ELECTRIC AND MAGNETIC FIELDS 2/2
--------------------------
John Pazmino
NYSkies Astronomy Inc
nyskies@nyskies.org
www.nyskies.org
2012 July 18 initial
2012 November 20 current
Introduction
----------
This article is the second of the two-part set on electric and
magnetic fields. Part one discussed electric fields and forces. We
now look at magnetic fields and forces.
I trust that you studied part one of this two-part set of
articles. That part has the background for vector maths, a necessity
for understanding this second part. If you missed part one, you better
read it first for the explanation of vector maths. It's at
www.nyskies.org/articles/pazmino/elecmag1.htm
Magnetism IS a difficult subject because of its many interactions
with electricity. I confine generally to two cases of interaction,
with a single electric charge and a current thru a long straight wire.
Never the less, you may still find the text as confusing as all Hell.
That's because the subject IS really difficult to appreciate without
prior tuition or experience. But with the ever-more important role of
magnetic fields in many astronomy phaenomena, home astronomer must
acquire an awareness of the main features of magnetism.
Magnetic poles
------------
In the early years of navigation by magnetic compass the 'north
pole' of te compass needle was the end that pointed toward the
geographic north pole of Earth. During explorations of the Arctic
region we found the actual place where compasses aimed at, the north
geomagnetic pole. it is in northern Canada several hundred kilometers
from the geographic pole. The south geomagnetic pole is near the
geographic south pole in Antarctica.
With magnetism in lab work it was soon realized that opposite
magnetic poles attract and like poles repel, much like for electric
charges. Michell discovered an inverse-square law for magnetic poles
based on this property.
Right away we got a conflict frozen by history. In order that the
north pole of a compass point to north, either it must be a magnetic
SOUTH pole or the pole inside the Earth is a magnetic SOUTH pole!
The usual -- hardly official! -- definition of a magnetic north
pole is a pole that is attracted to the Earth's north geomagnetic
pole. A magnetic south pole is repelled from that pole or is attracted
to Earth's south geomagnetic pole. This is the convention I use
thruout this article.
The other definition is that the magnetic north pole is that which
attracts the north end of a compass needle. The south pole attracts
the south end of the needle. Using a compass to test polarity is
routine for its simplicity but you MUST take care how you state the
test result!
The conflicting definition of north and south magnetic poles
forces firms dealing in magnetism to go thru a detailed explanation of
their services and products. These caution about specifying the
magnetic polarity. If you get the polarity backwards, because you did
not understand the convention of the company or wrongly identified the
polarity you need, you'll get a product that will work erroneously.
Monopoles
-------
In electric fields we can actually place charge on an object. If
the source is charged with positive electric, the field around the
source points outward because a positive test particle is repelled
from it. A negatively charged source has a field that points toward
it, attracting the positive test particle.
Some simple explanations of magnetism discuss magnetic charges
where a body somehow is loaded with north or south magnetism to make a
magnetic field. In history, Michell did find an inverse=square law for
magnetism that parallels that for gravity and light. (The one for
electric charges, Coulomb's law, wasn't yet discovered.)
As far as we understand, there are no real bodies with only one
magnetic polarity on it. Every magnet in nature has both opposite
poles, north and south. There are no magnetic monopoles. Michell's
law, as tempting as it is to work with, really has no practical value.
It is commonly missed from modern texts about magnetism.
The lack of monopoles makes it much hadrer to define magnetic
quantities parallel to electric charge. We ended up working with
magnetism thru the effects of electricity. Except for small magnets in
toys or handheld tools, we make a magnetic field by electricity.
Biot-Savart's law
---------------
Natural magnets are way to small and crude for any substantial
human needs. Magnetism today is prevalently produced artificially thru
electricity, banking off of moving charges.
I give here the end result of what can be a complex theory, a
simplified version of Biot-Savart's law. The word is French, routinely
butchered in English. A closer sound is 'bee-YOH sah-VARR'. Given a
charge with steady speed, its magnetic field is
(mag fld strength) = (mu0 / (4 * pi)) * charge * cross(speed, rr)
/ (radius ^ 2)
At first this looks like a radial central force like electric or
gravity fields. The magnetic field does surround the charge but it
ranges in value with the angle of the target point from the line of
motion. That's taken care of by the cross product.
'rr' is a unit vector in the direction of the radius out from the
charge and has unity size. That way it does not affect the calculation
yet gives the proper vector direction to the field.
This may seem like an arbitrary trick of maths. To some extent it
is. However, without using such a unity vector, you have to make up
hard-to-apply rules, in rhymes or jingos, for the sense and direction
of the magnetic field.
