## Electricity and Magnetism

All you need to know to understand the electromagnetic telegraph

### Introduction

It isn't hard to understand the wonder of electricity, and only arithmetic is required to make useful calculations and predictions. As much as is necessary to appreciate the telegraph of the 19th century makes an excellent foundation, and should be in the possession of every educated person. It should certainly be taught in schools, for which it is a remarkably appropriate science. This paper is an explanation of the fundamental concepts for the nontechnical person, perhaps most useful to clarify a jumble of conceptions randomly acquired over the years. It is, however, fully rigorous, and shows how to design the elements of an electromagnetic telegraph rationally.

### Matter and Charge

Our material world consists of very small objects, smaller even than atoms, that strongly pull and push on each other. These objects possess the property of charge. There are two kinds of charge, called positive and negative, that attract each other, while repelling others of the same kind. The names reflect the algebraic property that in equal amounts they are perfectly satisfied with each other and do not affect other charges. It happens that the small objects have charge that is a multiple of a basic unit, so that small clusters of objects can have an exactly zero charge. An atom consists of a heavy cluster of charges of the same kind, positive, held together by even stronger forces than electric ones, but which act only on these objects and are of very short range, surrounded by a number of light negative charges, electrons, that are strongly attracted to the positive nucleus but cannot actually get to it because they repel one another and are too big, anyway. They are held only because of the strong attraction for the nucleus, in spite of an aversion to each other. Each electron has one elementary negative charge, and their aversion for one another is stronger than mere repulsion of charges. They will tolerate a companion of the opposite sex (spin), but still repel her. The complete atom is a neutral assembly that does not affect distant charges.

The first thing I did in my explanation just now was to describe the structure of matter, since it makes the rest of the job much easier. This explanation was not available when electricity was recognized in the 18th century, so the new phenomenon was very difficult to understand. People thought it was some kind of light fluid, as they then understood heat, light and life were. On this basis, it was very hard indeed to understand electricity.

Atoms can be quite satisfied with themselves, but often join in small associations, where the charges are even more comfortable. What essentially happens is that the additional possibilities furnished by the presence of two or more nuclei allow the electrons to cuddle up even closer to them. The two atoms in molecules of oxygen, and nitrogen, in air are very satisfied with their personal arrangements, and move about like footballs, bouncing off each other. Others associate in regular arrangments, packing together in solids. These things are held together only by the tag ends of the electrical forces between the charges. In all these things, the positive and negative charges are exactly equal in amount, and their forces are not apparent on the scale of our observation, though the forces between the equal charges are wholly responsible for the solidity of matter, which is really mostly empty space.

Some atoms are well satisfied to share their electrons with others in large communal groups that hold all their electrons in common, and permit them to roam about to ensure absolute neutrality at all points. The nuclei arrange themselves in a regular lattice, surrounded by a 'glue' of moving electrons. If this sounds like a flexible arrangment, it is. The material can be bent and squeezed and hammered out--it is, of course, a metal. The interesting thing is that charge can apparently move from one point to another. If we put a extra electron in one end of a wire, the whole wire has an extra charge, that can be taken out at any other point instantaneously to reestablish neutrality. Metals are called conductors as a result of this property, unlike most materials in which the charges are definitely associated with some individual atom or molecule, called insulators. The distinction was very mysterious before the structure of matter was known, but now it should be obvious. Incidentally, when you look at a shiny metal, you are seeing the electrons.

