Electromagnetic radiation
-----------------------
In 1864 Maxwell formulated his theory of electromagnetic radiation
(EMR), tying together the then separate studies of electricity and
magnetism. Maxwell believed a real medium was necessary for conveying
the EMR. Later, by Einstein, such requirement was proved false. EMR
can travel thru empty space and it is by it that we acquire virtually
all intelligence about the universe.
The amount of knowledge we get by meteorites, cosmic particles,
Moon rocks, solar wind, and the like is minuscule against the
information from the EMR. Or, in reverse logic, with no EMR our
awareness of the universe would be pretty much zilch.
Frequency versus Wavelength
-------------------------
We employ either wavelength or frequency to describe EMR. lambda
is the symbol for wavelength; nu, frequency. Letter f is also commonly
used for frequency. The two are interchangeable at will by
c = lambda * nu
+----------------------+
| FREQUENCY-WAVELENGTH |
| RELATION |
| |
| c = lambda * nu |
+----------------------+
The metric name for 'cycle per second' is 'hertz' but many older
scientists still use cycles/second. Some works actually use 'sec^-1'
to clarify this idea, like '92e6 sec^-1'.
The metric unit of wavelength is a multiple of the meter like
nanometer in the optical range or micrometer in the infrared zone. The
micrometer is prevalently stated by its older na,e 'micron'.
Because it's clumsy to write the formal symbol for 'micro-' as the
letter 'mu ', the letter 'u' is sometimes used. It kind of looks like
a 'mu'..
In astronomy an older unit, 1e-10 meter or angstrom is still in
wide circulation. 1 angstrom is 1/10 nonometer.
The choice to use is often a matter of tradition. Some regions of
EMR were explored as a function of wavelength and others of frequency.
Radiowaves are generally described in frequency while light is cited
in wavelength. We can use either one as we like for any region of
electromagnetic radiation.
The EMR comprehended between two limits of wavelength (or
frequency) is called a spectrum, after the visual apparition
discovered by Newton in his experiments on light. He saw, when letting
sunlight refract thru a glass prism, a splay, devolution, dispersion
of colors. Much later these were found to be the multitude of
wavelengths represented in sunlight, each refracted slightly
differently thru the prism. Our eyes register each wavelength as a
separate color.
Irregular jargon
--------------
along with the separate work in the various wavebands of
electromagnetic waves comes the inconsistent jargon describing the
waves. altho I here use the International system of units and terms,
you will read and hear other terms, words, units of measure.
The main alternative measurement system is the CGS, 'centimeter-
gram-second. Irs units are still in wide use, mainly in specialized
sciences. The units are mostly powers of ten multiples of SI units,
making translation reasonably easy.
Terminology, names, nomenclature is more varied, with terms in
one waveband having different meaning in an other. You do have to
study the jargon when working across wavebands.A common instance is
the designation of wavebands. There are no official boundaries between
the zones of wavelength, except for certain ones regulated by
government agencies, such as broadcast radio. It happens that one
scientists calls the band he works with 'far ultraviolet' but an
other scientist in his own work calls it 'soft X-ray'.
An other situation is that many combinations of units, in both CGS
and SI, have honorary names, like 'joule' for 'newton.meter' and
'volt' for 'newton.meter/coulomb'. 'newton' is itself the honorary
name for 'kilogram.meter/second2'. Most of these names honor
scientists who worked with associated principles, effects, concepts.
Electric charge
-------------
The elemental unit of electric charge is the electron. Irs charge
by history is 'negative' of strength one unit. The proton also carries
one unit of charge, this being 'positive'. charges of like signum
repel and those with opposite signum attract. In ordinary matter the
protons and electrons are equal in number, netting the overall charge
to zero.
The protons are locked in atomic nuclei and are not easily available
for useful work. The electron can move around among atoms to
accumulate static charge or become an electric current.
A body taking on a negative charge has an excess number of
electrons. The protons in the target are too few to net out all the
electrons, leaving the extra ones to give the body its net negative
charge.
A positively charged body has a decess of electrons, leaving the
extra nuclear protons to exert a net positive charge.
Transferring electrons to and from an object was primitively done
by rubbing the object with silk or fur. Certain materials gained
electrons to be negatively charged. Other lost electrons to be
positively charged. This effect is the practice of electrostatics or
static electricity, still in routine use for entertainment.
A charged body creates an electric field around it that presses an
force on test particles in the field.
Priestley in 1766 experimented with charged metal cans. He
found that within the can there is no electric field and correlated
this fact with gravity. Within a hollow vessel there is no gravity
field from the vessel's own mass.
This effect comes from the inverse-square law. of gravity. Charges
must also follow the same rule.
Coulomb in 1785 built a torsion balance, like the one from
Cavendish for gravity, to determine the force relation directly.
Coulomb verified Priestley's work and found that
force = K * charge1 * charge2 / distance^2
charge1 and charge2 are the two charges, with their positive or
negative signa. If their product is positive (+ times + or - times -)
the force is repulsion. A negative product (+ times -) is an
attraction.