Permeability
----------
mu0 is a property of the medium in which the magnetic field lives
and is called the permeability. I here deal only with free space but
it is assessed and tabulated in references of materials properties.
Its value for free space is
mu0 = 4 * pi * 1e-7 newton.seocnd2/coulomb2
The peculiar feature of mu0 is that it is NOT an independently and
experimentally measured quantity. In the early years of electric and
magnetic work it was determined by experiment for assorted materials.
Maxwell showed that mu0 and epsilon0 are linked. The weird figure here
comes from the correlation of units in the SI scheme. It is an exact
figure, mathematicly equal to 4pi times 1e-7.
mu0 resembles yet is distinct from epsilon0. Epsilon0 is
epsilon0 = 8.8542e-12 coulomb2/newton.meter2
Magnetic field strength
---------------------
The units of magnetic field strength are, by doing the maths on
the units of all the factors in the Biot-Savart formula,
(mag fld strength) = [newton.second2/coulomb2]
* [coulomb,meter/second]
= [newton.second/coulomb.meter]
= [newton/coulomb] / [meter/second]
[tesla] = [newton/coulomb] / [meter/second]
The first block of units comes from mu0; the second, the other
factors. This combination of units for the magnetic field strength is
the 'tesla'.
The final set of units implies that in addition to force per unit
charge, there is some kind of speed embedded in the nature of magnetic
fields. An other feature is that the units apply to electric charge
and NOT to a magnetic pole. There is no practical description of
magnetic fields using magnetic poles.
You may know of an older unit of magnetic field strength, the
gauss. This is a CGS unit, still in wide use. One tesla is 10,000
gauss. The Earth's magnetic field is about 1/2 gauss or 5e-5 tesla. A
medical MRI generates a magnetic field of 4 to 6 tesla.
Gauss's law for magnetic poles
-----------------------------
If we surround a collection of magnets with a closed envelope,
like a sphere, and sum up all over its surface the magnetic field
strength, the summation is the net magnetic poles inside the envelope.
sum(dot(magnetic field strength, delArea)) = 0
That zero is really zero. The net magnetic poles within the envelope
is truly zero. The reason is that we must have equal numbers of paired
north and south poles in any magnetic object. There are no monopoles.
They all null out pair by pair, leaving NO left over magnetic poles
inside the envelope.
This does not mean, as some simple treatments explain, there is no
magnetic field around the envelope. It means that fo every bit of
field aiming out, there is some where else a bit of field aiming in.
Taking over the entire surface the summation of all bits is zero.
The choice and features of the Gauss envelope are the same as for
electric fields, as discussed in part one of the series.
Cosmology
-------
We now work with electric and magnetic fields thruout the
universe. They are vital for assorted celestial phaenomena and
effects, ever more necessary for the home astronomer to understand.
In cosmology we generally ignore electric and magnetic forces.
In the early era of cosmology we neglected these forces because we
plain didn't know they prevail in space. Yet long since we learned of
these fields, we still miss them out of most cosmological models.
The reason is found in Gauss's law. It is hard in nature to
separate any bulk quantity of electric charge, due to the intense
attraction of the two opposite polarities of charge. In a given large
volume of space, enclosed by a Gauss envelope, the net unbalanced
charge tends to zero. As the Gauss envelope covers more and more
volume of the universe, the interior net charge rapidly goes to zero.
Applying Gauss's law to magnetism we already saw that the result
is always zero. This comes from the pairing of magnetic poles in
nature. There is no separation of poles.
Trying Gauss's law on gravity yields an entirely different result.
There is only one kind of 'gravity charge'. mass.
sum(dot((grav fld strength), delArea) = progamma * mass
where progamma is an ingredient of the Newton, or gravitational,
constant. In the usual Newton's law we use gamma. From Gauss's law
with a uniform gravity field over a spherical surface we have
dot((grav fld strength), (4 * pi * dist ^ 2)) = progamma * mass
(grav fld strength) = progamma * mass / (4 * pi * dist ^ 2)
= gamma * mass / (dist ^ 2)
Altho this means that gamma = progamma/(4*pi) we, similar to the
constant in Coulomb's law, almost never use progamma. We always cite
just gamma with the 4*pi embedded in it.
There is no distinction of a gamma0 for free space and a gamma for
other media. There is no medium that modulates gravity. Regardless of
intervening material between a particle of mass and the Gauss surface,
it produces the same bit of gravity field strength.