Water is a remarkable substance, but its peculiarities with respect to electric charges are amazing. Its molecules are electrically neutral, of course, but the hydrogen end is positive, and the oxygen end is negative. There can be strong electrical forces close to the molecule, but not far away. These molecular dipoles can cluster around an atom that has lost some of its electrons, or gained extras, and comfort it greatly. There is so much comfort, in fact, that the dipoles can separate the positive sodium atoms and the negative chlorine atoms in a crystal of salt, forming a solution of fuzzy negative and positive ions in equal numbers, but mobile. Such a solution can conduct electricity, because the ions can move around to ensure electrical neutrality at any point in the solution. A positive ion, say, can be added at any point, and then abstracted at some other point, and it is as if it had moved from the one point to the other. It is like putting money in one branch of a bank and withdrawing it at another. I have been very careful to explain that what a conductor actually does is to ensure electrical neutrality over its extent, and does not 'conduct' a charge from one point to another like a letter in the post. It is more like water in a filled pipe; put a little in at one end and take the same amount out at the other.

Since charge is positive or negative, and opposite charges compensate, a flow of positive charges in one direction has the same effect as a flow of negative charges in the opposite direction, as far as the equilibration of charge is concerned. In a metal, only the electrons are mobile, but in an ionic solution both signs of charge are mobile. Electrically, it does not matter which sign of charge is moving, just so there is net motion. This is conventionally represented as the direction of motion of positive charge, and the rate at which charge moves is the electric current. The amount of charge is measured in coulomb (symbol C), and about 96,500 C is the charge of 6 x 1023 electrons. This number of electrons, as you may recall from Chemistry, is the number of molecules in one gram-mole, which is 18 grams of water, for example. A rate of charge transfer of one coulomb per second is called an ampere (A).

### Electric Circuits

If a net positive charge moves from point A to point B, there will be an excess of charge at B and a deficiency at A, which will exert strong electrical forces opposing this unbalance. The excess charge must be bodily restrained if the unbalance is to be preserved. For a current to flow continuously, there must be no accumulation of charge at any point. That is, the charge must flow in a closed path, a circuit. A closed loop of wire forms such a circuit, but would be very uninteresting, since nothing would happen, the charge being evenly distributed so that there was neutrality everywhere. The electrons or ions would be bouncing about, but we would see nothing on our level. Excitement begins when we discover some way to tear charges apart and expose them to two ends of the wire. The wire will accept charge at one end, and release charge at the other end, in an attempt to re-establish neutrality.

Electrons in the wire collide with the nuclei as a result of their rapid random motion, exchanging energy back and forth. The energy of vibration of the nuclei is heat, so the electrons heat the nuclei, and the nuclei heat the electrons, and all normally balances. When there is a current flow, however, there is a net movement of the mass of electrons in one direction, and an asymmetry is created, resulting in more movement being given to the nuclei, and less given back to the electrons, so the metal becomes hotter, and the motion of the electrons is opposed, as if by friction. The electrons have a certain amount of energy when they are supplied to the metal at one end, and less energy when taken out at the other. Now, energy is measured in joule, so the amount of energy going to heat per coulomb will be so many joule per coulomb, called volt (V). The energy per coloumb is called potential, pressure, or simply voltage. In 1827 Ohm realized that the energy lost was proportional to the current for a given sample, which was what one might expect from friction. The constant ratio of the potential difference to the current is called resistance, and its unit was given Ohm's name, and the abbreviation Ω.

If V is the voltage difference in volt, and I the current in ampere, then V = IR, where R is the resistance in ohm, which is Ohm's Law for electric circuits. [The metric police say we have to use the singulars and small letters.] We also have that the energy lost per second, or the power P, is P = VI. This is simply watt = joule/second = (joule/coulomb) times (coulomb/second), which follows from the definitions themselves. These two simple formulas have many applications, and can be put into many useful forms. For example, combining the two we find that P = I2R = V2/R, which tells us that the power dissipated in a resistance is proportional to the square of the current through it, or to the square of the voltage across it. Note that currents are always through, and voltages across, some part of the circuit.