The unit of charge was is primitively the electron but this is a
in infinitesimal unit. The SI unit is the Coulomb, equal by
experiment and measurement to
1 coulomb = 6.4518e18 electrons
K is the Coulomb constant like the Newton constant.It was a single
number under older systems of units. In the SI it is diffracted into
K = 1 / (4 * pi * epsilon)
to coordinate with other electromagnetic units. epsilon is a property
of the neduyn carrying the electric field.
Today we start with the definition of electric current, with unit
ampere. The electric current flowing thru an electrical device is
marked on device's nameplates on electric devices as a flow of
electricity thru the device.
The ampere is
1 ampere = 1 coulomb/second
Batteries are sometimes specified by thee ampere.hours of charge they
hold. With 3600 seconds per hour,
1 ampere.hour = 3,600 coulomb
Before the electron was discovered as the sole charge carrier in
electricity, we spoke of the electric fluid flowing from the positive
charge to the negative charge. This convention embedded into the
nascent 19th century electrical industry and persists today.
Electricity flows from the positive end of a battery to the negative
end.
The flow of electrons ended up by luck being the opposite flow,
from negative to positive. This is part of electronics, developed in
the early 20th century.
Electric field
------------
An electric charge has around it an electric field. It is similar
to the gravity field around a mass. The electric field of a point
charge follows Coulomb's inverse-square law.
force = (1 / (4 * pi * epsilon)) * charge1 * charge2 / distance^2
In a way analogous to Newton's gravity field strength, a charge q1
sets up an electric field of strength E at distance r
(field strength) = (1 / (4 * pi * epsilon)) * charge1 / distance^2
epsilon for a vacuum has the smallest value of all media and is
given the symbol epsilon0. Its value is
epsilon0 = 8.854e-12 C2/N.m2.
The 4*pi factor is an effort to simplify calculations like for
illumination. It didn't work as hoped but we're stuck with it.
+-----------------------------------------------+
| COULOMB'S INVERSE SQUARE LAW |
| |
| F is force, E is field strength, q is charge, |
| r is distance |
| |
| F = (1 / (4 * pi * epsilon)) * q1 * q2 / r^2 |
| |
| E = (1 / (4 * pi * epsilon)) * q1 / r^2 |
| |
| epsilon0 = 8.854e-12 C2/(N*m2) |
+-----------------------------------------------+
Magnetic poles and fields
---------------------
It turns out that there is little need for a distinct parameter of
a magnetic pole strength. This would be symmetrical with the unit of
electric charge or of mass. Because wo can not actually have separate
magnetic poles and the whole purpose of magnets is to build a magnetic
field in a given volume, work is performed with just the magnetic
field strength. An other reason is that magnetic fields can be
created via electromagnets. There generally are no 'poles' in the
ordinary sense. In Ampere's law the magnetic field surrounds th wire
in closed loops. There is no actual 'poles'.
In history, Michell in 1750 did discover the inverse- square law
for magnetic poles. His equation parallels those for gravity and
light. (Priestley and Coulomb were still in the future.)
magnetic force = M * pole1 * pole2 / distance^2
where M is the Michell constant. No value is standardized for it
because we hardly ever use this inverse-square formula.
pole1 and pole2 are magnetic poles, north or south. Like poles
repel; opposite, attract. From the absence of separate magnetic poles,
there is no standard value for a pole strength.
It could be feasible to experiment with Michell's law with a long
bar magnet and keeping close to each end of it. This dilutes the
influence of the other end on the force.
The magnetic field strength is found by the action of the field on
a test electric current. In a lab setup of specified geometry and
operation within the magnetic field. The force on the apparatus is
measured. We have
(magnetic field strength) = func(force, current, geometry)
The result of such an experiment is that the magnetic field strength
is in newton/(ampere.meter). Note this incorporates the main factors
defining field strength: newton for force, ampere for current, meter
for geometry.
This is hardly symmetrical with the newton/coulomb units
for the electric field strength. Given the lack of separable magnetic
poles we are compelled to go thru this round-about definition.
The units newton/(ampere.meter) is the tesla. The CGS unit, gauss,
is still widely use. 10,000 gauss equal one tesla. The Earth's
magnetic field strength at ground level is 1/2 to 1 gauss or 5e-5 to
1e-4 tesla.
Aren't we engaged in circular reasoning? We first used a magnetic
field to define the ampere, the coulomb/second. Then we use the ampere
to define the magnetic field! In a way, yes, and philosophers debate
this feature of physics endlessly. For one thing it leaves us with no
explicit quantity for the strength of a magnetic pole!
Ampere and Fraday
---------------
In the early 1800s there were hints that electricity and magnetism
are somehow linked. Oersted and also Ampere in 1820 discovered that an
electric current can produce a magnetic field. Connect a wire
across a battery, source of electric current, with a small lamp bulb,
like fro a pocket torch. The lamp shows that current is flowing and
prevents the battery from discharging by a short circuit thru the
wire.
Put a scouting or hiking compass near the wire. The compass
deflects away from its normal alignment to magnetic north. Every where
along the wire the compass reacts this way. The electric current
generated around the wire a magnetic field that attracts the compass
needle.