As the Gauss volume enlarges, it encloses ever more mass with
nothing to offset or annul it. Since mass is the source of gravity,
the gravity field strength summed over the Gauss surface grows with no
other competing force from electric or magnetic sources. On the grand
scale of the universe gravity is the one force driving cosmological
processes.
Magnetic field lines
------------------
By the cross product in the Biot-Savart law the direction of the
magnetic field is perpendicular to both the motion of the charge at
all locations around the charge. The lines of force for this magnetic
field are circles looking like latitude parallels whose poles are on
the line of motion.
The field is a maximum at right angles to the motion and is zero
along that line. Because you usually want the strongest part of the
field, some simple works give a formula for only the field strength at
this right angle. The problem with this method is that the cross
process is skipped. It becomes a regular multiplication IF and ONLY IF
the angle between the speed and radius is 90 degrees.
With the field lines circular, they do not land on a physical
pole. For most magnetic fields in science and industry the field is
generated thru electricity, There really is no physical magnet.
There are also no mobile magnetic particles, like electric
charges. Magnetic fields act on electric charges but in ways not
comparable with those for hypothetical monopoles.
Because we lack isolated mobile magnetic poles, we work with
magnetism almost entirely thru its effects on electric charges. For
example there is no magnetic 'current' or 'circuit' like for
electricity. We can and do consider current and circuits of electric
charges impelled by magnetic fields.
A weird property of magnetic fields, recta mente from the absence
of monopoles, is that they always come in closed loops. There are no
dead ends and the lines can not break open. Even when the poles of a
magnet are well separated, they are connected together thru the body
of the magnet. The force lines continue thru one pole, thru thu
magnet, thru the other pole. They are continuous with the lines in the
workspace outside of the magnet.
Force on a charge
---------------
Besides generating a magnetic field while in motion, a moving
charge is affected by an external magnetic field. For a charge in
steady motion thru a stable uniform magnetic field the applied force
by the field on the charge is
(mag force) = charge * cross(speed, mag fld strength)
The force by RHR is perpendicular to the motion of the charge. The
magnetic field can not alter the charge's forward speed. The charge is
diverted into a curved path while continuing its linear motion.
The cross product shows that if the charge is at rest in the field
it suffers no force. If the charge is moving along a field line, it
also experiences no force. The initial linear motion is supplied by an
external agency, not the magnetic field.
Circular motion
-------------
Here to fore we never mentioned any property of a charge other
than its amount of coulombs. Charges are loaded onto physical bodies,
even if so small as an electron. The force from the magnetic field
acts on the mass of this body to make it describe a circular path on
top of the body's linear motion thru the field.
We can find the radius of this path by letting the magnetic force
balance the reaction force in the circular path.
(mag force) = (reaction force)
= mass * speed ^ 2 / (path radius)
(path radius) = mass * speed ^ 2 / (mag force)
= mass * speed ^ 2
/ charge * cross(speed, (mag fld strength))
= mass * speed / charge * (mag fld strength))
A more practical expression is used when the radius is known from
measurements like in the shatter spray of nuclear collisions. The
magnetic field strength and speed of the charge are also known or
measured. Then we have
(path radius) = mass * speed / (charge * (mag fld strength))
(path radius) * (mag fld strength) / speed = mass / charge )
Since there are only a couple dozen subatomic particles with charge,
finding the mass/charge value narrows the identity of the particle.
The charge, impelled by both its initial linear motion and the
imposed circular motion ends up spiraling around a magnetic field
line. Mind well that there is NO actual line, like a track or wire,
that the charge runs along. It is merely the streamline that happens
to be at the charge. Each point in the magnetic field has its own line
of force thru it.
Also mind well that the direction of the charge motion is NOT that
of the field line. The direction of the magnetic field strength is
defined for an imaginary north pole, and NOT a plus or minus electric
charge. A charge of either polarity may move in either direction along
a magnetic line of force. The only distinction of polarity is that the
direction of the circular motion is opposite.
What can happen is that the magnetic field is nonuniform along the
trajectory of the charge. The force varies in angle against the speed
vector. The charge tends to move in a spiral around a line of force,
altho the linear motion was already imparted to the charge.
Force on a current
----------------
When the moving charge is constrained within a wire the wire
suffers a force orthogonal to its length. I give here a pseudo
derivation from the force on a charge. The case here is the force for
a given length of a long straight wire in a uniform steady magnetic
field.
(mag force) = charge * cross(speed, mag field strength)
= charge * cross(length/time), mag field strength)
= length * cross(charge/time, mag field strength)
= length * cross(current, mag field strength)
For a unit length of wire we get the force per meter of length
(mag force/length) = cross(current, mag field strength)
The cross product makes the force orthogonal to the wire in the
direction given by the RHR.