Electric circuits are represented by circuit diagrams that use conventional symbols. A conductor is represented by a line. The symbols for a cell or battery, and for a resistor, are shown in the Figure. A current is represented by an I with a numerical subscript if there is more than one, and a direction arrow shows the positive direction. In the Figure, if I = 2 A, then the current is flowing in the direction of the arrow. If I = -3 A, it is flowing in the opposite direction. Voltages are represented by V, with subscripts if necessary, and polarity marks + and - (or just +) show which end is at the higher potential. An E is used for an emf. Two different resistor symbols are shown. The graph shows the potential energy of an electron (or the voltage) at different points around the circuit. Remember that electrons are attracted by + and repelled by -, and that the current is opposite to their direction of motion, since they are negative charges.

A circuit can also be a complicated network of branches and nodes. At a node, where wires meet, the sum of the currents in all the wires must be zero (otherwise charge would build up at the node). Around any closed loop, the sum of the voltage rises must equal the sum of the voltage falls (otherwise there would be a jump somewhere which would mean an infinite current). These rules, called Kirchhoff's Laws, are sufficient to find the currents in every branch, which is called solving the circuit. The Figure shows two simple but very useful examples of branched circuits. Here, resistances are represented by simple rectangles, which is a third way of showing them on a circuit diagram. In the series circuit, the same current passes through both resistors, and their voltages add. In the parallel or shunt circuit, the same voltage is impressed across both resistors, and their currents add. In the first case, a single resistor of the sum of the resistances would be equivalent to the pair, so far as everything outside them is concerned: R = R1 + R2. This resistor would be larger than either of the two. In the second case, a single resistor the reciprocal of whose resistance is equal to the sum of the reciprocals of the separate resistances would be equivalent: R = 1/(1/R1 + 1/R2). This resistor would be smaller than either of the two.

The resistance will depend on the shape of the conductor. For a wire, it is easy to deduce that the resistance will be proportional to the length of the wire l, and inversely proportional to the cross-sectional area A. Therefore, R = ρl/A, where the constant ρ depends only on the material, and is called the resistivity, with units Ω-metre. Its reciprocal, σ = 1/ρ, is called the conductivity. The resistivity of pure copper at 20°C is 1.678 x 10-8 Ω-m. The resistivity of pure iron is about 6 times greater, and of mercury 58.6 times greater. The diameter of copper wires is specified by gauge numbers. In the B. & S. gauge, an decrease of one gauge number increases the diameter by a factor of about 2-1/6 = 1.1225, and #10 gauge is 0.1019 inches in diameter, with a resistance of about 1 Ω per 1000 ft. The resistance of a certain mercury column was originally taken as the unit of resistance, and the voltage of a Daniell cell as the unit of potential. With slight redefinition, these later became the ohm and the volt, by sheer luck fitting into the metre-kilogram-second system of units.

### Sources of Energy

Now let's turn to the other part of the circuit, the part that gets the charges separated in the first place and puts them where they can move through the resistance. Energy is given by separating the charges against their attraction for each other, which requires work. At first, the only way to do this was actually to rub off some charge by friction on an insulator, then to do the work by separating the insulator with its burden of charge from its origin, which remained with the balance of opposite sign. This was frictional, or static electricity, and we notice it in in dry weather on separating dried laundry--polyesters from cottons, for example--or on walking across a carpet, or stroking a cat. Being charged, we can now draw sparks from pipes, doorknobs or cats, and experience the muscular spasms. Touching a piece of gold leaf draped over a wire, or a pair of pith balls hanging together, the forces of repulsion between the charges transferred are made evident.