When the circuit is opened, the current stops, the magnetic field
vanishes. The compass returns to its natural alignment.
In words Ampere's law is
(electric current)_ -> (magnetic field)
where '->' means 'can produce or generate'.
Faraday in 1838 found that a varying magnetic field can produce an
electric current. Connect the wire to the lamp bulb this time with no
battery..Sweep one end of a bar magnet over the wire.The motion of the
magnet changes its field along the wire.
As long as the magnetic field is varying, bu continuing to move
the magnet. an electric current flows in the wire and lights the bulb.
The bulb may glow weakly but a stronger glow is obtained by more
vigorously sweeping the magnet. .
When the magnet is removed or held still, the current stops.
A verbal statement of Faraday's principle is
(varying magnetic field) -> (electric current)
Note carefully that Ampere and Faraday laws are not symmetrical. While
a steady electric current produces a magnetic filed by Ampere, it
takes a changing magnetic field to make an electric current for
Faraday. A steady field does not yield electricity.
Faraday's principle quickly leaded to the invention of the
electric dynamo or generator. Until then elecctricity was made by
chemical batteries or by static charge. In a dynamo a coil of wire is
rotated within the poles of strong magnets. As the coil rotates it
'sees' a varying magnetic field and a current is impressed in it. This
current is drawn off to run machinery.
Such dynamos were built by the 1860s for on-site production of
electricity, like in factories and ships. The coil was rotated by
steam engines or water wheels.
By way of history, Edison did not 'invent electricity'. Edison was
the first to commercialize electricity for anyone to obtain its
benefits. In doing so he invented the electric service industry.
Edison's original company from 1882 is today's Consolidated Edison
Company of New York.
The discoveries by Faraday and Ampere was elaborated into
mathematical expressions, skipped here, as new laws of physics.
Lenz effect
---------
If under Faraday's law a magnetic field induces an electric
current, doesn't that current in turn under Ampere's law set up a
magnetic field? If so, can these two fields combine to produce more
current? And we could bleed off some of this current to run our
machinery without further input of energy to the wires!
Is this a perpetual motion machine in the rough?
No.
The resultant field set up by the electric current opposes the
stimulant field to temper the rate of change. The fields are
antagonistic, not sympathetic.
You may experience the Lenz effect with the wire and lamp. With
the circuit open you can freely sweep the magnet over the wire. When
the circuit is closed you feel resistance against the magnet. This is
the Lenz field opposing the magnet.
This is way a dynamo needs a continuous external force to keep the
magnet-coil moving and maintain the output of electricity.
Lines of Force
------------
A line of force associated with a field is not a real physical
feature in nature. Older texts sometimes spoke of lines of force as if
they were actual traces thru a field.
None the less they are an excellent aid for visualizing fields and
they in fact do have a physical foundation. A line of force is the
trajectory of a test particle placed in the field and released to move
under the force of the field. It's like a freefall trajectory in
astrodynamics. There is no initial velocity of the particle. The
particle is gently let go in the field, not or injected or propelled.
For an electric filed we use a positive charge, like a proton.
When released in the field it is repelled from other positive charges
and attracted to other negative charges.
The path of this particle is a line of force. In principle you can
actually map out the field.
The direction of motion of the test particle is commonly
indicated in diagrams by arrows along the line. We can speak of a line
'going from positive to negative' as a jargon, not a statement of real
movement of the line itself.
The number and density of lines of force qualitatively show the
strength of the field. More and denser lines indicate a field stronger
than where the lines are spread apart. The number of lines emanating
from or cascading into a charge qualitatively is the strength of the
charge. Stronger charges have more lines associated with them.
We can not actually place a magnetic pole into a magnetic field
and watch how it moves. There is no isolated pole, like an electric
charge. Poles always come in paris, north and south together, they
cancel out in the field and don't move under its force.
In thought experiment we place a north pole in the field and trace
out its trajectory. This is the magnetic line of force. Arrows along
it in diagrams show the motion of the test pole.
Gauss's Law for charge
--------------------
Imagine a spherical shell enclosing some electric charges, a mix
of positive and negative ones. They may be scattered inside the shell,
not bunched at the center. For simplicity, each charge has the same
strength and is assigned one line of force.
The force line from a positive charge aims outward. That from a
negative charge points inward. These are the directions of motion of a
positive test particle placed on the line.
All lines of force pass thru the enclosing sphere, some inward,
some outward. Tally the in and out lines, algebraicly, over the
-sphere. Each inward line offsets a outward line, leaving leftover
lines of one polarity. Toss the offfset pairs and keep the leftover
lines.
Each pair of lines represents a positive-negative pair of charges
inside the sphere. Any left over lines, with no paired opposite ones,
are attached to extra positive or negative charges in the shell.
Gauss's law for charges states that on an enclosing sphere with
charges within it, the algebraic count of lines of force, the electric
field, all over its surface equals the net electric charge within the
sphere. Gauss worked this out in 1835 for both electric charges and
magnetic poles.
