Electricity (Kapp)/Chapter 1

by Gisbert Johann Eduard Kapp

ELECTRICITY

 

CHAPTER I

ON FORCES ACTING THROUGH SPACE

The conception of a force as something which pushes or pulls is familiar to every one. Equally familiar is the conception of an intervening link by which a force is transmitted from one body to another. If I pull a bucket of water out of a well the push exerted by the water on the bottom of the pail is transmitted to my hand by a very simple series of links. The bottom of the bucket pulls at its sides, these pull at the handle, the handle pulls the rope, and that finally pulls at my hand. We have here a transmission of force by links, all of which are in bodily contact. Thus far the process of transmission presents no difficulty to our conception of a force, but when we come to inquire why the water presses against the bottom of the bucket, we have no complete answer. All we can say is that the push is due to the fact that the water is heavy. This means that the water in our bucket is attracted towards the earth, but what kind of intervening link there is which transmits a force from the earth to every particle of water and every particle of bucket and rope we are quite unable to say. Everyday experience has so familiarised us with the action of gravity that we have become accustomed to simply accepting it as a fact in nature, without further inquiry as to the machinery which is instrumental in the transmission of this force through the intervening space. We simply say that gravity is a force that acts at a distance, and since by direct experiment and astronomical observation it has been found possible to formulate a mathematical expression for this force, there is, from a purely practical point of view, no need to find an explanation of the machinery by which this force is transmitted through space, whether the space be quite empty or filled with other bodies.

The confession of ignorance as to the nature of this machinery of transmission is, however, not a denial that such machinery exists; on the contrary, the conception that physical action can take place without the intervention of physical causes is repugnant to the human mind, and therefore physicists have invented the ether. By this they mean a physical something which pervades all space, whether filled by bodies or not, and this ether forms the connecting link by which forces are transmitted across space. Once we assume the existence of this physical though imponderable, that is, weightless substance, we may regard it as instrumental not only in the transmission of gravitational forces, but also of electric and magnetic forces and as the carrier of light and heat rays. There is, indeed, very strong experimental evidence that light, electricity, magnetism and all other manifestations of energy are propagated by ether vibrations. Maxwell was the first to point out that a connection of this kind exists, and by adopting his "electromagnetic theory of light" it can be proved mathematically, what has also been experimentally verified, that the speed at which a telegraph signal travels along the wire is equal to the speed of light propagation. It is extremely unlikely that such an agreement should be a mere coincidence, and we are therefore justified in assuming that the ether, although originally invented to bridge a gap in our reasoning, has nevertheless a real existence.

The acceptance of the ether as the medium of propagation of all kinds of forces across space does not explain the mechanism of its action, but, by ascribing to the ether certain properties, we are able to express in concrete figures, by the use of any convenient system of measurement, the results of experimental investigation. The general law of action at a distance has been proved by direct experiment and by astronomical observation to be as follows: Let two active masses be concentrated in two points a certain distance apart; then the force acting between them, that is, the force which is being transmitted from one point to the other by the intervention of the ether, is proportional to the product of the two masses and inversely proportional to the square of their distance. In the case of gravity this force is always attractive, that is, tending to bring the masses nearer together; in the case of electricity or magnetism it may be attractive or repulsive according to the nature of the active masses.

It should be noted that the term "active mass" is merely conventional as far as electricity or magnetism is concerned. It is not to be taken in its literal sense as applying to something which has bulk and weight. A steel bar, after being magnetised, weighs exactly the same as before, yet its ends exhibit certain properties which we may conventionally ascribe to accumulations of magnetic matter which, if brought near magnetic matter adhering to the end of some other magnetised bar, becomes "active matter" in the sense of producing either attraction or repulsion. The force is attractive if the ends of the bars brought near each other contain magnetic matter of opposite sign, and it is repulsive if the magnetism is of the same sign.

