Download Maxwell`s equations

Maxwell’s Equations and their
meaning for modern electrical
engineering:
How humans can deal with things they
cannot see
Bjarte Hoff
PhD Candidate
Institute of Electrotechnology
UiT The Arctic University of Norway
Maxwell’s equations
Qencl
 E  d A  є0
(Gauss's law)
 BdA  0
(Gauss's law for magnetism)
dE 

B

d
l


i

є
0 C
0
 (Ampere's law)

dt encl

dB
E

d
l


(Faraday's law)

dt
Electricity: Magic and entertainment
Explain the invisible
How to explain something you cannot see?
Analogies
Fluids as analogy
Flow of fluid through a pipe
Flow of electricity through a conductor
Can electricity be stored?
Conductor
Electrostatic
generator
Insulating layer
Leyden jar
Pieter van Musschenbroek (Leyden)
Ewald Georg von Kleist
1749 -> 1854
Alessantro Volta (1745-1827)
1749 -> 1854
Hans Christian Ørsted (1777-1851)
Electric currents create
magnetic fields
André-Marie Ampére (1775-1836)
Laid the fundation of
«Electrodynamics»
Ampéres law
1749 -> 1854
Georg Simon Ohm (1789-1854)
Ohm’s law
Michael Faraday (1791-1867)
Electromagnetism,
electrochemistry,
induction
1749 -> 1854
William Thomson (1824-1907)
Lord Kelvin
Electricity
Thermodynamics
James Clerk Maxell (1831-1879)
Electromagnetic waves
Maxwell’s equations
Maxwell’s work – The beginning
In a letter to William Thomson in 1854:
“Suppose a man to have a popular knowledge of electrical show and
little antipathy to Murphy’s Electricity, how ought he to proceed in
reading and working so as to get an little inside into the subject which
may be of use in future reading?
If he wish to read Ampere, Faraday, et cetera, how should they be
arranged and at what stage and in what order might he read your
articles in the Cambridge journal?”
Maxwell’s 1st paper:
«On Faradays Lines of Force»
Maxwell:
«By referring everything to the purely geometrical idea of the motion of an imaginary
fluid, I hope to attain generality and precision, and to avoid the dangers arising from a
premature theory professing to explain the cause of the phenomena»
Faraday’s Lines of Force -> Tubes of Force
Incompressible fluid used as an analogy
Maxwell’s 2nd paper:
«On Physical Lines of Force»
Maxwell:
«I propose now to examine magnetic phenomena from a mechanical point of view,
and to determine what tensions in, or motions of, a medium are capable of producing
the mechanical phenomena observed»
Mechanical bipolar molecular vorticies (or eddies) seperated by a layer of
particles are used as an analogy.
Maxwell’s 2nd paper:
«On Physical Lines of Force»
Maxwell:
«I propose now to examine magnetic phenomena from a mechanical point of view,
and to determine what tensions in, or motions of, a medium are capable of producing
the mechanical phenomena observed»
“The conception of a particle having its motion connected with that of a
“We have thus obtained a point of view from which we may regard the
vortex by perfect rolling contact may appear somewhat awkward. I do not
relation of an electric current to its line of force as analogous to the
bring it forward as a mode of connexion existing in nature, or even as that
relation of a toothed wheel or rack to wheels which it drives.”
I would willingly assent to as an electrical hypothesis.”
Mechanical bipolar molecular vorticies (or eddies) seperated by a layer of
particles are used as an analogy.
Maxwell’s 2nd paper:
«On Physical Lines of Force»
Maxwell:
«I propose now to examine magnetic phenomena from a mechanical point of view,
and to determine what tensions in, or motions of, a medium are capable of producing
the mechanical phenomena observed»
“We have thus obtained a point of view from which we may regard the
relation of an electric current to its line of force as analogous to the
relation of a toothed wheel or rack to wheels which it drives.”
Mechanical bipolar molecular vorticies (or eddies) seperated by a layer of
particles are used as an analogy.
Maxwell’s 3rd paper:
«A Dynamic Theory of the Electromagnetic Field»
•
Part III lists Maxwell’s original 20 equations for the electromagnetic field:
Three equations of:
- Magnetic Force
- Electric Currents
- Electromotive Force
- Electric Elasticity
- Electric Resistance
- Total Currents
•
One equation of:
- Free Electricity
- Continuity
Part VI contains electromagnetic theory of light
«We now proceed to investigate whether these properties of that which
constitutes the electromagnetic field, deduced from electromagnetic
phenomena alone, are sufficient to explain the propagation of light through the
same substance.»
Reformulated into a electromagnetic theory, without any sort of mechanical analogy
From 1865 to today
1865
Maxwell publish his 20 equations and 20 variables in:
«A Dynamical Theory of the Electromagnetic Field»
1873
Maxwell correct a sign error and include more equations in:
«A Treatise on Electricity and Magnetism»
1884
Heinrich Hertz publish his derivation of Maxwell’s Equations:
«On the Relations between Maxwell’s Fundamental Electromagnetic Equations
and the Fundamental Equations of the Opposing Electromagnetics»
1885-1887
1940
Oliver Heaviside reformulated 12 of Maxwell’s 20 equations into 4:
Several papers: Electrical Papers, vol. 1 and 2, London, UK: MacMillan and Co., 1892.
Albert Einstein referred to Maxwells equations in:
«Consideration Concerning the Fundamental of Theoretical Physics»
From 1865 to today
1865
Maxwell publish his 20 equations and 20 variables in:
«A Dynamical Theory of the Electromagnetic Field»
1873
Maxwell correct a sign error and include more equations in:
«A Treatise on Electricity and Magnetism»
1884
1885-1887
1940
«The so-called special or restricted relativity theory is based on
the
fact that
Maxwell’s
equations
(andofthus
the lawEquations:
of
Heinrich
Hertz
publish his
derivation
Maxwell’s
propagation
of light
in space)
are converted
into
equations of Equations
the
«On the Relations
between
Maxwell’s
Fundamental
Electromagnetic
same
form,
when they
undergo
Lorentz
transformation.»
and the
Fundamental
Equations
of the
Opposing
Electromagnetics»
A. Einstein, 1940
Oliver Heaviside reformulated 12 of Maxwell’s 20 equations into 4:
Several papers: Electrical Papers, vol. 1 and 2, London, UK: MacMillan and Co., 1892.
Albert Einstein referred to Maxwells equations in:
«Consideration Concerning the Fundamental of Theoretical Physics»
Maxwell’s equations today
 EdA 
Qencl
0
 BdA  0
(Gauss's law)
(Gauss's law for magnetism)
dB
(Faraday's law)
 E  d l   dt
dE 