Gauss's law itself can not tell the separate number of charges,
only the residual unpaired ones. You could have 20 gazillion electrons
and 20 gazillion and two protons. You 'see' only the two extra
PROTONS. In words:
(electirc lines thru sphere) = (net charges inside sphere)
The sphere does not have to be a sphere. It may be any shape so
long as it is closed, ensuring that lines of force from inside must
pass across its surface and not thru a rip, tear, other opening. It
may have folds! A line passing thru one ply of a fold in one
direction, passes an other ply in the opposite direction. Eventually
it leaves the enclosure and is then counted. The spherical shape is
routinely used because of its simple geometry and maths.
Gauss's Law for poles
-------------------
Repeat the setup of the enclosing sphere with magnets inside. Let
each north or south pole emit one line of force. The north pole lines
point outward; south pole, inward.
Tally the lines passing the sphere. pair off the in and out lines
and toss them and keep any left over unpaired lines.
Some north poles pair with south poles and their lines cancel in
the tally.
We find that ALL lines papir with NO left over ones! No matter
what mix of magnets are in the sphere, every un line is offset by an
out line, leaving none as a net excess.
Gauss's law of poles says that the net count of lines, the
magnetic field, is always zero over the sphere.
The reason is that so far as we know there are no lone magnetic
poles. EVERY pole has its mate of equal strength but opposite
polarity. EVERY enclosing sphere contains EXACTLY equal numbers of
north and south poles and the net magnetic field on the sphere is
ZERO.
Like for Gauss's law for charges the separate number of poles
inside the sphere is not determined. Only the excess of north or south
poles can be found, which is always none.
This in words is:
(magnetic lines thru sphere) = (zero net poles inside sphere)
Maxwell's Equations
-----------------
In 1864 Maxwell issued his theory of electromagnetism, uniting the
two separate disciplines into a new single construct, electromagnetic
wave or radiation. He demonstrated that all of the phaenomena of
electricity and magnetism can be reduced to four fundamental
equations. They are elaborations of Gauss's law for charge, Gauss's
law for poles, Faraday's law, Ampere's law.
Maxwell's equations in words are:
+----------------------------------------------------------------+
| MAXWELL'S FOUR EQUATIONS OF ELECTROMAGNETISM |
| |
| electric field on Gauss sphere = net electric charge in sphere |
| |
| magnetic field on Gauss sphere = zero magnetic poles in sphere |
| |
| changing magnetic field around wire -> elec current in wire |
| electric current in wire -> magnetic field around wire |
+ ---------------------------------------------------------------+
These equations divide into two groups. The Gauss equations relate
to poles and charges at rest. They cover electrostatics and
magnetostatics.They deal with each field separately without
commingling them.
The Ampere and Faraday equations relate to charges in motion or
magnetic fields in motion. They deal with electrodynamics and
magnetodynamics.
Gauss's Law for mass
------------------
We divert to examine an other Gauss sphere and the field on it.
This is sometimes left out of home astronomy dialog on cosmology but
it is one of its most fundamental principles.
Same Gauss sphere with some particles of mass inside. Each
particle has a single gravity line of force. Because gravity is an
attracting force, a small test mass moves inward along the line.
Tally the lines over the sphere, netting the inward oneS with
outward ones.
We find that ALL lines are inward! There are no outward lines.
Gauss's law for mass states that the count of lines, the gravity
field, over the sphere is the TOTAL mass within the sphere. Not a net
of 'positive' and 'negative' mass but ALL of it as only 'positive'.
Entirely unlike Gauss spheres for charges and poles, we can add
only mass of the one kind and increase the count of lines on the
sphere. There is no way to add mass that cancels out some lines, like
for charges, or all of the them, like for poles.
Cosmology is often a part of relativity. In such work gravity is
the force that governs the behavior of the universe on the larger
scales of volume.
There are vast magnetic and electric fields in space. They are
crucial for the study of nebulae, stars, galaxies. Astrophysicists
need a solid grounding in electromagnetic theory.
look again at the three Gauss's laws. On the scale of many
millions of lightyears, like within a galaxy cluster, there are inside
a Gauss sphere immense numbers of charges, poles, particles.
Because every magnetic pole has a matching opposite pole, there is
no imbalance of poles inside the sphere.
no net magnetic poles. The force lines thru the sphere cancel to
zero. Altho locally, within stars or nebulae, magnetic forces are
important, on the grand universal scale they are absent.
There are individual charges, such as electrons and protons, in
space. Within stars and nebulae we have humongouss flows of protons or
electrons separate from each other. A Gauss shell around these local
volumes can contain an unbalanced number of charge.
On the larger scale it is excedingly tough to sustain charge
segregation. The forces between opposite charges is incredibly strong,
overcoming other natural forces that try to keep the charges apart. A
Gauss sphere tends to enclose less and less imbalance of charge, more
and more equal numbers of positive and negative charges. The electric
field and its force tend to zero.
Looking at enclosed mass we have a whole different situation.
There being only one kind of mass, a Gauss sphere of ANY size will
contain an imbalance of mass, all 'positive'.
The larger the sphere the more mass it embraces as the sphere
ropes in ever larger volumes of space. The gravity field on its
surface also increases, this being the ever greater count of gravity
lines of force from the interior atoms.
