ELECTRIC AND MAGNETIC FIELDS 1/2 -------------------------- John Pazmino NYSkies Astronomy Inc email@example.com www.nyskies.org 2012 July 18 initial 2012 November 20 current s
Introduction ---------- On and off thru the years home astronomers struggled to understand what are electric fields and magnetic fields. Ever since we learn how crucial are these fields in interplanetary and interstellar space it was important for home astronomers to know about them as part of their astronomy upbringing. But because few home astronomers were tutored in formal sciences beyond the high school level, they had no good foundation in electric and magnetic fields. Even those who took physics in college soon lost fluency in them. The result is that these crucial aspects of astronomy are tough to interpret and explain to newcomers into the profession and to the public. Explanation often amounts to parroting a news article or simplified description from elementary textbooks. In July 2012 The NYSkies Astronomy Seminar held a two-part sequence on solar physics and its geomagnetic affects. Part one was on July 6th and part two comes on July 20th. After part one many attendees realized that while the dialog at the Seminar was comprehendible they felt inadequately grounded in electric and magnetic fields. They could recite what they learned at the meeting but really didn't absorb the basic physics. This article is the first of a pair to offer a logical explanation of electric fields, charges, forces. Part 2 dealing with magnetic features, is at
Home experiments -------------- Probably all of us played with magnets in science kits and devices. We stepped thru the instructions and repeated the experiments. After all was finished, we felt like we were merely playing with toys.We had little advancement in our understanding of magnetism. Welcome to the club. After picking up thumbtacks for the hundredth time with a toy magnet we gave up. Magnets were so dull. The greatest minds of science in the 19th century went thru the same experience. Yes, we exploited magnetism in the emerging new industries of telegraphy, alarms, locks, signals. But we really didn't know what is magnetism. Off hand it seemed that we should be able to manipulate magnetism like electricity and its rules and operations should be parallel to those of electricity. They weren't. On the other hand, as simple as electricity was for home astronomers engaging science kits, it was disgustingly hard to work with electrostatics. On some days, mostly humid ones, nothing worked The gadgets refused to take on or hold their electric charge and the expected results of experiments didn't show up. On other days, usually cool dry ones, every thing worked perfectly. But there was no assurance that on the next ay we would enjoy similar success. Yet for circuit electricity, like simple radios and telegraphs and door buzzers, we gained confidence thru the experiments. Batteries run down but were easily and cheaply replaced. We obtained good appreciation for electricity. What's more we could actually apply our new knowledge to practical functions. We repaired a broken door buzzer and rigged up a room-to- room intercom unit and heard radio broadcasts from out own circuits, built a headlight for our bikes, wire power for our toy trains.
System of units ------------- Today the standard set of weights and measures is the International System, SI by the French initials. This bases all measurements on the three fundamental units of meter, kilogram, and second. For electromagnetism and other radiations, four supplemental units are defined: ampere, kelvin, mole, and candela. The SI took effect in 1960 to overlay and partly displace the older flavor of metrics. This was the CGS system founded on the centimeter, gram, and second. It, too, had supplemental units for radiation and other disciplines of science. Like in SI many unions of CGS units had special names, like the gauss for magnetic field strength. Scientists like to honor their heros by naming derived or compound units after them. In the CGS and SI systems we find many units named for famous scientists like the newton, tesla, ampere, pascal, joule. The method to apply a special name to a combination of units has no apparent rhyme or order. Some trivial combinations have special names while some complex ones do not. A sad consequence of shifting from CGS to SI is that many worthy persons were already honored in the CGS system. They can not easily be honored again in the SI system without causing gross confusion. Perhaps we'll revisit this situation in an other hundred years when the CGS system moved well behind us in history? Altho CGS is no longer the standard metric system, there is a deep legacy litterature written in CGS and many older scientists still work in this system. Most SI units are tenfold multiples of the older CGS units, making conversion easy. The SI unit of magnetic field strength is the tesla, being 10,000 gauss. The newton of force in SI is 100.000 dyne from CGS. Home astronomers must be versed in both systems, at lest to the extent of looking up unfamiliar weights and measures and converting between the two sets.
