Maxwell’s Equations & Hertz Waves

Dr. Bill Pezzaglia

Updated: 2013Aug15

XI. Maxwell & Electromagnetic Waves

A. Maxwell’s Equations

B. Hertz Waves & Poynting

C. Polarization

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A. Maxwell’s Equations

1) Light is Electromagnetic

2) Displacement Current

3) Maxwell’s Equations

31. Hints that Light is Electromagnetic

A number of experiments suggested that there was a connection between electricity, magnetism and the phenomena of light.

• 1834, 1857 Speed of electricity in wire measured to be very fast (close to speed of light)

• 1850 Speed of light is measured accurately 1850 by Foucault.

• 1844 Faraday rotates the polarization of light with a magnetic field (implies light has magnetic properties).

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Charles Wheatstone (1802-1875)

•1834 discovery by English physicist Charles Wheatstone that current traveled through long lengths of wire with great velocity – almost 288,000 miles/second

•A bit off, it can’t travel faster than speed of light (186, 282 miles/second)

•1837 Developed an early telegraph (5 needles)

5 Michael Faraday

•1844 Faraday rotates the polarization of light using a magnetic field. Suggests light is a transverse magnetic disturbance

•1857 Wilhem Weber shows Amp of current is a Coulomb per second, gets characteristic speed of electrical signals to be speed of light.

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Gustav Kirchhoff (1824-1887)

•1857 Telegraphy Equations

•Derives (based on earlier work by Faraday & Thomson 1854) that speed of electrical signal in cable should be close to speed of light.

72. Field Induction

•Recall Faraday’s law is that voltage (emf=electric firld times circumference) in a wire loop was generated by changing magnetic flux Φthrough the loop.

•Maxwell shows that the law is more general. A changing magnetic field generates a circular electric field even if there is no wire!

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James Maxwell (1831-1879)

tV

∆∆Φ−=

b. Ampere’s circulation Law

• Recall : Ampere’s law says that a circular magnetic field is generated by a current.

• Or: B field multiplied by circumference of a circle is proportional to current flowing through circle

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IrB 0)2( µπ =

c. Maxwell’s Displacement Current

1861: Maxwell makes Ampere’s law look like the complement of Faradays Law. A changing electric flux will generate circular B field.

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Note “ c” is the speed of light, and Electric Flux is defined:

AErr

•=Ψ

tIrB

c ∆∆Ψ+= 2

10)2( µπ [details, you can

ignore equation]

3. Maxwell’s Equations

(a) The General Laws of Maxwell

• Gauss’s Law shows that charge is the source of electric fields (electric flux through a closed surface is proportional to net enclosed charge)

• Gauss’s Law for magnetism states that there are no magnetic charges (magnetic flux through a closed surface is zero).

• Faraday’s Law : changing magnetic fields create electric fields

• Ampere’s Circulation Law : current is the source of magnetic fields. Maxwell adds the “displacement current” to this equation such that changing electric fields create magnetic fields.

11b. Differential Form of Maxwell’s Equations

•1884 (with Gibbs) Heaviside reorganizes Maxwell’s equations compactly into 4 “vector”equations

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+∂∂+=×∇

∂∂−=×∇

=•∇

=•∇

Jt

EB

t

BE

E

B

00

0

0

εµ

ερ

Oliver Heaviside (1850-1925)

For completeness, here they are, but don’t worry about them.

c. Relativity and Maxwell’s Equations

• 1905 Einstein’s Relativity shows that time is the 4th dimension.

• In our ordinary “3D” view of the world, electric fields are different than magnetic fields, however we see they are complementary

• In “4D” we see that they are both the same thing, i.e. we “unify” electricity with magnetism.

• We can write Maxwell’s 4 equations in just 2:

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0][ =∂

=∂

νκµ

νµνµ

F

jF For completeness, here they are, but don’t worry about them.

B. Hertz Waves

1) Equations predict waves2) Hertz Experiment3) Energy in Waves

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1. EM Wave Equation(a) 1865 Maxwell shows his equations predict that electromagnetic waves can exist in vacuum (note E & B are perpendicular to each other and direction of wave)

151b. Prediction of Electromagnetic Waves

• The Theoretical speed:comes out very close toknown speed of light “c”

• Magnitude of electric and magnetic fields are simpl y related by wavespeed:

00

1

µε=v

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cBE =

2. Making EM Waves

(a) 1891 (1888?) Hertz demonstration that electromagnetic waves can be transmitted and then received. Proves existence of waves with frequencies of 100 million cycles per second.