In the same way there is repulsion between two conductors both positively or both negatively electrified, and there is attraction if one is positively and the other negatively electrified. Also in this case electrification does not alter the weight of the conductor, although we may consider the electricity carried by the conductor as an "active mass" in the sense that the force of attraction or repulsion acting through space is due to it. In gravitation the force is always attractive, whilst with magnetism or electricity as active matter the force may be either attractive or repulsive; in all cases, however, the same law applies as to the action through space.

The reader should note that the above statement of this law is no complete answer to the question as to the actual magnitude of the force. Experiment only teaches us that the force is proportional to the product of the two masses divided by the square of their distance, but if we wish to state the actual magnitude of this force in a definite figure we must agree on a system of units. As far as the attractive force between ponderable masses is concerned, such units are quite familiar; we know what is meant by the mass of a pound weight, and we also know how to measure a distance. With magnetic and electric forces the matter is not so simple. A distance we can measure in any length unit, but what about the unit for the "active mass"? We have seen that it is not a mass at all in the common acceptance of this term, and it can therefore not be expressed in any unit suitable for ponderable masses. We are thus compelled to settle the magnitude of the unit by the same formula which defines the force. The conception of unit active mass may then be derived from the following condition: If two equal masses one centimetre apart act upon each other with unit force, then each of them is a unit of active mass.

The same definition of unit mass must also fit if applied to gravitational attraction, but there is a difference. We know from experimental evidence (T. Erismann, Arch. d. Science, Jan. 1911, pp. 36-45) that the attractive force of gravity is not in the least influenced by the medium which fills the intervening space. Two bodies in air attract each other with exactly the same force as in water; nor would the force be altered if we placed a wall between them. There would of course be an additional attraction between each body and the wall, but no additional force of the attraction between the bodies themselves.

With magnetic and electric forces it is different. If the force acting between two electrically charged bodies be measured, first in air and then when immersed in oil or separated by a wall of glass, we should find a decrease of force in the latter cases. In these cases the whole or part of the medium which at first was air has been replaced by some other substance with the result of an alteration in the force. We thus find that not only the magnitude of the charges and their distance, but also the physical nature of the intervening medium has an influence on the force, and the mathematical formula expressing the magnitude of the force must take account of this. We must therefore introduce into the formula a coefficient, the numerical value of which will not only depend on the system of units chosen, but also on the medium filling the space through which the force acts. We thus arrive at the following mathematical expression—

$F=f{\frac {Mm}{D^{2}}}$

where $M$ and $m$ are the two masses, $D$ is the distance and $F$ is the force, all expressed in any system of units which may be convenient for the particular case in hand. The coefficient $f$ will naturally depend on the magnitude of the units chosen; on their nature, that is, whether we deal with gravitational masses, electric charges or magnetism; and on the medium filling the space through which the force acts.

We do not know what electricity is any more than we know what magnetism is; all we know is that they are not of the nature of ponderable masses, and that under certain circumstances they may become the vehicle for the transmission of energy in a similar manner to the ether itself. We might, in fact, consider them as ethereal manifestations without any attempt to explain the exact nature and mechanism of these manifestations. Such a conception is quite compatible with the practical use of our general formula; it simply means that we must look upon $f$ as a kind of ethereal coefficient, the numerical value of which has to be found experimentally.

The conception of an ethereal coefficient, stated in this general way, is perhaps a little difficult to grasp. To make the matter clear I start by applying it to the familiar phenomenon of gravity, and then proceed to investigate the more unfamiliar phenomena of electric and magnetic forces. At the outset we must agree on the units we are going to use in giving a numerical expression to the attractive force between two bodies. If one of these bodies is the earth and the other a stone, this force is simply the weight of the stone. If the two bodies are the earth and the moon, the force is that which just balances the centrifugal force experienced by the moon in flying round in her orbit. The formula for the attraction given on p. 14 refers to masses concentrated in points, and it might thus at first sight appear that its application to such a bulky object as our earth is not permissible. Neither moon nor earth can be considered infinitely small as compared to their distance. Nevertheless we may use the formula, for a mathematical investigation shows that in the case of spheres the summarised effects of all mass particles is the same as if the total mass were concentrated in the centre. Astronomy gives us all the data required for our calculation; all we need to get a definite numerical result is to agree on a definite system of units in which to express a force.