Maths on units ------------ A very powerful lifestyle skill that seems wholly lacking in the general education is the maths on units of measure. You know how to do math on numbers but you can also do maths on units of measure, like meters and grams. In fact, by applying this faculty in every day life you can block nast -- costly and dangerous -- mistakes. You can add and subtract only quantities with the SAME units, the 'apples and oranges' rubric. By seeing the units explicitly attached to two quantites you immediately catch blunders of adding and subtracting disparate numbers. It is a good habit to purposely write the units with every number in your math.SO very much of orthodox lifestyle math skips this vital step and play only with the numbers. It then instruct with hard-to- remember rules, rhymes, jingos to make the units work out correctly. You may multiply, divide, and do other operations on any units and this is crucial to understand in any technical subject. One common blunder is to do a times in the stead of a divide for a conversion factor. To change oldstyle gallons to liters you multiply -- no, divide? -- the liters by 3.8. With out the units attached to the '3.8' it's hard to remember which to do. Writing on purpose 1 gallon = 3.8 liters forces you to do the proper math. Watch.
1 gallon = 3.8 liters
1 = 3.8 liter/gallon
Now we need the liter equivalent of 21 gallons
21 gallon = (21 gallon) * (3.8 liter/gallon) = (79.8) * (gallon * liter / gallon) = 79.8 liter
The gallon and liter/gallon are taken into the maths, with 'gallon' cancelling on top and bottom of the fraction. What's left is liter. If we goofed and did the divide
21 gallon = (21 gallon) / (3.8 liter/gallon) = (5.53) * (gallon * gallon / liter) = (5.53 gallon2/liter)
which we immediately catch as a nonsense answer. With only the number 3.8 and ordinary ignorance you may never notice the mistake and your car runs out of gasoline after an hour's drive. In all the work in this, and the second, article I show the maths on the units, even if it adds extra steps in the calculation.
Signa and symbols --------------- One of the most frustrating features of home study of electricity and magnetism is the chaotic set of symbols and signa. It is tough to compare formulae between sources because characters used by one author do not match those of the other. There is no generally agreed set of symbols in spite of the oversight for the definitions and names of the units of measure. While the unit of electric potential is the volt with symbol V, the quantity itself has no standard symbol for it in formulae. In these two articles I use no symbols. I write out, in full or abbreve, the very name of the quantity. This strategy does away with the burden of figuring out what a given character in a formula stands for.Ido distinguish the name of the unit by flanking it between bumpers. An other major headache is the use of the minus signum in formulae. This is a function of the way the parameters in the formula are treated but commonly the author doesn't make this convention clear. Conflicts hit you when you mesh formulae across authors. Most likely each author is correct within his own text, as long as he logicly and mathematicly stays on track.
Electric current -------------- It seems evident that we should define electricity from a unit of electric charge, the electron. History went an other way with electrical experiments underway a century and more before the electron was discovered. Actually the use of 'current' had to wait for the perfection of the battery. Until then, with machines producing static electricity, the flow of electric was impulsive, uncontrolled, erratic. The slow steady action of the chemicals in a battery made a smooth stable flux of electric. It was presumed that electricity was a flow of a fluid made of positive charges. This fluid issued from the plus pole of a battery, went thru the electrical device, returned to the battery by its minus pole. The chemicals in the battery somehow refreshed the current, with chemistry still in an immature state, and sent it back out thru the plus pole for a new lap of the circuit. This soon became so embedded in society that today we still say that the electric current leaves the battery's plus pole and returns to it thru the minus pole. The positive pole is the 'high' or 'hot' side of the circuit. Eventually we learned that am electric current is the movement of electrons, which carry a minus charge. There is no mobile particle with positive charge. The proton,with a plus charge, is locked in place within the nucleus of atoms and can not move under conditions safe to have around you. When we work with the movement of electrons, as we do in microelectronics, we must use the electron current. Electrons flow from the negative pole to the positive pole, exacta mente opposite from the conventional current. Context within the task to hand tells which current is intended, but you do have to know which it is.
Electric current -------------- We define the unit of electric current, the ampere, with two thin straight parallel wires spaced one meter apart. Each has equal electric current in them.
1 ampere--> ----------------------- --- \ | \ | / 2e-7 newton | 1 meter / | ----------------------- --- 1 ampere-->
The currents induce a magnetic field around the wires that applies a force to the wires. The force either pulls or pushes the wires, according as the currents run in the same or opposite directions. The wires are of indefinite length and the force is measured over a one meter reach of them. When the current is adjusted to make the force 2e-7 (1/20,000,000) newton per meter reach of the parallel wires, the current in each wire is declared to be one ampere. A current of so many amperes is declared to be the flow of that same number of coulombs of charge per second. Inversely, since the prime unit is the ampere, the coulomb is one ampere.second
1 ampere = 1 coulomb/second
1 coulomb = 1 ampere.second
Note well that unlike for gravity the FLOW of charge is the prime metric unit and the AMOUNT of charge is a derived quantity. In gravity the fundamental unit is the kilogram, the AMOUNT of mass. There is no 'gravity current' associated with a flow of mass like a river or wind. We simply speak of so many kg/s of mass flux. An other point to notice is that we defined electric current, a fundamental quantity in metrics, thru the medium of a magnetic field. This field and magnetism in general, is not explicitly described in the definition of electric current. We see here an example of a seemingly trivial special name. There's clumsiness to say 'ampere.second' but for some reason this combo of units got the special name 'coulomb'. Alternatively, the flow in the wires could have been declared as one 'coulomb/second'. There would be no need to name it 'ampere'.