Ampere's law
----------
Before the nature of electric current was learned to be a flow of
electrons, the generation of magnetic field by electricity in wires
was discovered. This is Ampere's law. Ampere's law is Biot-Savart's
law applied to a stream of charges in a steady flow in a long wire.
By recognizing that for a wire the only radius of use is the
perpendicular distance from the wire and that any point around the
wire receives the combined magnetic field of charges all along the
wire, we have
(mag fld strength) = (mu0 / (2 * pi)) * cross(current, rr)
/ radius
In this case 'rr' is always at right angle to the wire. The cross
product becomes a regular multiply operation. In many treatments the
cross product and 'rr' are left out with no explanation.
The unit for field strength is the tesla, as seen by working with
the units in the formula
(mag fld strength) = [newton.second2/coulomb2]
* [coulomb/second] / [meter]
= [newton.second/coulomb.meter]
[tesla] = [newton/coulomb] / [meter/second]
which is the same as for the pure Biot-Savart's law. Ampere's law is a
special case of that law, altho in history is was found first.
Being that the magnetic field is produced by an electric current
and that [coulomb/second] = [ampere], an other unit for the tesla is
[tesla] = [newton.second/coulomb.meter]
= [newton/meter] * [second/coulomb]
= [newton/(meter.ampere)]
The magnetic force lines are circles centered on the wire. The
direction of the force along the lines is given by the cross product
and right hand rule. Rotating the current vector, for the traditional
current!, into the radius vector moves the imaginary bolt in the
direction of the force.
As example, a current in a horizontal wire flows to the right. One
radius points up. The RHR on the radius directs the field out of the
paper. Applied on a radius pointing down (not shown) the lower lines
of torce point into the paper. The field lines loop out of top, to
downward in front of the wire, to in to the paper at the bottom. They
continue under the paper to form closed rings centered on the wire.
. . . . . --- . . .
. . . . . . |
. . . . | radius .
. . . . . | . . . .
==============wire===========
x x ---current--> x x
x x x x x x x x
x x x x x x x
A variation of the RHR is to pretend to grab the wire with the
right hand so that the thumb points in the direction of the
conventional current. The fingers curl around the wire in the
direction of the force lines.
Michael Bloomberg
---------------
Michael Bloomberg, mayor of New York, sits a radio talk show on
station WOR. On Friday 2 March 2012 he fielded several questions sent
in via Twitter. One asked about 'F*cking magnets, how do they work?'.
It was a joke about the song group Insane Clown Posse.
The mayor didn't catch the joke and proceded to give a simple
explanation of electrons orbiting in atoms with paired opposing
magnetic poles. Magnetic materials don't have fully paired electrons.
The net extra poles line up to create a magnetic field!
This description is more or less correct, being that Bloomberg was
an electrical engineer in his early career. Show host John Gambling
whispered to him the real meaning of the question and the two broke
into chuckles.
Faraday's law
-----------
We looked at the force acting on moving charges within a magnetic
field. We played with a single charge impelled into the field and a
stream of charges in a current thru a long wire.
We now have a segment of wire, short enough to fit entirely within
the magnetic field. This wire has no current but it has mobile
charges. At rest in the field the wire feels no magnetic force.
To get moving charges we bodily displace the wire with a steady
speed. The electrons, now moving, are constrained to the sire and can
move only along it. The magnetic force pushes electrons toward one end
of the wire where they must stop. This end is negatively charged. The
other end, where electrons are withdrawed, is positive.
The diagram here show the geometry of the wire and field. I leave
out the 'x' or '.' to reduce clutter but the field is perpendicular to
the paper. The wire segment '===' sits in the field and moves with
steady speed to the lower right. I deliberately do not make the motion
at right angles to the wire.
positive charge negative charge
+ + wire segment L - -
+ ============================== -
+ #+ \ - #
# \ #
sweeped area--# delD/delT--\ #
cross(L, delD/delT) # \ #
# # # # # # # # # # # # # # # #
The '#' outline a slice of area sweeped out by the wire in an
increment of time delT due to its speed delD/delT. The area itself is
the cross product of the wire length and the speed. delArea =
cross(length, delD/delT). As the wire continues its motion the area
keeps changing, in this case getting ever bigger,
The migration of electrons continues until their own electric
field force balances the outside magnetic force. There is then a final
separation of charges in the wire.