The definition of a mechanical force is: something which produces acceleration of a ponderable mass. Acceleration is the rate at which speed increases in respect of time. Thus, if an electric tramcar starting from rest attains its full speed of 24 miles per hour in the time of 20 seconds, its average acceleration is 1.2 miles per hour in each succeeding second, or "1.2 miles per hour per second." If instead of giving the speed in miles per hour we give it in metres per second, the acceleration of this car would be 0.535 metres per second per second. Taking the metre as the unit of length, the second as the unit of time, and the kilogram as the unit of force, we have thereby also settled what the unit of mass must be.

A stone weighing one kilogram, and in fact any stone, when starting to fall from rest, acquires in the first second a velocity of 9.81 metres per second. Its acceleration or gain of speed is therefore 9.81 metres per second per second. Since the force which pulls the stone towards the earth is one kilogram, and since in any system of units the product of mass and acceleration represents force, the mass of our stone is the 9.81th part of unit mass. Therefore in the system of units chosen in this example, a stone weighing 9.81 kilograms has unit mass.

Expressing now the known masses of earth and moon in this system, and remembering that the average radius of the moon's orbit is 385,080 km., and the length of the month 27 days 7 hours 43 minutes, it is easy to calculate the centrifugal force from the well-known relation between mass, radius and time of revolution. The result is in round numbers 20,000 million million tons. It is difficult to grasp the meaning of so prodigious a force, but we may get an idea of its magnitude by calculating the diameter of a cylindrical bar of the strongest steel able to just support the application of such a force longitudinally. It comes out at 320 miles in diameter. Having thus found the force, we can now determine the numerical value of the ethereal coefficient of mass attraction for the particular units chosen, namely, the metre as the unit of length, the kilogram as the unit of force, and a mass of 9.81 kilogram weight as the unit of mass. The result of this calculation is

$f={\frac {6.47}{10^{10}}}$

The symbol $10^{10}$ means that 10 is to be multiplied 10 times with itself. The numerical expression for $f$ may also be written in the form

$f=6.47\times 10^{-10}$

where the minus sign of the exponent signifies that 6.47 is not to be multiplied, but divided by $10^{10}$.

In the above example showing how the ethereal coefficient may be determined for any arbitrary system of units, I have taken as the unit of mass the mass of 9.81 kg.; this was merely done as a matter of convenience, so as to be able to regard the kg. as the unit of force, as is customary in engineering. It is, however, more in consonance with first scientific principles not to fix arbitrarily a unit for the force, but derive it from the three fundamental units of mass, length and time, since every physical quantity may be expressed by reference to these three units. If we choose the centimetre as the unit of length, the gram as the unit of mass and the second as the unit of time, we adopt what physicists call the centimetre-gram-second system of measurement. For this the abbreviated designation c.g.s. is customary. In this system force is a so-called derived unit, namely that force which, acting steadily in the same direction for a second on the mass of one gram, will give it an acceleration of one cm. per second per second. This unit is called the dyne, and from what has been said above it is obvious that 981 dynes go to one gram, or 981,000 dynes (approximately one million dynes) are equivalent to the kg. If we now repeat the calculation, using the c.g.s. system, we get the force in dynes if we express in the general formula

$F=f{\frac {Mm}{D^{2}}}$

the masses in grams and the distance in cm. The ethereal coefficient then has the value

$f=6.6\times 10^{-8}$

The knowledge of this coefficient enables us to determine for any two bodies the attractive force if their masses, configuration and relative position are given. For spheres the calculation is quite simple, but for bodies of more complicated shape it is very difficult, and sometimes only possible in rough approximation. It would, for instance, hardly be possible to accurately calculate the mass attraction between two Dreadnoughts lying side by side, but by using the general formula and the coefficient $f$ as here determined we get as a rough approximation a force of $7$ lb.