In 1800, Volta discovered that this can be done for us by chemistry, and copious amounts of charge can be separated. The work is done over a much smaller distance, so the voltage is lower, but the amount of charge is hugely greater. To see what happens, suppose we dip some metallic zinc in water. The busy dipoles struggle with the mobile electrons in the zinc for the affections of the zinc nuclei, and liberate some of the nuclei, forming positive zinc ions, with two elementary charges, that are now free in the water. This leaves excess negative charge on the zinc, and excess positive charge in the water, and this attraction soon produces an equilibrium, in which some zinc is dissolved, and some potential difference exists between zinc and solution. The water is usually made somewhat acid, with hydrochloric or sulphuric acid, to prevent the zinc ions from reacting with the water to make a gummy precipitate of zinc hydroxide that would coat the zinc and prevent further action. Every metal would behave just like the zinc, but to different amounts. If copper is immersed in a solution containing copper ions, also doubly positively charged like the zinc ones, the electrons in the metal win out and some of the ions join the metal, adding positive charge to it, and leaving the solution negatively charged. For each metal, the end point of the contest is different. Even the hydrogen ions formed in very small amounts by the water can be enticed by the electrons in a metal electrode, and changed to hydrogen gas, which appears as small bubbles. Whatever is most likely happens in each case.

Now let us suppose we have zinc in zinc ion solution, and copper in copper ion solution, with the solutions made electrically neutral by an equal charge of negative ions (chloride or sulphate, for example). The zinc will be more negative than its solution, and the copper more positive than its. If we put the two solutions into electrical contact, so that they can exchange charged ions, the ions will move until the two solutions are at the same potential, which will necessarily make the copper more positive than the zinc, in fact by about one volt. This is the case in the Daniell Cell, shown in the Figure. If a wire is connected from zinc to copper, electrons will be attracted from the zinc, and communicated to the copper, in an attempt to restore electrical neutrality. It will be a vain attempt, since no sooner is the potential reduced, but more zinc dissolves, and more copper is plated out, and this will continue as long as zinc metal exists. We have an electric circuit, in which chemical energy is given to electrons, which then appears as heat in the resistance of the external circuit. The zinc-copper cell creates an electro-motive force, or emf, that is opposite in direction to the potential difference across its terminals, driven by the relative urges of zinc and copper to dissolve in water. The symbol for an emf is usually E. Around any circuit, the sum of the emf's E must be equal to the sum of the voltage drops V, and this rule allow us to calculate what the current will be. When current is drawn from a cell, its voltage decreases slightly, because of the resistance to the current of ions in the electrolyte and for other reasons. This can be thought of as an internal resistance in series with the battery.

### Currents and Magnetism

If you place a compass needle just below a wire oriented north-south, and then pass a current through the wire directed from south to north, the north pole of the needle will be deflected to the left. If the wire is below the needle, then the deflection will be in the opposite direction. If you pass a wire through a hole in a card sprinkled with iron filings, the filings will cohere into circles surrounding the wire when a current is established (this requires a very strong current with just one wire). All this shows that an electric current is accompanied by a magnetic field that encircles the wire in the same sense as the fingers of the right hand grasping the wire with the thumb pointing in the direction of the current. The strength of the field is inversely proportional to the distance from the wire. Imagine the field as stretched rubber bands forced away from the wire by the current, and collapsing on it when the current is reduced.

By winding the wire into a coil, the magnetic effect can be multiplied by the number of turns of wire, so that it will affect the needle or the iron filings much more strongly. A coil wound around a long cylinder, called a solenoid, has a field that runs through the cylinder from end to end, then diverges at the ends to loop around and join, just like the field of a bar magnet of the same shape. These solenoids and coils exert forces and torques on each other exactly like permanent magnets. These forces always tend to make the most compact and smoothly-varying magnetic fields possible.

An even greater strengthening of the magnetic field results from inserting a core of soft iron in a solenoid. The increase can easily be a thousand-fold, so we have an easy way of making strong magnetic fields that is much more convenient that the use of permanent magnets. Only iron (and its relatives nickel and cobalt) behave this way, which is due to a peculiarity of their electronic structure. What happens is that the electrons are themselves small magnets, and the weak magnetic field created by the currents in the winding trigger the co-operative alignment of these electronic magnets, head to tail, with the same effect as if the currents had been multiplied hundreds or thousands of times. Without this effect, magnetism would be a mere curiosity, not the basis of the electrical industry as it is today.