The electric field in the wire over the length of the wire is an
electric potential, more commonly called a voltage. If the wire is
connected to a load, by leads extending beyond the magnetic field, the
device is energized as if from a battery. The ends of the wire are the
poles of the battery with plus and minus polarity. As long as the wire
keeps moving thru the field, assumed of indefinitely large extent, the
potential stands in the wire.
When the wire stops or runs out of the magnetic field, the
potential vanishes and we have an inert piece of wire.
Now we must be attentive to the maths. The force on the charges is
taken from the force formula
(mag force) = charge * -dot(speed, mag fld strength)
The speed is that of the wire carrying its embedded charges with it
(mag force) = charge * -dot(delD/delT, mag fld strength)
We waited until the electric and magnetic forces balance to achieve a
stable charge separation in the wire
(electric force) = (magnetic force_
(elec force) = charge * -dot(delD/delT, mag fld strength)
We now seek the force per unit charge
(elec force) / charge = -dot(delD/delT, mag fld strength)
And this acts over the length of the wire segment
(elec force) * length / charge
= length * -dot(delD/delT, mag fld strength)
Recall that the cross product is geometricly the area of a
parallelogram formed by two vectors.
(elec force) * length / charge
= -dot(mag fld strength, delArea/delT)
The left side is the energy per unit charge, or potential
(elec potential= -dot(mag fld strength, delArea/delT)
And this is one version of Faraday's law. It describes that a moving
wire in a magnetic field generates along its length a voltage. This
effect is magnetic induction.
I must stress that this version is only one of many ways to
expLain the concept of magnetic induction to produce electric
potential. This phaenomenon is the basis of our entire electromagnetic
industry.
A more general statement of Faraday's law allows that the magnetic
field itself may change with time, also to produce a potential. An
other series of Faraday's law applications has both field change and
area change.
(elec potential = -del(dot(mag fld strength, area) / delT
I used the straight moving segment method because it follows recta
mente from the magnetic force equation.
Lenz's law
--------
The minus signum in Faraday's law is significant. When the
magnetic field induces a potential in the wire,the resulting current
itself creates, by Ampere's law, a magnetic field. How do the two
fields interact? Will they combine to make a stronger field and
increase the potential by positive feedback?
The potential is set up by the external field in such a way that
its own current's field will oppose, impede, resist the change that
produces it. The current's field tries to stop the change of the prime
field or sweeped area to turn off the Faraday action.
This is Lenz's law. An actual application is specific to the
geometry of the fields and wires but the minus signum calls attention
to the contrary magnetic fields.
In order to maintain the voltage under Faraday's law an external
energy must continuously be supplied to the mechanism that changes the
area or field. This is how such outside energy, typicly mechanical, is
converted into electricity.
Dynamos
-----
Faraday's law enabled the invention of the dynamo, a device that
converts mechanical energy into electricity. It is an arrangement of
wires within a magnetic field. The usual method is to let the
magnetic field be stable and just alter the area it intercepts in the
coils. This is easiest done by a rotary motion where the coil is
faceon and edgeon in turn to the field. The rotation is available by a
variety of means, such as a water wheel or steam engine. This change
over time in the presented area makes the voltage at the ends of the
coil, where the associated current is taken off.
As the need for electricity grew, so did the size and complexity
of dynamo. With mechanical power so abundant, the dynamos could be
made as large as necessary to produce electricity for large factories
and towns.
Dynamos were in use by the 1860s for in-house electric generation,
like factories and ships. Edison's fame comes from making electricity
available to any customer as a commericial service. A premises wanting
electricity paid Edison to have wires brought to it from the street
mains, analogous to telegraph and telephone service. The original
Edison dynamo station on John Street, Manhattan, was torn down in the
mid 1980s. A plaque honors the location on the successor building.
Transformers
----------
We now can understand the operation of an electric transformer,
like the ones in power adaptors and on utility poles. A transformer
converts incoming current of one potential, or voltage, into an
outgoing current of an other potential.
It consists of a steel or iron armature, solid or laminated. The
armature may be a closed ring or square for compact volume. Such
construction is why the device is so heavy for its size.
Around this armature are winded two wire coils, one for the
incoming current and the other for the outgoing current. The diagram
here schematicly shows a transformer, with the primary winding on the
top and secondary on the bottom..They are electricly insulated to keep
the electron flow separated in their own coils. In a real transformer
the coils may be intertwined or overlapping. The diagram has them
separate for clarity.