It is, of course, out of the question to check such a calculation by direct experiment, since disturbing causes, such as the slightest breath of wind striking the side of the ship, will produce a disturbing force many times greater than the force to be measured. If, however, we could eliminate all disturbing forces, then a direct determination of $f$, quite independent of astronomical observation, would be possible. Such determinations have been made by Cavendish, Maskelyne, Airy and others, the most recent being Poynting's, carried out in the Birmingham University. Professor Poynting has measured, by means of an exceedingly delicate balance, the attractive force between two lead spheres of known mass, and has thus determined $f$, and from this value he found the mass of the earth to be $6.6\times 10^{27}$ grams. In popular language, he has weighed the earth.

The reader may perhaps ask what all this has to do with electricity. Nothing directly. I have merely introduced the subject of gravitation, which is familiar to all, as a starting-point, so as to familiarise the reader with the conception of the ethereal coefficient; and I now go back to the consideration of electric and magnetic forces acting across space.

I assume that the reader is familiar with the usual textbook explanation of how bodies may be electrified, or, as it is also termed, charged with electricity. Imagine then that we have given electric charges to two spheres which are suspended from silk threads. Such suspension is necessary, for if we were to handle the spheres or lay them on to the table their charges would leak away; if we wish a body to preserve its charge for a sensible time we must support it by an insulator—such as silk, glass, mica, ebonite, which does not allow electricity to flow along or through it. Metals offer a very easy path for the flow of electricity, and are therefore called conductors. There is no sharp line of demarcation between insulators and conductors. Dry wood, for instance, is not a perfect insulator; and when damp it is not a perfect conductor. Dry air at atmospheric pressure is almost a perfect insulator, but if rarefied or at high temperature it becomes more or less of a conductor. All metals are conductors, but they are not all equally good conductors. Mercury is not so good a conductor as iron, iron is not so good as copper, and silver is still a slightly better conductor than copper. The difference between the two last-named metals is, however, not great enough to justify commercially the use of silver instead of copper wire in the construction of electrical machinery. For the present we need not inquire further into any fine gradations between conductors and insulators.

We assume that the silk threads used for the suspension of the charged spheres and the air surrounding them are perfect insulators, so that the spheres will retain their charges as long as they do not come into actual contact with each other or some other conductor. If we suspend the spheres near each other we find that they do not hang plumb. If they are both positively or both negatively charged the distance between their centres will be greater than the distance between the points of suspension. If the charges are of opposite sign, the opposite will be the case. This shows respectively that a repulsive force and an attractive force is causing the deviation from the vertical. If we know the weight of the spheres, measure their distance and the angle of deviation of the suspending threads from the vertical, the force acting between the two charges can be calculated from well-known mechanical principles in quite a simple manner.

It is, however, necessary to avoid disturbing influences. The spheres must hang in the middle of a very large room, so that floor, ceiling and walls are far removed, and we must make the observations by telescope, as otherwise the presence of the body of the observer near the spheres would disturb the electrical equilibrium. I need hardly say that such an experiment would be expensive and difficult; in reality it need not be made, as there are other far more practical methods of investigation available, but it is convenient to imagine such an experiment, because it will enable me to explain in the simplest possible way certain first principles. Suppose then that we are not deterred by questions of cost and have overcome all the technical difficulties. Let us first, without altering the amount of charge on each sphere, merely shift their positions so as to get different distances. Measuring the force in each case, we will find that this force varies inversely as the square of the distance. We have thus verified part of our general equation. Now let us retain one particular distance and change the amount of charge, first on one sphere only and then on both. We find that the force varies directly as the product of the two charges. This experiment confirms the rest of the equation.

