, inducing an electric field that supplies the grid (see
Figs.10 & 11).
Similarly, the strings on an electric
guitar vibrate when plucked. As they
oscillate, they cut through the magnetic field produced by the pickups.
This changing magnetic field induces a
voltage in the pickup windings, which
is amplified by a circuit to drive the
speaker(s).
#4) Ampere’s law
∇ × B = μJ
Here, B is the magnetic field, J is the
electric current density in amperes
per square metre (A/m2) and μ is the
magnetic permeability of the material
or medium.
The original form of Ampere’s law
states that the flow of electric current
produces an orthogonal magnetic
field. The strength of this field is proportional to the current flow and the
magnetic permeability of the material
(Fig.12).
Ampere’s law is the magnetic equivalent of Gauss’ law. We know that electric charge is the source of the electric
field but Ampere’s law shows that the
movement of electric charge is the
source of a magnetic field.
This phenomenon is most apparent
in an electromagnet, where a wire is
wrapped into a coil. As electric current flows, a magnetic field is produced orthogonal to the wire (Fig.13).
Suppose a high permeability material
such as iron or ferrite is placed in the
coil’s core (Fig.14). In that case, magnetic dipoles orientate themselves in
the direction of the magnetic field,
increasing its strength.
Using an ironbased core to increase
magnetic field strength is very common in many magneticallydriven
devices. For example, silicon steel
is widely used in transformers and
the field windings of most electric
motors or generators. It is also used in
hair clippers, where the 50Hz mains
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Fig.10 (left): Faraday’s law of induction on a simplified threephase alternator. The permanent magnet rotor spins,
providing a changing magnetic field. An electric field is induced in the top coil, as shown by the voltmeter.
Fig.11 (right): the same arrangement as Fig.10 but the rotor has rotated 90°, so the top coil sees no change in the
magnetic field. The voltmeter shows no deflection. If the rotor continues to spin, the south side of the magnet will soon
be near the coil, inducing an electric field with opposite polarity. Through a full 360° rotation, a sinusoidal waveform
is generated, ie, AC voltage.
waveform is used to induce a changing magnetic field in cutting teeth,
providing an oscillatory motion to
trim the hair.
Ferrite is another common ironbased material widely used in magnetic products. It is favoured for its
unique properties as a poor electrical
conductor but a good magnetic conductor (high permeability). That is
why it is widely used as a former for
highfrequency inductors, in permanent magnets for hobby DC motors
and as a source of magnetic fields in
loudspeakers.
This magazine also commonly features AM ‘loopstick’ antennas in its
vintage electronics section, which
often have an adjustable ferrite core.
By rotating the screw, the ferrite can
be moved in or out of the coil, providing an inductance adjustment to ‘slug
tune’ the receiver.
μ is the permeability of the material
or medium, ε is the permittivity of
the material or medium and E is the
electric field (so ∂E/∂t is the change in
electric field over time).
The additional term includes the
property that a timevarying electric
field produces an orthogonal magnetic
field. Put simply, the strength of the
magnetic field is proportional to the
permeability and permittivity of the
material, as well as the electric field’s
strength and rate of change.
When considering this relation,
together with Faraday’s law of induction, it can be seen that a timevarying
electric field produces a magnetic field
Fig.12: an example
of Ampere’s law.
Current flowing in
a wire produces an
orthogonal magnetic
field.
Maxwell’s addition to
Ampere’s law
Figs.13 & 14: if the length
of wire from Fig.12 is
coiled, the magnetic fields
constructively interfere,
producing a stronger
field (left). If a high
permeability material
is used in the core,
magnetic dipoles orientate
themselves in the direction
of the field, increasing the
field strength (right).
The original form of Ampere’s law
only relates electric current to magnetic field strength. Significantly, Maxwell added a term that relates electric
and magnetic fields, termed “Maxwell’s addition”:
∇ × B = μ(J + ε∂E/∂t)
Here, B is the magnetic field, J is
electric current density in amps (A),
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and a timevarying magnetic field produces an electric field (see Fig.15). It
is a remarkable property; as Faraday
so eloquently phrased, “nothing is too
good to be true if it be consistent with
the laws of nature”.
