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

0

0 0

(Gauss's law)

0 (Gauss's law for magnetism)

(Ampere's law)

(Faraday's law)

encl

EC

encl

B

Qd

d

dd i

dt

dd

dt

E A

B A

B l

E l

є

є

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?

Electrostaticgenerator

Insulating layer

Conductor

Leyden jar

Pieter van Musschenbroek (Leyden)Ewald Georg von Kleist

1749 -> 1854

Alessantro Volta (1745-1827)

1749 -> 1854

André-Marie Ampére (1775-1836)Hans Christian Ørsted (1777-1851)

Electric currents createmagnetic fields

Laid the fundation of«Electrodynamics»

Ampéres law

1749 -> 1854

Michael Faraday (1791-1867)Georg Simon Ohm (1789-1854)

Ohm’s law Electromagnetism, electrochemistry,

induction

1749 -> 1854

James Clerk Maxell (1831-1879)William Thomson (1824-1907)Lord Kelvin

ElectricityThermodynamics

Electromagnetic wavesMaxwell’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 ofparticles 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»

Mechanical bipolar molecular vorticies (or eddies) seperated by a layer ofparticles are used as an analogy.

“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.”

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 ofparticles are used as an analogy.

“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.”

“The conception of a particle having its motion connected with that of a vortex by perfect rolling contact may appear somewhat awkward. I do not bring it forward as a mode of connexion existing in nature, or even as that I would willingly assent to as an electrical hypothesis.”

Maxwell’s 3rd paper:

«A Dynamic Theory of the Electromagnetic Field»

Reformulated into a electromagnetic theory, without any sort of mechanical analogy

• 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

• 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.»

One equation of:

- Free Electricity

- Continuity

From 1865 to today

1865 Maxwell publish his 20 equations and 20 variables in:«A Dynamical Theory of the Electromagnetic Field»

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 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.

1940 Albert Einstein referred to Maxwells equations in:«Consideration Concerning the Fundamental of Theoretical Physics»

1873 Maxwell correct a sign error and include more equations in:«A Treatise on Electricity and Magnetism»

From 1865 to today

1865 Maxwell publish his 20 equations and 20 variables in:«A Dynamical Theory of the Electromagnetic Field»

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 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.

1940 Albert Einstein referred to Maxwells equations in:«Consideration Concerning the Fundamental of Theoretical Physics»

1873 Maxwell correct a sign error and include more equations in:«A Treatise on Electricity and Magnetism» «The so-called special or restricted relativity theory is based on the fact that Maxwell’s equations (and thus the law of propagation of light in space) are converted into equations of the same form, when they undergo Lorentz transformation.»A. Einstein, 1940

Maxwell’s equations today

0

0 0

(Gauss's law)

0 (Gauss's law for magnetism)

(Faraday's law)

(Ampere's law)

encl

B

EC

encl

Qd

d

dd

dt

dd i

dt

E A

B A

E l

B l

Gauss’s law for electric fields

+

E

q

0

enclE

Qd

E A

0

q

Total flux through any closed surface,is proportional to the total charge insidethe surface:

12

0 8.8541.. 10 - Permittivity in vacuumF m

0

0

A

B

C

D

Q q

Q q

Q

Q

Gauss’s law for magnetism

0B d B A

Total magnetic flux through any closedsurface, is always zero (no monopoles):

N

S

B

A magnet will always have twopoles, hence total flux is zero.

Faraday’s law

Bdd

dt

E l

A changing magnetic field is accompanied by a changingelectric field at right angles to the change of the magneticfield

B

Bdl

EMF

B

E

E

EE

E

E

Changing magnetic flux resultsin a electric field and therebya current around the loop:

i

i

1[ ] [ ] [ ]E d E Vm l m V V l

Faraday’s law

NS

A

0

Ampere’s law

0 0E

C

encl

dd i

dt

B l

An electric current is accompanied by a magneticfield whose direction is at right angles to the current flow B

B

B

B

Ci

B

Ci

Ampere’s law – Maxwell’s extension

0 0E

C

encl

dd i

dt

B l

B ?B

Q Q

Time-varyingelectric field

Maxwell extension:A changing electric field is accompanied by a changingmagnetic field

Displacement current

Capacitor

Battery

Electromagnetic waves

0 0E

C

encl

dd i

dt

B l

Ampere’s law - Maxwell extension:A changing electric field is accompanied by a changing magnetic field

Bdd

dt

E l

Faraday’s law:A changing magnetic field is accompaniedby a changing electric field

0 0 2

1µ

c

Faraday’s law and Ampere’s law - Transformer

Alternating current(50 Hz)

Changing electric flux Bd

dt

Ideal magnetic core material

Alternating current(50 Hz)

Bdd EMF

dt

E l

0 Cd i B l

Ampere’s law:Faraday’s law:

B

Electrical machines

N

S

Bdd EMF

dt

E l

Faraday’s law:

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

0

0 0

(Gauss's law)

0 (Gauss's law for magnetism)

(Faraday's law)

(Ampere's law)

encl

B

EC

encl

Qd

d

dd

dt

dd i

dt

E A

B A

E l

B l

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.

?Questions?

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