Electric charge ------------- To accumulate charge on a body we can attach the body to an electric current of known number of amperes and let it run for so many seconds. The piled up charge is the number of amperes times the number of seconds it was engaged. In the very early days of electric studies there were attempts to define a base amount of charge as the unit. It proved impossible to agree on how to arrange the charging of a body and the size of the unit of charge. There arose many sets of electric units, none easily convertible into the other, In time we determined the equivalence between the coulomb. As far as we know the electron has the smallest discrete amount of charge and, perhaps in an other history, could have been used as the fundamental unit of charge.
1 coulomb = 6.2415e+18 electrons
i electron = 1.6022e-19 coulomb
Charge deployment --------------- When charge is loaded onto a conducting material, usually metal, the mobile electrons repel each other in the material. They try to recede as far as possible from each other and not deploy uniformly thru the volume of the conductor. As the electrons push apart they migrate to the farthest places on the conductor, which is generally the exterior surface. The total charge resides only on the outer surface of a conductor while the interior remains neutral. One example is a closed box, shape not important, made of sheet or mesh metal. When this box is loaded with charge, the electrons stay on the outside of the box and there is no charge any where inside the box. This is one way to protect vulnerable electric equipment from possible harm due to outside electric effects. A prime frustrating fact of home experiments with electric charges and static electricity is that the charge on an object often does not 'stick'. The experiment is a dud or gives only weak results. This happens if there is a conducting medium to drain off the charge, leaving the object neutral. The grandest enemy of home electric experiments is moist air. Rain and humidity are good wicks to soak off the charge. In New York humid moist weather is prevalent in summer. When the air is cool and dry, mainly in autumn and winter, your experiments work correctly.
David Blaine ----------- A couple days before the October 5th NYSkies Seminar the Liberty Science Center announced an electrical demonstration by magician David Blaine. From October 5th thru the 8th he would stand on a 6-m tall insulated pole while getting zapped by one million volts of lightning bolts. This event was a bonus for the Seminar as a vivid illustration of electric fields. The show was on pier 54, foot of 13th St at Hudson River. A 4- story tall cubical shed was set on the pier to house Blaine and his electrical apparatus. It was covered in what looked like black cloth with one side open to face spectators on the pier. Blaine was dressed in a mesh metal suit with grounding cable. He could open the face cover to let visitors see him but had to careful never to touch his face during this peek-a-boo. Music played and once in a while a voice-over comment was issued. i went to see him on the 8th a little before 2 PM. The ringmaster noted that Blaine already exceded his own record of 63 hours of for isolation. At 2 PM he will pass 66 hours.A countdown and cheering marked this instant. Alas, due to rainy or humid weather, the electrical show was dampened. The lightning bolts shot out only intermittently at unpredictable moments. I could not capture any in photographs because they were just too brief to get my camera on them. Other astronomers reported similar results for their visits. On my visit the air was cool but filled with moisture with a thin fog. A couple hundred spectators were on hand. You got a wrist band at the entry gate but this was never checked on the pier and apparently you could come and go with no fuss. To guard against excessive noise form the lightning generators, really large Tesla coils, ear plugs were offered. The zapping was loud but no close to needed the plugs. David Blaine was safe from the strong electric field generated by the lightning machines. The metal garment was intensively charge only on the outside. No charge or field penetrated to the interior to harm him. The real trial for him was the endurance of standing in place for three days. He could take sugared water but nothing else. He voided into the suit. This event illustrated two points from the Seminar. One was the problem of moisture for home experiments with static electricity. In moist air just about no experiment works as intended. The charge dribbles off the props into the air. The other is that charge stays on the outer surface of a closed box, in this case a completely enclosing metal garment. The show was free and open 24/7 during its run. A few Seminar attendees hopped the 14th Street bus after the meeting to see Blaine on the opening night. Other, like myself, went on other days over the Columbus Day weekend.