A common example is in the transmission of radio waves by an antenna.
Alternating current in the antenna
produces a timevarying magnetic
field around the conductors, which in
turn produces a timevarying electric
field that continues to propagate in
free space. Some distance away, these
fields induce a current in a receiving
antenna, allowing the wireless transfer of information.
Australia's electronics magazine
November 2024 93
Ideal Bridge Rectifiers
Choose from six Ideal Diode Bridge
Rectifier kits to build: siliconchip.
com.au/Shop/?article=16043
28mm spade (SC6850, $30)
Compatible with KBPC3504
10A continuous (20A peak),
72V
Connectors: 6.3mm spade
lugs, 18mm tall
IC1 package: MSOP12
(SMD)
Mosfets: TK6R9P08QM,RQ (DPAK)
21mm square pin (SC6851, $30)
Compatible with PB1004
10A continuous (20A peak),
72V
Connectors: solder pins on
a 14mm grid (can be bent to
a 13mm grid)
IC1 package: MSOP12
Mosfets: TK6R9P08QM,RQ
5mm pitch SIL (SC6852, $30)
Compatible with KBL604
10A continuous (20A peak), 72V
Connectors: solder pins at
5mm pitch
IC1 package: MSOP12
Mosfets: TK6R9P08QM,RQ
mini SOT23 (SC6853, $25)
Width of W02/W04
2A continuous, 40V
Connectors: solder
pins 5mm apart
at either end
IC1 package: MSOP12
Mosfets: SI2318DSGE3 (SOT23)
D2PAK standalone (SC6854, $35)
20A continuous, 72V
Connectors: 5mm screw
terminals at each end
IC1 package:
MSOP12
Mosfets:
IPB057N06NATMA1
(D2PAK)
Fig.15: Maxwell’s addition to Ampere’s law
models the propagation of an electromagnetic
wave. A changing electric field induces an
orthogonal magnetic field, which in turn induces an electric field. The wave
propagates in a direction normal to both the electric & magnetic fields, at the
speed of light. Source: https://tikz.net/files/electromagnetic_wave001.png
This is also how our sun can power
the Earth’s biosphere. As tiny atoms
such as helium and hydrogen undergo
nuclear fusion inside the sun, they
emit electromagnetic waves. These
waves propagate through free space as
timevarying electric & magnetic fields,
eventually reaching Earth, where they
are used as an energy source by the
flora & fauna on this planet.
Theory of relativity
Years after Maxwell’s publication, a
young Albert Einstein expanded these
equations in his own papers. Einstein
was fascinated by the concept of light
as an electromagnetic wave. The significance of this for him was the notion
that the speed of the wave depends
only on the permittivity and permeability of the medium it travels through
and is therefore invariant of the rela
tive speed of the source (Fig.16).
This understanding led Einstein to
publish his groundbreaking theory of
special relativity in 1905, as well as the
wellknown mass/energy equivalence
formula, E = mc2, where E is energy,
m is mass and c is the speed of electromagnetic waves (light).
This work was further expanded by
Einstein’s theory of general relativity
in 1915, which included the force of
gravitation in addition to the electromagnetic concepts introduced in special relativity. Maxwell’s equations are
so central to this theory that they can
be derived from Einstein’s general relativity formulas.
Einstein paid tribute to Maxwell
later in his career when asked whether
he “stands on the shoulders of Newton”, to which he replied, “no, on the
SC
shoulders of Maxwell”.
TO220 standalone (SC6855, $45)
40A continuous,
72V
Connectors:
6.3mm spade lugs,
18mm tall
IC1 package: DIP8
Mosfets:
TK5R3E08QM,S1X
(TO220)
See our article
in the December
2023 issue for more details:
siliconchip.au/Article/16043
94
Silicon Chip
Fig.16: the speed of electromagnetic waves is proportional only to the permittivity
and permeability of the material they pass through. In this prism, red light
travels at a different speed than blue (because their wavelengths differ), so they
are refracted at different angles. This inspired Albert Einstein to derive his
groundbreaking theories of relativity. Source: www.vectorstock.com/35129206
Australia's electronics magazine
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