On the scale of the whole universe the only governing force is
gravity. Cosmology is truly driven by the action of the mass, and not
the poles and charge, within the universe.
Maxerll's waves
-- ---------
When Maxwell studied the four cardinal formulae, he found that
they are properties of single new entity, a wave of perpendicular
interacting electric and magnetic fields. We examine some major
features of this new wave.
A general wave function, such as a mechanical wave,, looks like
d^2(amplitude) / dx^2 = (dt^2 / dx^2) * (d^2(amplitude) / dt^2)
The amplitude of the wave is the displacement of the wave
perpendicular to the direction of the wave's motion. It is a math
expression, like a sine function.
The 'd's are part of the symbols for 'derivative' in calculus,
which we won't actually handle as such.
The left side relates amplitude to the downrange place along the
direction of wave motion, or the x-axis. The right side relates
amplitude to the time elapsed since the wave started from the zero
point of the x-axis.
In his investigations of the interaction of electric and magnetic
fields Maxwell developed an expression for the strength of the
electric field
d^2(strength) / dx^2 = mu * epsilon * d^2(strength) / dt^2)
He compared this with the general wave equation and matched the terms.
The amplitude in the general case becomes the electric field
strength.
The dt^2/dx^2 becomes (mu*epsilon). These two parameters are the
ability of the medium, in which the wave travels, for sustaining a
magnetic (for mu) or electric (for epsilon) field.
Some substances let a magnetic field penetrate them well while
others are good shields against the field. This permeability of the
medium is expressed by mu.
A similar reasoning applies to an electric field. epsilon is the
permittivity of the medium.
Before Maxwell, these parameters were experimentally measured for
a huge assortment of materials as basic properties of these materials.
When the material was a vacuum the mu and epsilon are denoted
epsilon0 and mu0 is used.
mu0 and epsilon0
--------------
The Maxwell's wave formula we see the matching of terms and
letting the medium be a vacuum.
dt^2 / dx^2 = epsilon0 * mu0
= (dt / dx)^2
= 1 / speed^2
speed^2 = 1 / (epsilon0 * mu0)
speed = sqrt(1 / (epsilon0 * mu0))
When Maxwell plugged in the a;ready known values for mu0 and
epsilon0 he came up with a speed of his electromagnetic waves. It
matched virtually exactly the a;ready known speed of light!
From his other work with electromagnetic waves, he already
explained optical effects, including refraction of light across media.
In 1876 Maxwell revealed that light was not a distinct phaenomenon
in nature. Light was an electromagnetic wave behaving according to his
theory.
we have here three basic parameters, all historicly determined by
experiment by three different schools of physics. The speed of light
was mechanicly measured by physicists dealing with light.
mu0, the permeability of a vacuum, was measured by scientists
working with magnets.
epsilon0, the permittivity of a vacuum, was measured by scientists
in the electricity discipline.
The values of all three were published in the physics litterature.
Values for epsilon0 and mu0 are now folded into the metric system
of measures as
+----------------------------------------------+
| ELECTROMAGNETIC CONSTANTS |
| |
| c = 2.998e8 meter/second |
| |
| mu0 = 4*pi*1e-7 newton/ampere2 |
| = 12.567e-7 newton/ampere2 |
| |
| epsilon0 = 8.854e-12 coulomb2/newton.meter2 |
+----------------------------------------------+
Where's the medium?
-----------------In Maxwell's time light was believed to travel thru
a physical medium, like other waves then known. altho a vacuum
contained nothing, it just had to be filled with a substance, the
aether, that conveyed light thru outer space. c, the speed of light in
vacuum, was the speed of light thru this aether.
Attempts to detect the aether were tried in the 1880s. The first
was the Michelson-Morley experiment to measure the speed of light at
various directions from the Earth's orital motion.
The thinking was the light would travel faster or slower as Earth
moved into or away from a beam of light, like a boat's speed is found
by tracking a floating object thrown forward and then rearward.
All attempts failed to yield changes in c with orientation. c was
the same with one nasty interpretation being that Earth did not move,
but in fact stood still in space.
We don't take up relativity here but we can appreciate how
lightspeed can be a constant for every one. It comes from two physical
properties of nature, epsilon0 and mu0. c is an algebraic combination
of these parameters, tehmselfs constants. The properties of a vacuum
don't change with motion of the observer.
Physicists wrestled with this situation with many curious
explanations. In 1905 Einstein showed that light did not need an
actual medium and c was the same for all observers. From this
realization he developed the theory of special relativity.
Refraction of Light
-----------------
Maxwell found that he could explain all the phaenomena of light
which were previously known only thru experiment and empirical rules.
For rxample, Snellius in 1621 figured the law of refraction. Until
thrn trgtavtion wwas a tabulated function. Ptolemaeus in about 130 AD
first compiled tables of refraction. Ptolemaeus, lacking good
instruments and trigonometry, could not come up with a proper
analysis.
