John Pazmino 
 NYSkies Astronomy Inc
 2012 July 18 initial
 2012 November 20 current 
    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
    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. 
    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. 
    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. 
    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 
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 
    (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   . 
              .  .  .  .  .  |  .  .  .  . 
              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 
    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. 
    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. 
    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 
         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 
    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 
    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.
    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.