ELECTRIC AND MAGNETIC FIELDS 2/2 -------------------------- John Pazmino NYSkies Astronomy Inc email@example.com 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
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]
[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.