A new ampere? ----------- Until about year 2000 we didn't have really precise value for the charge of the electron and, by extension, the number of electrons comprising a coulomb. In the 2-thous as metrology improved we secured better measurements of the elctron charge. It was then plausible to ask if the ampere should be redefned as so many electrons per second. There is a proposal at the International Bureau of Weights and Measures to do exactly that. The group is discussing the plan and will vote on it at its 2014 convention. Such a redefinition of the ampere as so-many electrons per second is part of a larger project to shift definitions of metric units more to the properties and parameters of nature and less reliant on human- based operations. To do this, certain of the physical properties of nature will be redimensioned to exact values, no longer subject to experimental measurement. If the redefinition plan goes thru, the ampere will be
1 ampere = 6.2415e+18 electrons/second
where the count cited here is approximate. The official value will have more decimal places. One consequence of defining the ampere this way is that it freezes the value of the electron charge. It is no longer a subject of future measurement.
Speed of electricity ------------------ One very common belief is that electricity, the flow of electrons in circuits, travels at the speed of light. This is evidenced by the instantaneous response of electrical devices to signals they receive. The transmission of the signal initiated at one end of a wire is very rapid, but not lightspeed. The very electrons themselfs travel orders more slowly. We have a wire of a given cross section, made of copper, with a current of so-many amperes. The speed of the electrons is
(elec speed) = (current) * (electron/coulomb) / ((electron density) * (cross section))
The electron density for copper is that of copper atoms in the wire, obtained from the manufacturer, along with the cross section of the wire. Each copper atom has one mobile electron for the electric current. I use here a nominal density of 8.45e+22 atom/cm3. As example, the wire carries 0.4 ampere (0.4 coulomb/second) and has section of 0.03cm2. What is the speed of the electrons?
(elec speed) = (0.4 coul/sec) * (6.2415e+18 elec/coul) /((0.03cm2) * (8.45e+22 elec/cm3)) = 9.85e-4cm/sec = 9.85 micron/second
This is an incredibly slow speed! For higher current, the speed is proportionaly greater. This increases internal friction in the wire to heat it. Wire has a maximum ampere rating to prevent overheating.
Electric field ------------ An electricly charged body sets up a field of force around it that acts on other charges placed in it. For simplicity sake I deal here only with point seats of charge. Charge, like mass, can be distributed in any geometry such as lines, sheets, slabs, spheres, shells. The force on a target charge placed in an electric field is given by Coulomb's law, a parallel to Newton's law for gravity.
force12 = -kappa0 * charge1 * charge2 * rr / (distance12 ^ 2)
The force of charge 1 on 2 is the negative of the force of charge 2 on 1. This reflects the rule of action and reaction.
force12 = -(force21)
The unit of this force is newton, like for mechanics and gravity. Force is a vector, having direction along the line between the two charges. By long convention the force of a positive charge is directed outward from it. A negative charge's force is directed inward to it. In order that the maths work out properly for this convention, the minus signum is needed. Some authors attach the minus signum to kappa0.Thiskappa0, Coulomb's constant, is
kappa0 = 8.988e9 newton.meter2/coulomb2
The force calculated by Coulomb's equation may be attractive (positive) or repulsive (negative). This is the rule that 'like charges attract; unlike, repel'. The maths put the correct signum of the force according as the polarity of the two charges. One feature you may notice, if you be versed in vectors, is that the left side, force, is a vector, but as written the right side is a scalar or regular number, We have to put a vector into the right side. We do this by inserting on the right a radius vector of unity length so it really only imparts a vector direction. That's the 'rr' I have in the formula above. It aims outward from the charge, in the positive direction.
Electric field strength --------------------- Depending on the depth of charge on the home body the force on a given target will vary. We say the strength of the electric field is the force it applies to a unit charge on the target. The electric field strength is
(elec field strength12) = force12 / charge2 = -kappa0 * charge1 / (distance12 ^ 2) Permittivity ---------- The medium around an electric charge weakens the electric field it sets up. The parameter of a medium to permit an electric field is permittivity, epsilon. This was experimentally measured for assorted substances and entered in tables of properties. In this article I work only with electricity and magnetism in vacuo. Epsilon for vacuum is epsilon0. Over the years we found that epsilon0 was actually a component within kappa0. The relation is
kappa0 = 1/(4 * pi * epsilon0)
Epsilon0 is so much a more fundamental parameter of nature that many physicsts put aside Coulomb's constant and write his rule as
force = -(1/(4 * pi * epsilon0)) * charge1 * charge2 * rr / (distance12 ^ 2)
This looks clumsy only because I wrote it in ASCCI text. The value of epsilon0 is
epsilon0 = 8.8542e-12 coulomb2/newton.meter2
The Coulomb constant kappa0 is
kappa0 = 1/(4 * pi * epsilon0) = 1/(4 * pi * (8.8542e-12 coulomb2/newton.meter2) = 8.988e9 newton.meter2/coulomb2
Electric potential ---------------- Electric potential is the potential energy of a point within an electric field, measured relative to a chosen zero or base location.