Snellius's law is
n1 * sin(a1) = n2 * sin(a2)
a1 is the angle of the light from medium 1 into medium 2. a2 is
the angle of the light in medium 2. Both are measured from the
orthogonal on the boundary of the media at the entry. point.
n1 and n2 are empiricly determined properties of medium 1 and
medium 2, the index of refraction. It expresses the 'strength' of a
medium for bending the path of light thru it. A larger index implies
that light is more steeply bent than a medium with a smaller index.
The index is a relative one, banking off of a base medium as
unity. For most earthly work, the base medium is air, with n - 1.000.
For optics nuilt for use in outer space, the vacuum is the medium as
1.000 and air is a bit more.
The refraction index varies with wavelength, which makes possible
prism spectrometers, but here we ignore this feature.
In the diagram here the boundary between the two media is the
vertical line '#'. The horizontal line is the perpendicular at the
entry point of the light ray. The ray comes from the left in medium 1,
index n1, at an angle of incidence a1. It enters medium 2, index n2.
It continues in medium 2 with angle of refraction a2,.
# /
(n1) # / )
# / a2 )
- - - - - -#- - - - - - - -
( a1 /#
( / # (n2)
/ #
Maxwell thru his EMW theory worked out that refraction across two
media is
sqrt(1 / (mu1*epsilon1))) * sin(a1) =
which is the same form as the empirical Snellius law. Matching terms
from Snellius and Maxwell, we see that
sqrt(1 / (mu1*epsilon1))) = n1
sqrt(1 / (mu2 * epsilon2)) = n2
From the wave equation
sqrt(1 / (mu1*epsilon1)) = speed1
sqrt(1 / (mu2*epsilon2)) = speed2
These are the velocities of light in each medium.
speed1 * sin(a1) = speed2 * sin(a2)
Because medium 1 is prevalently vacuum or air, speed1 is c itself.
c * sin(a1) = speed2 * sin(a2)
c / speed2 = sin(a2) / sin(a1)
speed2 / c = n1 / n2
= 1.000 / n2
-> 1 / n2
The indices of refraction are the ratios of the speed of light in
vacuum (or air) to that in medium2. n1 for air or vacuum is set to
unity, so the ratio is always 1/n2.
The sense is that in a medium other than a vacuum light travels
SLOWER than c.
Maxeell by this analysis showed that refraction index is a direct
feature of electromagnetic waves. And it applies to EMW of any
wavelength, not just those exciting vision.
Cherenkov effect
--------------
The diminished speed of light within media other than a vacuum
leads to a peculiar effect described by Cherenkov in about 1950. In
nuclear experiments particles are accelerated to very nearly the
lightspeed. When they are sent into a detection medium, such as liquid
helium, they emit light in a pattern resembling the wake of a boat in
water! The light LAGS the particle and is left behind! The nuclear
particle is moving faster than light! This is the Cherenkov effect.
The particle is moving faster than the REDUCED lightspeed within
the helium. The light radiates away at the actual speed of light for
that medium and slower than particle itself.
No Einstein rules are broken. It remains true that no matter can
excede the speed of light in vacuo, but it is also true that it can
excede the slower speed of light within the medium it passes thru.
Propogation of EMW
----------------
Altho Maxwell's theory explained optics and radiation with
electromagnetic eaves, these waves were only a maths model. Was light
the only electromagnetic wave? Can other kinds exist? Can we
deliberately make an electromagnetic wave?
Hertz in 1886 was the first to generate artificial
electromagnetic waves. He built an electric circuit that oscillated
electric and magnetic fields. he rigged up an other circuit to
receive them. This was the birth of the electromagnetic industry.
Today we produce RMW by electronic devices, like at broadcast
transmitters. They produce oscillating electric and magnetic fields
which are released into the air as EMW thru an antenna. In the sketch
here '#' is the transmission house, '=' is the antenna.
++ A -- <- B <- -- C ++ -> D ->
+===#-==- ===#=== -===#===+ ===#===
>>>>>>> ....... <<<<<<< xxxxxxx
>>>>> ........ <<<<<<<
>>>>>>> .......
>>>>>>>
Scene A has the base oscillator sending electrons to the right end
of the antenna and sucking them out of the left end. The positive
charge there comes from the atomic protons that no longer have
electrons to offset them. The separated charges produce an electric
field pointing right, '>'.
Scene B has the base 1/4 oscillation cycle later when it stops
pumping electrons. The electrons at right end of the antenna racing
left. '<-'. This current sets up a magnetic field pointing out, to
you, '.' suggesting an arrow point. It pushes away, sown in the
sketch, the electric field from the previous 1/4 cycle.
Scene C is at the 2/4, 1/2, cycle when the base pumps electrons
from the right end. This separated charge yields an electric field
pointing left , '<'. It pushes the previous electric and magnetic
fields away.
Scene D is at 3/4 cycle of the oscillator. Pumping stops and
electrons race right along the antenna, '->'. They generate a magnetic
field pointing away from you, 'x', alluding to arrow feathers. This
field shoves away the earlier electric and magnetic fields.
After scene D the transmitter circuit begins a second cycle back
at scene A. The fields travel away at lightspeed to constitute the
electromagnetic wave.