Writing now $Q$ and $q$ for the quantity of charge on each sphere the general equation takes the form

$F=f{\frac {Qq}{D^{2}}}$

In the case of both spheres containing equal charges this may also be written as an equation between the product of $F$ and $D^{2}$ on the one hand, and $f$ and $Q^{2}$ on the other—

$F\times D^{2}=f\times Q^{2}$

Suppose we have succeeded in so adjusting the charges that $F\times D^{2}$ is unity; this might be the case for $D=10cm$. and $F={\frac {1}{100}}$ dyne, or $D=1cm$. and $F=1$ dyne. The product $f\times Q^{2}$ will then also be unity. All that our experiment tells us is that the product of two things is unity, but it does not tell us the separate value of each of the two things, which is only another way of saying that we do not know and probably shall never know what electricity really is any more than we can know the real nature and value of the ethereal coefficient. We can, however, choose one of the factors, and then the other is also determined. If we adopt the definition of unit mass given on p. 13, then $Q$ is 1 and $Q^{2}$ is also 1. From this it follows that $f$ is also 1, and our general formula simplifies to

$F={\frac {Qq}{D^{2}}}$

In adopting this formula we have arbitrarily settled the magnitude of the unit of electric quantity. It is such a quantity of charge as will give the force of one dyne, if acting on an equal charge at a distance of one cm.

It will be noticed that the train of reasoning followed here is different from that we followed in the case of gravitation. There we started by adopting a particular quantity of ponderable matter as the unit, namely the gram. This is the obvious way, because we know what the mass of a gram is and we can reproduce it at any time. A cubic cm. of water at four degrees C. has the mass of one gram. Having thus settled the magnitude of the mass unit we determined the numerical value of the ethereal coefficient. In the electrical case we settle arbitrarily the value of the ethereal coefficient as unity, and determine on this basis the magnitude of unit electric quantity. In our experiment the spheres are at rest, there is no flow of electricity, and the system is in static equilibrium.

The unit of charge thus denned is therefore called the electrostatic unit of electric quantity in the c.g.s. system. In our experiment the room was filled with air. Let us now fill the room with oil. Since oil is an excellent insulator the spheres will retain their charges, but we shall observe a diminution of the force. The charge on each sphere has not altered, but the force acting between them has become smaller. We have settled the magnitude of unit quantity in such way that the coefficient $f$ in air shall be unity, but after filling the room with oil we find that this coefficient is only say ${\frac {1}{2}}$. Whether it is exactly ${\frac {1}{2}}$ or some other fraction depends on the particular kind of oil used. To treat the matter quite generally let us call the fraction ${\frac {1}{K}}$. The force will now be expressed by the formula

$F={\frac {1}{K}}{\frac {Qq}{D^{2}}}$

$K$ being a number depending on the medium in which the spheres are suspended. This numeric indicates the degree of attenuation of the force brought about by the presence of an insulating body between the spheres. This body, which separates the two electrified bodies, is called the dielectric. To bring back the force to its old value we must increase the charges. By using a dielectric we have enabled the spheres to hold a greater charge without exerting on each other a greater force. We have increased their capacity for storing a charge, and for this reason $K$ is called the specific inductive capacity of the medium, or also the dielectric constant of the medium. The value of $K$ is about $2$ for oil, $2$ to $3$ for paper, $6$ for mica, and may go up to as much as $10$ for glass. The larger $K$, the greater is the charge with a given force pushing the electricity on to the conductor. This force must, however, not be confounded with the mechanical force of attraction or repulsion with which we have been concerned hitherto; there is a relation between the two, as will be explained in Chapter III, but they are not identical.

The same reasoning as above applied to electric attraction and repulsion may also be applied to forces produced by magnetism, but if we attempt an experimental verification of the general law of forces acting through space we encounter some difficulty. When dealing with electricity it is quite easy to isolate a positive from a negative charge each on its own conductor, or, as we may also term it, it is possible to accumulate free electricity of one sign on a conductor. It is not possible to accumulate only north magnetic matter, or only south magnetic matter on one piece of steel. We always get magnetic matter of both signs simultaneously on the steel. If this has the form of a bar we can, by stroking it with a loadstone, make one end of the bar a north pole and the other a south pole, but if we break the bar in halves we do not get one half all north and the other all south. Each half again shows north at one end and south at the other. In experimenting on magnetic forces we are, therefore, always disturbed by the presence of magnetic matter of the opposite polarity. Another difficulty lies in this, that the magnetic matter is spread over the whole of the bar—more dense at the ends, but still of sensible density at points towards the middle. Thus it becomes difficult to estimate the average distance of action. These difficulties are so great that the same methods of experimenting, which we supposed to be used when investigating electric forces, become quite impracticable, and other methods have to be devised.