(pot energy12) = -kappa0 * charge1 / distance12)
Electric potential is energy per unit charge, joule/coulomb, with the special name of 'volt'. It is the work needed to move a unit charge from infinity to given distance against the force of the field. Altho infinity is the natural zero point of potential energy it is virtually always practical to assign zero to a point in the workspace to hand, like the chassis of an electric device or the negative pole of a battery. We work with differences from a local zero to all other points in our workspace. The electric field is a conservative field, one where the potential energy between two points depends only on the location of the endpoints, not on the path traveled between them. A field of ocean currents is not conservative because the path between the endpoints is affected by friction. The work done moving an object along each path is different for a given pair of endpoints.
Conservation and superposition ---------------------------- Electric field, like gravity fields, enjoy superposition and conservation. Superposition means that fields do not interfere with each other. They act independently but their dorces add, by vectors, at all points. The net force form all the superposed fields is the vector addition of the individual fields. Conservation means that the diels potential energy between any two points is a function only of the location of the points within the field. The path taken from the one to the other doesn't matter. We calculate the potential between the points by the path of our choice to simplify the maths. For a single point seat of field ths path is a radial one and we merely take the algebraic difference between the potentials of the two points.
strength versus potential ----------------------- Do not confuse field strength with potential, altho they are sometimes treated alike. Field strength is measured in newton/coulomb and varies with, for a point seat, 1/r2. Potential energy is in joule/coulomb and is a function of 1/r. Potential at a point requires a stated reference or base or zero level, usually that of an other stipulated place in the field. This is why knowing the potential at one point can't determine by itself the potential at any other point. Look again at the units for both properties
(field strength) = [newton/coulomb]
(field potential) = [joule/coulomb]
The joule is a newton.meter of work or energy. To move a charge from a one to an other point in the field work is done on it by forcing its motion thru a certain displacemnt. We have
(field potential) = [joule.coulomb] = [newton.meter/coulomb] = [newton/coulomb] * [meter] = (field strength) * [meter]
(field potential) / [meter] = (field strength)
[volt/meter] = [newton/coulomb]
The volt/meter is common in geophysics and meteorology to describe electric field strength like in the atmosphere.
Electric power ------------ Electrons from a battery deliver their energy to the attached device. The amount of energy per coulomb's worth of electrons is the potential of the electrons, stated in volts. The rate of flow of coulombs per second in the device is the current in amperes. The rate of delivering, producing, consuming the electric energy is the joules per coulomb times the coulombs per second
[joule/second] = [joule/coulomb] * [coulomb/second] = [volt] * [ampere]
This quantity, the flow of energy, the power, is named the 'watt'. It is also used for any other kind of energy flow, not just electric. Electric devices are labeled with their volt and either the watt or ampere ratings, This formula stricta mente is valid for direct current, DC, like that from a battery, and not to alternating current, AC. It turns out that for most situations,the difference, due to the interaction of magnetic fields and synchronizing of the cycles with in the circuit, is small. You usually can in AC go with watt = volt*ampere.
Lines of force ------------ A line of force is merely the path followed by a test particle placed in the force field and left to move by the force impressed on it by that field. These are also called force lines and field lines. Beyond this simple property a line of force has no actual reality. By inserting a test into the field you place it on a unique path that becomes the line of force passing thru the initial position of the particle. Placing the particle at a new location makes a new line of force. In diagrams the field lines are placed as convenient for showing the shape and strength of the field. There is no quantitative method for laying down these lines. You may draw as many or as few as you like. Because a plus and minus charge move in opposite directions under the force of an electric field, the direction along its path is by convention that for a plus charge. The direction of the electric force at a given point is tangent to the force line thru that point. Arrows placed along the line of force are always the direction toward which such positive charge moves. This is why foeld lines are shown pointing out of a plus seat of the field and into a negative seat. Lines of force are orthogonal to the lines of potential within a field. A test particle falls in the field across potential levels. You can sketch the potential lines by always drawing them perpendicular to the field lines they cross. A force line has endpoints on the source of the electric field, Field lines can not dead end or break apart but are continuous. They can not cross each other but always align along distinct paths. In the early era of electric and magnetic studies there was a notion of field strength in terms of line density, so-many lines per unit area. This method of stating field strength is no longer used.