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Heinrich Hertz (1857-1894)

2b. Nikola Tesla (1856-1943)

•1891 (1893?) Chicago World’s fair, demonstrates wireless telegraphy (30 feet)•1894 Lodge transmits 150 yards

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2c Guglielmo Marconi (1874-1937

•1899 Marconi “steals” Tesla’s design and broadcasts across the English Channel

•1901 Across the Atlantic

193. Wavespeed Phenomena

(a) Index of refraction:• Speed of light “v” in media is slower

where “n” is index of refraction (about 1.5 for glass).

• Index can be calculated from the electrical permittivity (ε) and magnetic permeability (µ) properties of the media.

• Index usually depends upon wavelength of the light (e.g. in glass red might have n=1.50 while for blue n=1.53)

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εµ1=

=

n

n

cv

(b) Reflection & Transmission

• As a wave (such as light) in media 1, with index n1, enters a denser media (index n2) where the speed changes, part of the wave will be reflected.

• Proportion given by formula:

• The rest is transmitted.

• For glass (n=1.5) we calculate that 4% is reflected, 96% transmitted

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2

12

12

+−=

nn

nnR

(c) Absorption

• Good conductors: reflect nearly 100%

• Poor conductors: wave penetrates into media to “skin depth” and is absorbed (energy turned into heat). Wave exponentially decays with distance.

• For AC Signals traveling through a wire, at higher frequencies the skin depth is very small, and so electricity will travel only on the outside of a conductor (hence increasing its resistance).

• At 60 cycles the skin depth is 8.5 mm for copper, so making a bigger diameter wire is a waste of metal.

• Instead, use Litz wire, made of many small wires.

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C. Polarization

1) Linear Polarization2) Birefringence3) Circular Polarization

23Linear Polarization

[1812 Fresnel develops wave theory of transverse polarized light, well before the electromagnetic nature was known]

light has two perpendicular linear polarizations (electric field) can be horizontal or vertical

1888 Hertz shows electromagnetic waves have transverse polarization (equivalent to “light”)

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• 1808 Malus’s Law: Reflected light is often polarization

• 1812 Fresnel develops wave theory of transverse polarized light

• 1815 Brewster’s angle: at this angle of incidence the reflected light is entirely “s”polarized such that electric field is parallel to the interface surface

Polarization by Reflection25

nb =θtan

Note: “Plane of incidence” is the plane defined by the three beams above.The normal also lies in this plane. The plane is perpendicular to the surface.

• Linear polarizer can be used to detect polarized light, only lets one polarization through!

• 1808 Malus’s Law: Linear polarized light passing through a second polarizer tilted at angle θ to first will be attenuated:

• Hence no light gets through “crossed polarizers” (θ=±90°)

Detecting Polarized Light26

20 ][cosθII =

•Optically active materials can rotate the polarization

•If such a substance is put between “crossed polarizers” (90º angle) you will often see interesting colors.

Optical Activity27

•1669 Erasmus Bartolinus (Denmark) discovers the birefringence (double refraction) of calcite crystals.

•When polarization was understood better, it was realized the two different polarizations took different paths (they are “refracted”differently, or the index of refraction is dependent upon polarization)

•Index of refraction: n=c/v, so the different polarizations travel at slightly different speeds.

Birefringence28

Quarter Wave Plates

A quarter wave plate retards horizontal polarization by 90º to vertical. It can be used to make circular polarized light from linear polarized light.

29Circular Polarized Light

Another type of polarized light can be left or right handed circular polarized

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Detecting Circular Polarized Light

A quarter wave plate will turn circular back into linear, which can be detected by a linear polarizer

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References•http://maxwell.byu.edu/~spencerr/phys442/node4.html

•http://en.wikipedia.org/wiki/Timeline_of_Fundamental_Physics_Discoveries

•http://www.sparkmuseum.com/GLASS.HTM

•http://keelynet.com/spider/b-103e.htm

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Things to Do•Find tesla museum stuff

•Who first predicted circular polarized light?

•Can we make a 3D image for students using polarized light? Need two projectors?

•Ideally we’d use circular polarized light, but one test so far shows either the transparency projector or the screen does not preserve the circular polarization.

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