These methods are based on the conception of the magnetic moment, that is, the product of the distance of poles into their strength. Any physical magnet can then be considered as a bundle of mathematical magnets, each carrying magnetic matter only at the extreme ends. We observe experimentally the summarised effect of all these elementary magnets, and by mathematical reasoning we are able to deduce the law under which magnetic forces act across space. Experiment shows that the law stated in the beginning of this chapter also holds good for magnetic forces. Moreover, the magnitude of the unit of magnetism may be determined in the same way. If we find that two equally strong poles placed one cm. apart exert on each other a force of one dyne, then each contains unit of magnetic matter. This definition again means that we have arbitrarily assumed the ethereal coefficient of air to be unity.

When making the experiment with electric charges we found that by filling the space between the active charges with a substance such as oil or glass, the force was diminished. No such effect is observable with magnets. We may put them under oil or water, or we may put a sheet of glass between them, and we shall find precisely the same force. If, however, we immerse them in liquid oxygen there will be a decrease of force, and if such a thing as an iron atmosphere were possible, the decrease in such an atmosphere would be very great. We may therefore say that the ethereal coefficient for magnetic forces is unity for air, oil, wood and any so-called non-magnetic substance; and smaller than unity for magnetic substances such as iron, nickel and cobalt. If we try the experiment with a plate of bismuth we shall find a slight increase of the force, showing that the magnetic ethereal coefficient for bismuth is a shade greater than unity, the value assumed for air. For iron it is enormously smaller. We may say that iron is more permeable to the transmission of magnetic forces than air or brass or wood, and the degree to which this transmission is facilitated is called the magnetic permeability. In physical and engineering calculation the permeability (which is nothing else than the reciprocal of the ethereal coefficient) is indicated by the Greek symbol $\mu$, so that for magnetic forces the general equation for action over a distance $D$ becomes

$F={\frac {1}{\mu }}{\frac {Mm}{D^{2}}}$

The suitability of any particular kind of iron for use in electrical machinery depends on the value of $\mu$, and the exact determination of this ethereal coefficient thus becomes a matter of practical importance. In making such determinations advantage is taken of certain relations which exist between electricity in the flowing state, commonly called electric currents, and magnetic effects. Since any flowing current represents energy, that is to say, is a dynamic phenomenon, such experiments have an electrodynamic character, and the unit of magnetic matter as defined above, under the arbitrary assumption that the magnetic ethereal coefficient for air is unity, is called the c.g.s. unit of magnetism in the electrodynamic system.

We have thus two different systems of measurement, the electrostatic and the electrodynamic. They have been adopted as a matter of convenience in order to be able to regard in both the ethereal coefficient of air as unity. The result of this is that the absolute magnitude of electric quantity in the two systems is very different. In the electrostatic system unit quantity is exceedingly small as compared to the amount of electricity which goes to make up one unit of charge in the electrodynamic system. It requires 30,000 millions electrostatic units to make up one electrodynamic (or, as it is also called, electromagnetic) unit of electricity. The speed of light is 30,000 millions cm. per second. It is highly improbable that the agreement between the speed of light and the numerical ratio between the units should be a mere coincidence; but if it is not, then the ratio between the units is not merely a numeric but something which has a particular character, namely, the character of velocity, that is, a length divided by a time. Further, if we rule out the idea of a merely accidental agreement between two numbers, we are driven to the conclusion that the ether is the actual carrier of force and energy; and this is the conception on which the modern science of electricity and magnetism, and in fact the whole structure of electrical engineering, is founded.