Electron-volt ----------- The electric potential energy is in joule/coulomb, so the total energy of a particle is this energy times the amount of charge on this particle. One very common charged particle, particularly in electronics and atomic physics, is the electron. It has a charge of 1.6022e-19 coulomb. If this electron is subject to an electric potential of one volt, one joule/coulomb, it acquires an energy of 1.6022e-19 joule. This amount is the electron-volt, the energy of one electron in a electric potential energy of one volt.
i electron-volt = 1.6022e-19 joule
1 joule = 6.2414e+18 electron-volt
The spelling of electron-volt differs to include or not the hyphen. Since electron-volt is not an official SI unit, there is no standard spelling.
Batteries ------- With the concept of charge and potential in hand we can understand how a battery works. A battery is a jar, can, cell with chemicals that load electrons with energy and sends them out to do useful work. They leave the battery from the minus pole and return by the plus pole. Remember that the usual rule that electricity flows from he plus to the minus pole refers to the traditional POSITIVE current flow. The changes in energy content of the electrons as they flow thru the battery and device is illustrated here. Within the battery at the left the electrons are loaded with energy from the chemical reactions inside the battery and then they leave thru the minus pole. Note that this is the electron flow, opposite to the traditional current flow.
energy full load ^ +----------+ | | | /| | \ | | plus / | | \ | | | / minus | \ | plus / next lap | / | | \ | empty | / +--------+----------+---------+----------+---- |battery-|wire------|device---|wire------|battery--
From the minus pole the electrons travel, with almost no loss of energy, thru wires to the device. By design and build, wires are made to conduct electric current with minimal loss of potential along them. In the device the electrons deliver their energy to perform useful work. I draw the delivery as a smooth decline but in a real electrical device the extraction of energy takes place at various points within the circuit,bit-by-bit until the electron is empty of energy The electrons travel back to the battery with empty load thru wires and enter the battery thru its positive pole. This completes one lap of the current. The electrons take on a fresh load of energy from the battery's chemicals and are sent out for a new round of the device. For the duration of activity of the chemicals this process continues. An ordinary battery does not store electric charge directly, not even a 'storage battery'. There is no stock or reservoir of electrons or coulombs inside the battery that can run out like water in a tank. It contains a mix of chemicals that slowly and steadily release energy thru reaction. The chemicals eventually deplete, exhausting their reactivity, and the battery runs down. The total energy packed into the chemicals is a measure of the total amount of charge that can be supplied by the battery. This could be stated in coulombs. The more common measure is ampere-hour. Most people think of a battery as lasting for a period of time before going dead. During this time is is delivering so many coulomb/second or amperes. One ampere- hour is 3,600 ampere-seconds or 3,600 coulomb. When a battery is recharged it is really refreshing its chemicals to once again load energy onto electrons. Only batteries with chemistry specificly made for refreshing can be recharged and even then for only a finite number of cycles. After that the battery no longer holds its charge, meaning that its chemicals no longer can be refilled with energy. You can find how many electrons circulate thru the battery when attached to a device of known ampere rating, say 500 milliampere. We have
(electron/second) = [coulomb/second] * [electron/coulomb] = [ampere] * (6.2415e+18 elec/coul) = (500e-3 coul/sec) * (6.2415e+18 elec/coul) = 3.1208e+18 electron/second
This is a fantasticly huge number of electrons! Yet there is no real accumulation of charge in the device or battery because the electrons are circulating in a closed loop. The device and battery remain neutral in net charge. If the device is rated in watts, one watt is one joule/second. The electrons flowing thru the battery is then
(elec/sec) = [joule/sec] * [elec/coul] / [joule/coul] = [watt] * (6.2415e+18 elec/coul) / [volt]
Voltage and amperage ------------------ It is not a wise habit to name the qunatity or concept by the name of the unit for it. In spite of this rubric, it is so very prevalent to call the electric potential of a device its 'voltage'. The current thru the device is called its 'amperage. By similar logic the power consumed by the device, in joule/second or watts, is its 'wattage'. In good science we simply do not use such words, not any more than we say 'grammage' for the mass or 'newtoage' for force. We tolerate the '-age' terms in casual discourse. In oldstyle lingo as examples 'poundage' is used for weight; 'footage'. length or distance.
Electromotive force ----------------- An other term commonly used is 'electromotive force' or EMF. It is not a force at all and there is really no need for it as a distinct term. It is usually applied to the electric potential at free ends of an open circuit like the poles of an battery. there is the capability, in the lay person's mind, of forcing a current to flow when the battery is connected to an electric device. It is really the same principle as the volt rating of the battery. A common loose phrasing is that the volts of electricity leave a battery from the positive pole and return thru the negative pole. Volts, electric potential, does not itself move. It's a potential to do work if allowed to by hooking up the battery.