We now stand downrnge from the broadcast base and watch the wave
pass by. The base sends out in alternation an electric and then a
magnetic field in a sequence of repeating cycles. The fields are 1/4
1/4 cycle out of step. As one is cresting to its maximum strength the
other is passing thru its zero strength.
Stand alongside an EMW passing in front of us, we see something
like this, where the wave is traveling to the right along the x-axis
at speed c.
This is a snapshot of several wavelengths of the EMW showing the
direction of the two interacting fields. The base is off-page at the
left and wave travels by us left to right. Prior segments move to the
right farther downrange.
^^^ ... vvv xxx ^^^ ... vvv xxx ^^^ ... vvv xxx ^^^ ... vvv xxx
^^^ ... vvv xxx ^^^ ... vvv xxx ^^^ ... vvv xxx ^^^ ... vvv xxx
^^^ ... vvv xxx ^^^ ... vvv xxx ^^^ ... vvv xxx ^^^ ... vvv xxx
| | | |
|<-wavelength-->| |<-wavelength-->|
^ = positive pointing electric field (arrow up)
. = north pointing magnetic field (arrow point)
v = negative pointing electric field (arrow down)
x = south pointing magnetic field (arrow feathers)
The distance between corresponding points on successive cycles is
the wavelength of the EMW. This distance is NOT only that between
peaks or zero points of successive waves, as some books assert.
The number of wavelengths passing by us per second, traveling at
speed c, is the frequency of the wave. We ,au designate this wave by
wither its wavelength or frequency according as local practice.
For waves sent out by AM radio stations, the frequency is in
hundreds of kilohertz. In the FM band it's tens of megahertz
By standing within the wave and facing into it, we experience a
cyclical succession of electric and magnetic fields, shown here in
steps of 1/4 cycle.
^^^ <<< vvv >>> ^^^ <<< vvv >>> ^^^ <<< vvv >>>
^^^ <<< vvv >>> ^^^ <<< vvv >>> ^^^ <<< vvv >>>
^^^ <<< vvv >>> ^^^ <<< vvv >>> ^^^ <<< vvv >>>
0 1 2 3 4 5 6 7 8 9 10 11
|<----one cycle---->| |<----one cycle---->|
^ = positive pointing electric field (arrow up)
< = north pointing magnetic field (arrow left)
v = negative pointing electric field (arrow down)
> = south pointing magnetic field (arrow right)
0, 1, 2, ... = successive quarter cycle interval.
Energy of a Field
----------------
An electric field carries energy coming from the energy spent to
create the field.. Without going thru the derivation, we note that
given a field of strength E, the density of energy, joule/meter3, is
densityE = (epsilon0 * E^2) / 2
where E is the electric field strength
A parallel formula gives the energy density of a magnetic field
densityB = ((1(1 / mu0) * B^2) / 2
also in joule/meter3.
The electric field strength is measured in newton/coulomb, parallel
to the gravity field strength in newton/kilogram.
The magnetic field strength is in (newton.second)/(coulomb.meter)
or newton/(ampere.meter). Recall that coulomb/second = ampere. This
odd unit comes from the way we define magnetic field strength by way
of an electric current.
Energy in EMW
-----------
As an EMW passes over an observer he sees the electric and
magnetic fields in alternation. The energy densities of these fields
are added together to give the total energy density of the EMW.
The two fields alternate in cresting and zeroing. The energy
producing the fields is exchanged between them as each field rises and
falls in strength during a cycle. The sum of the two portions is
constant, equal to the input energy of the generator. We have
We have for an electromagnetic wave
densityEMW = densityE + densityB
= ((epsilon0 * E^2) / 2) + (((1 / (mu0) * B^2) / 2)
= (((epsilon0 * E^2) + ((1 / mu0 ) * B^2)) / 2
This does not mean the two individual densities are always the
same. Each varies as the amplitude of its field, 1/4 cycle offset from
te other field. The SUM of the two is constant over the entire cycle.
Also this equation deals with the energy content of the fields,
NOT strengths. The two fields have entirely different field strengths
and their sum over a cycle is not constant.
Root Mean Square
--------------
The energy equations are based on the instantaneous strengths of
the electric and magnetic fields. In an EMW the two fields vary in
strength from a maximum positive value to a maximum negative value in
a given cycle. It is normally not practical or important to measure
the fields at a given instant and work out the energy densities.
If the wave has any considerable frequency, the oscillation of
fields is so rapid that it looks like a steady field of equivalent
strength as the real one. For most motors, heating, lighting,
household applications don't need a detailed profile of the wave.
They can be built to accept an equivalent steady energy flow.
From long experience in engineering a fair and overall good
equivalent wave energy density is sqrt(2)*(maximum). In decimals this
is (0.707)*(maximum). For rough work you may use 0.7. This is the
'root mean square', or 'RMS', field energy treated as tho it were a
steady, not varying, field. We have
densityRMS = densityErms + densityBrms
= epsilon0 * Erms^2) / 2 + ((1 / mu0) * Brms^2) / 2
= ((epsilon0 * Erms^2) + ((1 / mu0) * Brms^2)) / 2
Just about all uses of EMW work with the RMS value of its energy
density, often without actually stipulating it as such. If you really
mean the peak value you have to deliberately say so.