\ \|/ /---magnetic field at end of armature
\|/
#--armature
======#>
#>
incoming <#====== current <#>
<#> outgoing
<#> current
<#======
#>
======#>
/|\
/ /|\ \---magnetic field at end of armature
The incoming, also called the primary or ingredient, winding is
energized with electricity. For the moment it comes from a battery or
other direct current source. The primary coil by Ampere's law creates
a magnetic field that surrounds it and the abutting secondary or
egredient coil. The iron core, a ferromagnetic material, constrains
and concentrates the field to the vicinity of the windings.
However, altho there is a magnetic field produced by the steady
current entering the transformer, there is no current created in the
secondary coil. A steady magnetic field around a wire does not impel
the electrons in the coil to start a current flow. The transformer
merely gets hot from the current in the primary winding.
A varying magnetic field or presented area of the secondary coil
is needed. Since in most transformers the coils are tightly secured in
the armature there is no movement of them. The magnetic field must be
made to change continuously over time. to get the Faraday action.
The easiest way is to supply the ingredient current as alternating
current, one that varies in voltage cyclicly. In the United States the
frequency of this variation is 60 cycles per second.
The magnetic field made by the alternating current is changing in
strength in step with the current. This time-varying field can now
induce a voltage in the secondary winding and generates the egredient
current in it. The outgoing current is also alternating current, with
the same frequency, 60 cycles/second, as the incoming current. The
reversal of current comes from the alternating reversal of the
magnetic field producing it.
In the ideal transformer there is no loss of energy within the
coils, which almost can not be fully achieved. Power adaptors and
pole-top transformers get hot from some of the incoming energy being
radiated away as heat.
The power, watt, coming in equals that leaving. With [watt] =
[volt]*[ampere], we have
[watt]in = [watt]out
[volt]in * [ampere]in = [volt]out * [ampere]out
By the internal constrctuion of the transformer the windings are
arranged to shift the ratio of amperes to volts on each side, always
keeping the product of the two the same. A 'step-up' transformer is
built to make the egredient electricity of a higher voltage than the
incoming electricity. A 'step-down' unit outputs a current of lower
voltage than the incoming one.
I better add that for almost all small gadgets running off of an
adaptor the current needed is direct current like from a battery. In
fact, many devices run from a real battery as well as from an adaptor.
Since the output of the transformer is alternating current, a
further processing is done inside the adaptor, by an electronic
circuit not shown here, to rectify the electricity to direct current.
Transformers on utility pols don't rectify their output because
the current sent to your premises is alternating current. Any direct
current needed there must be provided by your own rectifier unit.
A transformer must never be connected to a DC input. The primary
coil will not induce current in the secondary because it makes a
steady magnetic field over that coil. There is no Faraday's law in
action. The primary will overheat and possibly burn out the unit.
A transformer should, properly connected to an AC input, not run
with no output load attached. While there is a voltage across the
secondary coil, with no current taken from it the transformer will
overheat and possibly burn out. This is a common cause of failure in
gadget adaptors. The gadget is turned off or disconnected but the
adaptor is still taking input current from the wall socket.
Some shuffling
------------
We developed Ampere's and Faraday's laws from simple geometry to
demonstrate their principles. The laws are very general, applying to a
very wide range of situations. Here we shuffle the formulae a bit to
better resemble the form you see in other texts about electricity and
magnetism.
In our version of Ampere's law the magnetic lines of force were
circles centered on the wire and the magnetic field strength was the
same around the line. The field decreased only radially from the wire.
The line of force in an arbitrary arrangement of currents may not
be circular or have a simple formula to describe them. The field
strength may vary along the field line. We handle these cases by
loosening the way Ampere's law is stated. Start with our version
(mag fld strength) = (mu0 / (2 * pi)) * cross(current, rr)
/ radius
= 1 / (2 * pi * radius)) * mu0 * cross(current, rr)
(mag fld strength) * (2 * pi * radius/rr) = mu0 * current
The 2*pi*radius/rr is the length of the circular field line but if
that line is not circular or has no other formula for its length, we
replace the 2*pi*radius/rr with 'length'.
dot((mag fld strength), length/rr) = mu0 * current
This length really doesn't have to along a field line but along any
closed path enclosing the wire. If we ignore the lines of force the
magnetic field along the path may vary from place to place. We then
have to sum up the field bit-by-bit over the path. delL is the piece
of path we sum over as we walk around the path. It insumes into it the
rr unit vector to keep its vector quality.
sun(dot((mag fld strength), delL)) = mu0 * current
This is more like the way you find Ampere's law stated in technical
works. It is still a simplified form yet far more general than the one
we started with.
Faraday's law must also be enlarged to handle arbitrary situations.