Angles and areas -------------- We must digress to discuss vectors and their maths. I give here only a reminder about vectors, not a full discourse. Most home astronomers can grasp the concept of vectors with obvious examples like velocity and force. An other kind of vector is an angle and an area. At first this sounds strange since an area is just a bounded piece of a surface and an angle is just the opening between two lines. Look at this diagram of an angle
o------------------------------a ---- | --- | --- / theta --- / --- / --- ---b
The angle between lines oa and ob is theta. It can have direction in the sense that theta can be counted from a to b or from b to a. Theta can be either a plus or minus angle. You may draw the angle with an arrow to show which way to count it. Now imagine that Pretend the angle is a handle or knob on a right-hand bolt. The bolt stands orthogonal to the paper. Now turn the knob. If you turn it clockwise, from a to b advances the bolt into the paper away from you. The direction of the bolt's movement is a vector associated with the angle. In this case it aims into the paper. The value of the vector is the size of the angle. More commonly this size is the value of a trig function of the angle, like its sine or tangent. Turn the handle anticlockwise from b to a. The bolt retracts out, toward you. The vector direction is out of the paper. The size is that of the angle or its trig function. After you think about this weird treatment of angles, you probably already applied them as vectors. If you work with hardware, machines, tools, you know there is one hell of a difference between turning a screw so-many degrees clockwise and anticlockwise. The direction of travel is opposite and can do very undesirable things if it's wrong way round. areas are a bit more tricky. Here is a square element of area on a large surface like a tile.
a +-----------+ b | | | | | | | | | | | | c +-----------+ d
Trace the square clockwise from a thru b, c, d, and back to a. We enclose an area within the square. Imagine that this travel clockwise turns the handle of a bolt, like for the angle. The bolt moves away from you into the paper. The direction of movement is the vector associated with this area. Walking the area anticlockwise from a thru d, c, b, and back to a moves the bolt out of the paper toward you. In this way we attach a vector property to a surface area. It points orthogonally on the area in either the inward or outward direction, according as the sense of circulating around the area. The size of the vector is the value of the enclosed area.
Dot product --------- We look at two features of vector math that are crucial for a good appreciation of electric and magnetic behavior. We got two ways to multiply vectors, one called the dot product; other, cross product. The names come from the symbol used in typeset maths: a fat dot, like a typographic bullet, and a thick X, like a railroad crossing. In the dot product the two vectors after multiplication lose their vector quality and the result is a regular number. A cross product of two vectors is a new vector that stands at right angles to both original ones. The dot product is
C = dot(vec(A), vec(B)) = val(A) * val(B) * cos(theta)
'Vec' is my ASCII method to say that A and B are vectors. In typeset work a vector is written in script or bold or is topped by an arrow. 'Val' means the value, size, magnitude of the vector. Theta is the angle between the two vectors. The angle is measured FROM vec(A) TOWARD vec(B), in the order that A and B are stated in the formula. The cosine of theta carries its own signum that is processed with the signa of the original vectors. An example for vec(A) = 74.5; 21.3 deg, vec(B) = 32.7; 41.1 deg.
C = dot(vec(A), vec(B)) = val(A) * val(B) cos(theta) = (74.5) * (32.7) * cos(+41.1 - +21.3) = 2436.15 * cos(+19.8 deg) = 2436.15 * +0.9409 = 2292.13
The subtraction in the cosine part shows that theta is rotated FROM vec(A) TOWARD vec(B), from 21.3 deg to 41.1 deg. Note also that I'm careful to insert the signum of the angles and the cosine. It is easy to leave it out and you could get into a heap of trouble downstream in your work. Put in the signum explicitly. Believe it or not you already use the dot product when you do orbit calculations. See the 'r2' in the denominator of Newton's formula? In one part of the book you read about orbits you learn that r is the radius vector from the Sun to a comet. In that very same book you work thru Newton's law and put values for r into that r2 term. What happened to the vector property of the radius? The secret is that r, vec(r), was dotted against its own self.
r2 = dot(vec(r), vec(r)) = val(r) * val(r) * cos(theta)
The two vectors are counts of the one signle vector for r. The angle between r and itself is zero. Theta = 0 and cosine(theta) = +1.
r2 = val(r) * val(r) * cos(0) = val(r) * val(r) * (+1) = val(r) * val(r) = r * r = r ^ 2
Cross product ----------- The cross product is
vec(C) = cross(vec(A), vec(B)) = val(A) * val(B) * sin(theta), then apply right-hand-rule
Theta is counted FROM vec(A) TOWARD vec(B). While the formula for cross product resembles that for dot product, it uses the vector property of angles. The vector for theta has size sin(theta) and direction given by the imaginary bolt. Here's an example for vec(A) = 231.5, 32.1 deg and vec(B) = 0.49, 9.7 deg.