By removing, for practical purposes, the varying values of
electric and magnetic energy density, replacing them with the level
RMS value, we have
densityRMS = densityErms + densityBrms
= (epsilon0 * Erms^2) / 2 + ((1 / mu0) * Brms^2) / 2
= ((epsilon 0 *Erms^2) + (((1 / mu0) * Brms^2)) / 2
= ((epsilon0 * Erms^2) + (epsilon0 * Erms^2)) / 2
=*2 * epsilon0 * Erms^2 / 2
= epsilon0 * Erms^2
We can simply add the two rms values and then equate them because
the RMS value is a levelizing of the field variation into a steady
uniform field.
Electric Field Overwhelms
-----------------------
Recall that the RNS electric and magnetic energy densities are
equal, we have
(epsilon0 * E^2) / 2 = ((1 / mu0) * B^2) / 2
epsilon0 * E^2 = (1 / mu0) * B^2
epsilon0 * E^2 / B^2= 1 / mu0
E^2 / B^2 = 1 / (epsilon0 * mu0)
E / B = sqrt(1 / (epsilon0 * mu0))
= c
This is an extraordinary result! The electric field strength is
some 300 million times greater than the magnetic field strength in an
electromagnetic wave.
Any authors scramble this statement really good. The energy
densities of the fields are the same in an EMW. The field strengths
themselfs in that EMW are vastly different.
What this means is that we routinely depict in textbooks an EMW as
a SINGLE field, the electric field, orthogonal to the direction of
propagation. In optics, for instance, a lightwave is a single
oscillating field, the electric field This one-field model does help
explain polarization easier than with the both fields.
To see how vastly larger the electric field strength is, make a
scale drawing of an EMW with the two intersecting fields. Let the peak
magnetic field strength be one millimeter. What is the height of the
electric field strength on the same scale? E?M = 300 million and M =
1mm, making E be, hold your hat, 300 KILOMETERS tall! You need a hell
of a large paper for the drawing.
Irradiation
---------
When an electromagnetic wave is intercepted by a target it
delivers its energy to the target. This energy can be wasted as heat
or utilized to produce work. 'Work' is a very general term meaning any
action useful to humans. One action crucial for astronomy is exciting
the eye to produce vision. The eye pupil is the target.
As the EMW flows over the target, a certain volume of it passes
thru the target per second. The energy within this volume is imparted
to the target as energy per unit time, or power. time.
The irradiation on the target is this energy/meter3 times the
speed of the wave, which is c the speed of light.
irradiation = densityEMW * c
Irradiation is in joule/meter2.second, or watt/meter2. The
joule/second has the special name 'watt', the unit of energy flow, or
power. The target has an area facing the EWMW in meter2. The total
power captured is the irradiation times this area,
power = irradiation * area
This power is converted by the target into work. In this way
electromagnetic waves allow us to accomplish useful work from a remote
station without wires.
Sun's irradiation
---------------
the Sun's irradiation is nearly enough 1,350 watt/meter2. This is
from satellite observations at the Earth's distance from the Sun. It
varies with Sun distance from Earth, sunspot cycle, solar eruptions,
secular changes. It is attenuated by the atmosphere to around 1,3200
W/m2 at the ground.
What is the strength of the Sun's electric field?
The energy density of the EMW is the irradiation divided by speed
of light
densityEMW = irradiation / c
= (1,350 W/m2) / (2.988e8 m/s)
= 4.518e-6 W.s/m3
= 4.518e-6 J/m3
The electric field strength is
densityEMW = epsilon0 * E^2
E^2 = densityEMW / epsilon0
= (4.518e-6 J/m3) / (8.854e-12 C2/N.m2)
= 5.103e5 J.N.m2/m3.C2
= 5.103e5 N.m.N.m2/m3.C2
= 5.103e5 N2.m3/m3.C^2
= 5.103e5 N2/C2
E = sqrt(5.103e5 N2/C^2)
= 7.144e2 N/C
-> 714 newton/coulomb
A more familiar statement is, with the definition of the 'volt' as
one joule/coulomb is
newton/coulomb = newton.meter/coulomb.meter
= joule/coulomb.meter
= volt/meter
The energy density of the electromagnetic radiation is in
joule/meter3. One joule m the unit of energy or work,is itself one
newton.meter. We have
joule/meter^3 = newton.meter/meter^3
= newton/meter^2
which is a force over a unit area, or pressure.
An incident EMW of a given energy density exerts on a target
surface a pressure equal in value to that energy density.
If we have the irradiation of the wave, in watt/meter
for the Sun at Earth distance away the radiation pressure is
pressureEMWW = densityEMW
= 4.685e-6 N/m2
+-----------------------------------+
| IRRADIATION PARAMETERS OF SUN |
| |
| irradiation = 1.350 W/m2 |
| |
| elec field strength = 7.144e2 N/C |
| |
| energy density = 4.518e-6 J/m3 |
| |
| radiation pressure = 4.5185e-6 N/m2 |
+------------------------------------+ -