Start with our formula
(elec potential) = -del(dot(mag fld strength, area) / delT
We note that the potential is the electric field strength times the
length over which it acts. In our case this was the straight piece of
wire but it doesn't have to be. The wire may be of irregular
curvature, even knoted. Since we already have 'length' we allow it to
be the length of what ever shape of wire we have.
dot((elec fld strength), length) = -del(dot(mag fld strength,
area) / delT
We also allow that the field along this wire can vary from place
to place, calling for us to do the summation by increments of delL
along the path.We then have a generalized Faraday's law
sum(dot((elec fld strength), delL)) = -del(dot(mag fld strength,
area) / delT
This is closer to what other works present for Faraday's law.
Maxwell's equations
-----------------
We're now in place to collect certain of the equations we looked
at into a unifying set for both electricity and magnetism. We looked
at these equations as separate features of the two topics.
We take first the two versions of Gauss's law
sum(dot(elec fld strength, delA)) = charge / epsilon0
sun(dot(mag fld strength, delA)) = zero
We next assemble Ampere's law and Faraday's law. You will find various
statements of these laws among authors. I use the ones we played with.
sum(dot(mag fld strength, delL) = mu0 * current
sum(dot(elec fld strength, deL))
= -del(dot(mag fld strength, area)) / delT
These four equations, with appropriate manipulation, fully
describe every thing about electricity and magnetism and are as a set
the Maxwell equations. Maxwell massaged these four to find that
electricity and magnetism are really two aspects of a single entity
elctromagnetism.
They contain the properties of a medium to support fields, the
epsilon0 and mu0. These were until Maxwell merely experimentally
measured properties with no relation to each other.
They express the absence of isolated magnetic poles and the
preservation of electric charges. This means that in a given workspace
charges can not be created or destroyed, only netted out.
They show how to make magnetism from electricity and electricity
from magnetism. And that to get electricity we need an outside source
of energy to drive the Faraday action. Electricity is not free for the
taking from nature.
Speed of action
-------------
Of significant importance is the fact that the influence or action
of a electric or magnetic field is not instant. A remote point feels
the field after a delay as the field's influence or action travels
from the source to the target point.
Until Maxwell, light was the only electromagnetic field, altho not
at all recognized as such. Its speed was measured to be, in modern
value, about 300,000 kilometer/second. Maxwell showed that all the
properties and behavior of light are explained thru electromagnetism.
Light is merely one kind of electromagnetic field.
In particular, the speed of action of electromagnetism is also the
speed of light, The usual way to show this is with electromagnetic
wave functions. This is not necessary for us here.
First let all quantities in the Maxwell equations be uniform and
constant to remove the need for the summation and incremnet functions.
This simplifies the maths substantially without losing physics rigor.
dot(mag fld strength, length) = mu0 * current
dot(elec fld strength, length) = -dot(mag fld strength, area)
/ time
dot(elec fld strength, area) = charge / epsilon0
Divide Faraday's law by Ampere's law.
dot(elec fld strength, length) / dot(mag fld strength, length)
= -dot(mag fld strength, area) / mu0 * current * time
elec fld strength / mag fld strength
= -dot(mag fld strength, area) / mu0 * current * time
elec fld strength / (mag fld strength) ^ 2 = area / mu0 * current
= -area / mu0 * (charge / time) * time
Remove 'area' thru Gauss's law for electric field. The minus
signum vanishes because it was associated with the change of area in
Faraday's law.
area = charge / (epsilon0 * elec fld strength)
elec fld strength / (mag fld strength) ^ 2 = area / mu0 * current
= (charge /(epsilon0 * elec fld strength))
/ mu0 * (charge / time) * time
(elec fld strength) ^ 2 / (mag fld strength) ^ 2
= (charge
/ epsilon0) / mu0 * (charge / time) * time
= (1 / epsilon0) / mu0
= 1 / (epsilon0 * mu0)
(elec fld strength) ^ 2 / (mag fld strength) ^ 2
= 1 / (epsilon0 * mu0)
In this last equation we removed every thing related to the human
aspects of electromagnetism involved with producing the fields or
laboratory experiments. What's left are properties of the medium and
the two field strengths. These prevail in nature with no need for
human presence,
We now look at the units of measure for the field strengths.