The magnetic field forms closed loops, like the closed electric circuit. Although it is not a flow of anything, the analogy is so useful that the fiction of a magnetic circuit has been created. The driving force is ampere-turns NI, or magneto-motive force, mmf, (symbol M) while the analogy of current is magnetic flux in weber (Wb), (symbol Φ) and the ratio of the two for the geometry considered is the reluctance (symbol R), the analogue of resistance. Thus, M = R Φ, the analogue of Ohm's Law.

Just as in the case of the resistance of a wire, the reluctance will be proportional to the length and inversely proportional to the cross-sectional area of the magnetic circuit, and so R = l/μA. The constant μ is called the permeability, and is characteristic of the medium. For everything except iron, μ = 4π x 10-7 Wb-m/A. For iron, μ can be thousands of times larger. There is nothing like a magnetic insulator, but a magnetic field is effectively confined to an iron core, and spreads out only in other media.

The tractive force in newton exerted by the magnetic field across an air gap of area A is about Φ2/2μA, where the flux Φ is in weber, and μ is the value for air. (9.8 newton is a kilogram force, and a newton is about a quarter of a pound.) Let us design the magnetic circuit for a sounder, where we want a tractive force of 1 lb, the length of the magnetic circuit is 6 in, the air gap 4 mm, its permeability 1000 times that of air, and its area 0.250 sq in. In metric, this is 4 newton, 0.152 m, and 0.00016 m2. The reluctance of the iron is 5 x 106 units, and that of the air gap four times larger, so the total reluctance is 2.5 x 107 units. The flux required, determined by the tractive force, is 4 x 10-5 Wb. Hence, the ampere-turns required is 0.00004 x 2.5 x 107 = 1000 A-t. Divided between two coils, that is 500 A-t each. If the operating current is taken as 1/4 A, then each coil needs 2000 turns of wire. This example shows quite clearly how the air gap is the controlling factor, not the permeability of the iron. If there were no air gap, the reluctance, and the ampere-turns required, would only be a fifth as much, just 100 A-t in each coil, or with the same ampere-turns, the flux would be five times larger, and the tractive force 25 times larger, or 25 lb. This illustrates a great problem in using simple tractive armatures, in that the range of motion must be small, and the variation in force large.

The early builders of electromagnets strove to make them as strong as possible, by maximizing the product NI, under the condition that there was a given space available for the windings. One could go from many turns of fine wire to a few turns of thick wire, but with a given battery, there was one choice that would make the strongest magnet. We can easily make the calculation with what we know, but this information was not available to the early experimenters, and they proceeded by trial and error. Finally, Professor Jacobi at St. Petersburg came up with the desired condition. The ampere-turns are maximized when the resistance of the winding is equal to the internal resistance of the battery.

When J. P. Joule tried to build an electric motor (like many others) in 1838, it dawned on him that Jacobi's condition made electric motors impractical, since the copious heat that was produced was simply a loss of energy, and any motor large enough to do useful work would also burn up, besides being wasteful of zinc. This led him to stop building motors and start experimenting with energy to establish its conservation. It was not for nearly fifty years that the discovery of back emf led to efficient and practical electric motors. Now the motor could have low resistance, and at the same time a limited current so losses would not be excessive, since the current was controlled by the back emf, not the resistance of the windings. Jacobi's Law was repealed for motors.

### Telegraph Lines

Jacobi's Law is still valid for telegraphy, however, since the currents are never large enough to produce excessive heat. The resistance of the receiver shold be equal to the resistance of the line. A 100-mile line has a resistance of about 1600 Ω, and this is the optimum resistance of the receiving relay. The total resistance of the circuit will then be 3200 Ω plus the internal resistance of the battery. If a current of 20 mA is necessary to operate the relay reliably, then a potential of 64 V is required. 65 jars of gravity would do the job. Of course, this calculation has not included the effects of leakage. If there are 30 poles per mile, and the leakage at each pole is 100,000 Ω, then the total shunt resistance would be about 33 Ω if all concentrated at one point. Fortunately, it is not, but a leakage of this magnitude would obviously have a large effect, probably putting the line out of service. A leakage of 10 MΩ, on the other hand, would result in a shunt resistance of 3300 Ω, and would have a much less negative effect on our line. The importance of good line insulation is obvious.