 B  d l  0  iC   0 dt encl (Ampere's law)
Gauss’s law for electric fields
Total flux through any closed surface,
is proportional to the total charge inside
the surface:
E 
 EdA 
Qencl
0

E
q
0
q
QA  q
QB  q
QC  0
QD  0
 0  8.8541.. 1012 F m
-
Permittivity in vacuum
+
Gauss’s law for magnetism
Total magnetic flux through any closed
surface, is always zero (no monopoles):
B 
 BdA  0
B
N
A magnet will always have two
poles, hence total flux is zero.
S
Faraday’s law
E
dB
 E  d l   dt  EMF
B
A changing magnetic field is
accompanied by a changing
electric field at right angles to
the change of the magnetic
field
dl
B
Changing magnetic flux results
in a electric field and thereby
a current around the loop:
i
E
E
E
i
B
E  d l  E [Vm ]  l [m]  V [V ]
1
E
E
Faraday’s law
0
A
N
S
Ampere’s law
dE 

 B  d l  0  iC   0 dt encl
An electric current is
accompanied by a magnetic
field whose direction is at right
angles to the current flow
B
B
iC
iC
B
B
B
Ampere’s law – Maxwell’s extension
dE 

 B  d l  0  iC   0 dt encl
Displacement current
B
Time-varying
electric field
Q
Q
B?
Capacitor
Battery
Maxwell extension:
A changing electric field is
accompanied by a changing
magnetic field
Electromagnetic waves
Faraday’s law:
A changing magnetic field is accompanied
by a changing electric field
dB
 E  d l   dt
Ampere’s law - Maxwell extension:
A changing electric field is accompanied
by a changing magnetic field
dE 

 B  d l  0  iC   0 dt encl
 0 µ0 
1
c2
Faraday’s law and Ampere’s law - Transformer
Changing electric flux d  B
dt
Ampere’s law:
 B  d l  0iC
B
Alternating current
(50 Hz)
Faraday’s law:
dB
E

d
l


 EMF

dt
Alternating current
(50 Hz)
Ideal magnetic core material
Electrical machines
Faraday’s law:
dB
 E  d l   dt  EMF
N
S
Maxwell’s Equations today?
Robert «Bob» Scully, former President IEEE EMC Society:
«Truly, Maxwell’s Equations are the heart and soul of our discipline»
Maxwell’s Equations and Electrical Engineers
 EdA 
Qencl
0
 BdA  0
(Gauss's law)
(Gauss's law for magnetism)
dB
(Faraday's law)
dt
d 

 B  d l  0  iC   0 dt E encl (Ampere's law)
 Edl  
Bibliography
•
J. C. Maxwell, «On Physical Lines of Force, part 1-4,» London-Edinburgh-Dublin Philosph. Soc., vol.
21-23, 1861-1862.
•
J. C. Maxwell, «A Dynamical Theory of the Electromagnetic Field,» in Philosophical Transactions of
the Royal Society of London, UK, 1865, pp. 459-512.
•
A. Einstein, «Consideration Concerning the Fundaments of Theoretical Physics,» Science, New
Series, vol. 19, No. 2369, The Science Press, New York, NY, 24 May 1940, pp. 487-492.
•
J. C. Maxwell, The scientific papers of James Clerk Maxwell, New York: Dover Publications, 1965.
•
H.D. Young and R. A. Freedman, University Physics, 11th ed. Texas: Pearson, 2004.
•
D. Fleisch, A Student’s Guide to Maxwell’s Equations, Cambridge University Press, UK, 2008.
•
R. Scully, «The Evolution of Maxwell’s Equations Through a Brief Critical Examination of the History
and Background of the Man and His Times – Part 1-4,» IEEE Electromagnetic Compatibility
Magazine 2013-2014.
•
R. Scully, «Maxwell’s Legacy: The Heart and Soul of the EM Discipline,» IEEE MTT-S International
Microwave Symposium, Phoenix, AZ, 2015.
•
D. Brooks, Maxwell’s Equations Without The Calculus, Kirkland, 2016.
?
S
Questions?