(elec fld strength) ^ 2 / (mag fld strength) ^ 2
= 1 / (epsilon0 * mu0)
[newton/coulomb] ^ 2 / [newton.second/coulomb.meter] ^ 2
= 1 / (epsilon0 * mu0)
1 / [second/meter] ^ 2 = 1 / (epsilon0 * mu0)
[meter/second] ^ 2 = 1 / (epsilon0 * mu0)
[meter/second] = 1 / sqrt(epsilon0 * mu0)
This is a very deep result! The units of speed on the left belong
to the electric and magnetic fields, NOT to any mechanism that
produced the fields. They are the speed of action of the fields,
The value of this speed of action is a function only of the
properties of the medium sustaining the fields, mu0 & epsilon0 for
vacuum or mu & epsilon for other media.
Maxwell put in values for epsilon and mu for various materials as
taken from references in physics and chemistry. He got huge values for
the speed of action. When he plugged in measured values for a vacuum
he got a speed equal to that of light, then a purely measured value.
This suggested that light was a form of electromagnetism. With
modern values we have
mu0 = 4 * pi * 1e-7 newton.seocnd2/coulomb2
epsilon0 = 8.8542e-12 coulomb2/newton.meter2
(speed of action) = 1 / sqrt(epsilon0 * mu0)
= 1 /sqrt ((8.8542e-12 coulomb2/newton.meter2)
* (4 * pi * 1e-7 newton.seocnd2/coulomb2))
= 1 / sqrt(1.113e-17 meter2/second2)
= sqrt(8.9875e16 meter2/second2)
(speed of action) = (2.9979e8 meter/second)
This is the speed of light in vacuo.
Optical applications
------------------
The huge speeds obtained by plugging in values for epsilon and mu
for other materials leaded to an amazing find. For transparent
material there was on record the refractive index, a number used in
optical design. The number was always greater than unity but for
unknown reasons.
With the idea that the epsilon-mu figure for these materials was
the speed of light in these media we learned that the ratio of the
vacuum lightspeed over the speed in the medium equals the refractive
index! The behavior of optical material was a direct result of light
being a form of electromagnetism.
We see that epsilon0 and mu0 are not independent properties of a
medium, but are tied to the speed of action in the medium. In fact,
when we redimensioned the unit of length, the meter, by declaring a
fixed speed for light, we removed both epsilon0 and mu0 from the realm
of measurement. Their values are now established with no further need
to measure them again.
Epsilon0 and mu0 are physical properties of the medium, with the
same value for all observers regardless of relative motion. Because of
this, we see that the constancy of lightspeed is NOT a premise or
hypothesis in Einstein physics. The speed of light, by the arithmetic
combination of epsilon0 and mu0, is itself an physical property of the
medium. It is naturally the same for all observers. It's part of the
principle that the behavior of nature is the same for all observers,
which is the real primitive foundation of Einstein's work..
Comparison of fields
------------------
It is well to look at the three major fields of force in astronomy
in a side-by-side comparison. The table shows several properties for
each type of field
----------------------------------------------
Property | Gravity | Electric | Magnetic
------------+-----------+-----------+----------
charge | mass | electric | not used
origin | natural | nat, artf | nat, elec
polarity | only one | + and - | N and S
current | not used | electrons | none
monopole | one | two | none
force law | Newton | Coulomb | Michell
Gauss law | tot mass | net +/- | zero
force seat | only mass | only elec | elec and magn
potl energy | negative | pos/neg | not used
force range | 1/r2 | 1/r2 | 1/r2
modulated | none | by matter | magnetic media
medium parm | none | epsilon | mu
induction | none | make magn | nake elec
field line | endpoint | endpoint | closed loop
-------------------------------------------------
See that electric and magnetic forces and fields are intermingled.
Often a one produces or accompanies the other. Gravity so far is by
itself, in spite of massive efforts to bring it under the same roof
physical structure as electricity and magnetism. String theory tries to
do this but as at 2012 has not succeded.
As at mid 2012 the elementary carrier of gravity, the Higgs boson,
is not in hand. Large Hadron Collider found suggestions and hints of
its existence but not for sure. Some theories assign properties to the
Higgs boson that help incorporate gravity into electricity and
magnetism as part of the Grand Unified Theory of uniting all the
forces into sectors of a single entity.
Conclusion
--------
These two articles as a set offer a grounding in the basic
principles of electric and magnetic fields. Altho I try to use simple
examples I did also try to make the treatment realistic and complete.
I left out electronics, electric circuits, motors, power systems,
radio, heliophysics, cosmic rain, neutron stars and
other fascinating themes that derive from the principles in this set.
Probably the hardest sections are in part 1 on vectors. While the
behavior of vectors is important in electric and magnetic subjects you
will need to know about vectors for many other areas of home astronomy.
They, for instance, make orbital mechanics a lot easier to work with.