The line with leakage at the poles can be solved as an ordinary series-parallel circuit, but this is hardly practical for a 100-mile line with 3000 poles or so. A very good approximation can be obtained if it is assumed that the series resistance and the shunt conductance are uniformly distributed along the line, and a solution for the voltage and current at any point found by using the calculus. This is a bit advanced, but at least I can show how the final equations are obtained. Let's measure distance x in miles, and consider a short section of line of length dx. If R is the resistance of the line wire in Ω/mi, this section has resistance Rdx, and the voltage drop across it is dV = -IRdx. If the leakage resistance at a pole is r, then the current that flows to earth at the pole is V/r. If there are n poles per mile, then the current is nV/r per mile, or GV, where G = n/r is the leakage conductivity per mile. The current that flows to earth in our short section is then VGdx, so dI = - VGdx. The changes in the line voltage and current in the distance dx are dV and dI. The first two equations in the Figure then follow, which are a pair of differential equations for V and I.

An equation for V alone is found by differentiating the first equation, and then using the second to eliminate dI/dx. An analogous thing can be done to find an equation for I, which will be the same. This second-order equation is easily solved by assuming an exponential dependence eax. Substituting this in the equation, we find a2 = RG, so there are two linearly independent solutions, for a = +(RG)1/2 and a = -(RG)1/2. The sum of these with arbitrary constants A and B is the general solution of the equation. They are determined by the conditions at the start and end of the line, and its length L. At x = 0, a voltage V0 is applied, and at x = L, a load resistance of value RL is connected from line to earth. That is, V(L) = RLI(L). These two conditions give V0 = A + B, and (R - aRL)e-aLA + (R + aRL)eaLB = 0, from which the constants A and B can be found. Now, V and I are known at any point along the line.

If the line is infinitely long, the positive exponential parts must vanish, and so we find that A = V0, B = 0. Then, V(x) = V0e-ax, and I(x) = (a/R)V(x). The result for I shows that the resistance "looking into the line" at any point is R/a, which is called its characteristic resistance. If the line is broken at any point, and a resistance of this magnitude connected between it and earth, conditions on the remaining part of the line will not be affected. All of these matters are of very great importance with alternating current, but we see them here in simple form.

As a practical example, let us assume that R = 16 Ω/mile, and that we have 30 poles per mile, with a leakage of either 10 MΩ (line A) or 100 kΩ (line B). G is then 3.0 x 10-6 S/mi (inverse ohm) for line A, or 3.0 x 10-4 S/mi for line B. For line A, a = 6.93 x 10-3 per mile, or 1/a = 144.3 mile. For line B, a = 0.0693 per mile, or 1/a = 14.43 mile. The characteristic resistance of line A is 2309 Ω, while the characteristic resistance of line B is 231 Ω.

For a 100 mile line A, V(x) = V0[1.213 exp(-ax) - 0.213 exp(ax)], so the voltage at 100 miles is V(L) = 0.181 V0. If the 400 Ω relay requires 20 mA to operate, then we must have V0 = 44.7 V. The current taken by the line at x = 0 is 23.5 mA, so 3.5 mA, or 15%, is leakage. Turning to line B, it is found that V(x) = V0 exp(-ax), practically the same as for an infinite line. At x = 100 miles, V = 0.000978 V0, so if V0 = 45 V, then the relay current at the end of the line is only 0.11 mA, which will be insufficient to operate the relay. The input current will be 0.195 A, practically all of which will be lost to leakage. These examples demonstrate quite clearly the importance of controlling leakage.