vec(C) = cross(vec(A), vec(B)) = val(A) * val(B) * vec(sin(theta)) = 231.5 * 0.49 * vec(sin(+9.7 - +32.1)) = 231.5 * 0.49 * vec(sin(-22.4)) = 231.5 * 0.49 * vec(-0.3811) = vec(-43.23)
When vec(A) and vec(B) are coincident or parallel, theta = 0, sin(theta) = 0. The cross product vec(C) is zero. When the vectors are orthogonal, theta is 90 deg, sin(theta) is +1. The cross product is maximized. The cross product has a geometric meaning as the area of the parallelogram formed by the ingredient vectors. When the vectors are at right angles the paralleogram becomes a rectangle and its area is the maximum. For parallel or coincident vectors the figure collapses into a line with zero area. The order of multiplying the vectors in the cross product is important. The operation is not commutative like for arithmetic. Reversing the order gets you a vector pointing opposite from the correct alignment. This can cause grief in later parts of your work.
Right hand rule ------------- The convention for vector directions is determined by the right hand rule, RHR.It is based on the action of the imaginary bolt in the section 'Angles and areas'. With the right hand curl the fingers in the sense of the turning, as if gripping the knob. Make sure the fingers extend around the area or angle in the SAME sense that you would do such turning of the bolt. Now stick out the thumb perpendicular to the fingers. It points in the direction of the angle or area vector. That's it, honest, it is. In most of physics we fix things so the vectors comply with the RHR. The one major caution is for electric current, a vector that flows with the POSITIVE charges and NOT the electrons. You must wrap the fingers in the sense of this conventional current, opposite from that of the electron flow. Else your resultant vector aims in the opposite direction and further work with it is all wrong. An example is given here for the direction of the force applied by a magnetic field on a wire. The wire has an electric current flowing to the right and the field is aiming into the paper. The 'X' looks like tail feathers on the vector arrows for the field. A field aiming at you out of the paper could be represented by dots for the arrow heads.
X X X X X X X S X X X X X X X ----------------------------- -- current--> X X X X X X X X X X X X X X X
This is the conventional current, like in household wiring or a battery circuit. The electrons flow is to the left. The force is a vector given by the current and field directions, skipping for now the magnitude of this force. It is orthogonal to BOTH the current and field but in which sense, up or down? Place the right hand on the paper to 'turn' the current onto the field, like turning the handle of the imaginary bolt. The thumb points up. The force direction in this example is up, not down.
Gauss's law for electric field ---------------------------- If we surround a collection of electric charges with a closed envelope, like a sphere, and sum up all over its surface the electric field strength, the summation is the net electric charge inside the envelope.
sum(dot(field strength, delArea)) = (net charge) / epsilon0
Because the field strength generally will vary over the surface due to the irregular deployment of the interior charges, we must sum up by small areas, deltaArea:
sum(dot(field strength, deltArea)) = dot(strength1, deltArea1) + dot(strength2, deltArea2) + dot(strength3, deltArea3) + ...
The size of delArea is chosen to allow the field strength to be uniform over it but not necessarily equal to that in the adjoining delArea. This choice comes from experience and trial. It also helps to choose a shape for the Gauss volume with a simple geometry. The net charge is the residuum after nulling out all plus and minus charges inside the envelope. If there be 84 protons and 82 electrons, there are 2 left over protons with 2 units of positive charge. These 2 units make the summed up field over the envelope. The separate plus and minus charges are not revealed thru Gauss's law. The envelope doesn't have to be a simple shape, altho having one makes the maths much easier. It can even have folds and pockets, just so long as it has no holes, tears, other openings. As the electric field penetrates each ply, outward or inward, it goes into the sum. The dot process takes care of the signum of each element in the sum. In the special case where the Gauss envelope is a sphere and the charges are huddled in its center, Gauss's law flows recta mente from the formula for the field strength. The field strength is the same all over the surface of the sphere because the charges are all the same distance away, the radius of the sphere.
(fld strength) = (1 / (4 * pi * epsilon0) * (charge / (radius ^ 2) = (charge / epsilon0) * (1 / (4 * pi * radius ^ 2))
The surface areaa of the Gauss sphere is 4*pi*radius2. Shifting this to the left side gives
(fld strength) * (4 * pi * radius ^ 2) = charge / epsilon0
which satisfies Gauss's law.
Conclusion -------- In this part one we clarified many of the loose descriptions of electric fields, forces,charges that confuse the home astronomer. I made no attempt to build a full course in electricity, only to sort out the various concepts and parameters for you. In part two we look at magnetism and its relation to electricity.