Electricity (Kapp)/Chapter 3

by Gisbert Johann Eduard Kapp

CHAPTER III

ON POTENTIAL

In the first chapter we investigated in a general way the force acting between two bodies, and we found that this force may be expressed by a mathematical formula which is the same for real masses, electricity and magnetism. The units as regards length, time and force may be the same in all cases, but the units in which we express the amount of active matter producing the force must naturally be different in each case. We have also seen that the nature of the medium across which the force acts is immaterial in the case of gravitation, but may modify the force in the case of electricity and magnetism. Let us now, without for the moment specifying any particular kind of active matter, assume that one of the bodies contains a large amount of active matter, say $M$ units, and that the other contains one unit only. For brevity I shall call them the large and the small body. We can then use the small body to investigate the properties of the space surrounding the large body. In this investigation the only variables are the force and the distance from the active centre of $M$, the force being always directed either towards or from that point.

The force may thus be considered as an attribute of space, and it becomes possible, even if $M$ itself is inaccessible, to determine its magnitude and location by measuring the direction and magnitude of the force experienced by unit matter in different parts of space. It is not even necessary that the measurements should be made on unity of active matter; any convenient quantity of active matter in the small body will serve. All we need do is to reduce the measured force in the ratio of the actual amount of active matter used to its unit value. It is in this manner that astronomers, by observing disturbances in the orbit of a known star, can predict the existence of some heavenly body not yet discovered by the telescope. The astronomical problem is exceedingly complicated because of disturbances from other active masses for which allowance has to be made, but in studying the same problem as applied to electric charges no such complication need arise. We can so devise the conditions of the experiment that no other force than that acting between the large and the small body is present. The condition that one body should contain a charge large in comparison with the other is not essential, only convenient, as it obviates the necessity of making mathematical corrections which would be necessary if the small body contained a charge of the same order of magnitude. In this case the assumption that the charge distributed over the surface of a sphere acts in the same way as if it were concentrated in the centre is no longer strictly true for small distances, so that certain corrections become necessary.

The first physicist who investigated quantitatively the action of electric and magnetic forces across space was Coulomb, who towards the end of the eighteenth century invented for this purpose an instrument known as the torsion balance. As applied to electric measurements, it consists essentially of a very light scale beam made of sealing-wax and glass fibres, and suspended horizontally from a thin wire attached to its middle. One end of the beam carries a gilded pith ball, and the other a mica disc as a counterweight. The beam cannot dip either way, but it can be set into different angular positions by giving a twist to the upper end of the suspending wire, or if the upper end of the wire is held in a suitable clamp, it may set itself into an angular position in accordance with any electric force acting on the pith ball. The wire is clamped in a so-called torsion head at the upper end of a glass tube. By means of the torsion head any desirable amount of twist can be given to the upper end of the wire and read off on a circular scale, whilst the angular position assumed by the beam is indicated on a second scale placed at the level of the beam. To protect the apparatus from air-currents the beam is enclosed in a cylindrical glass vessel. In making experiments with static charges it is important to minimise as far as possible dissipation of the charges through the air, and for this reason precaution should be taken to keep the air dry. This is done by placing into the glass vessel a saucer with chlorate of potassium. Any moisture originally in the air is thereby extracted and rendered harmless. The cover of the vessel has an aperture through which is lowered a second gilded pith ball so as just to touch that on the scale beam.

If now an electric charge is imparted to the two balls by touching the connecting wire of the fixed ball with a charged body, the beam is deflected, and the deflecting force can be calculated from the angular position at which the beam comes to rest. By twisting the torsion head the balls can be brought nearer, and a new position of equilibrium obtained. Observations of the deflection, with different amounts of twist of the torsion head, are taken, and from these it is possible to calibrate the balance, that is, mark out the scale, and then use the calibration for the exact measurement of the repulsive forces acting between the balls. Coulomb was thus able, by means of his torsion balance, to establish the law of electric action at a distance.

We may look on the torsion balance as the practical way of making the experiment described in Chapter I. There is, however, this difference. The purely experimental part of the work with the torsion balance is quite simple, and free from external disturbing influences, but the mathematical investigation of the experimental results is complicated. On the other hand, the mathematics of the experiment described in Chapter I are quite simple and elementary, but the practical carrying out of such an experiment would be very difficult and costly. As I am, however, not concerned with any actual experiment, but with the explanation of first principles, I prefer to base this explanation on the mathematically simple but technically impracticable experiment rather than on an experiment which, although easy to perform, requires a complicated mathematical interpretation.

Let us then revert to the hypothetical experiment of a large electrified sphere suspended by a silk thread in the middle of a large room. Let the sphere contain a charge of $Q$ electrostatic units of positive electricity. Let the small sphere contain unit positive charge, and let us assume that we may place this at, and measure accurately the force at, any point. We shall then find that the general law of action through space holds good if we measure distances between the centres of the spheres. This means that the distributed charge on each sphere acts as if it were concentrated in its centre, a fact which can also be proved mathematically, starting from the general law. The repelling force on unit charge is given in dynes by the expression

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

This force diminishes rapidly as we increase the distance; at 10 times the distance it is ${\frac {1}{100}}$ at 100 times the distance it is ${\frac {1}{10000}}$ the original value. Obviously if the room is large enough it may be practically zero close to the wall, and yet quite sensible within a foot or so of the large sphere.

Let us assume that somehow or other we pick up a pith ball charged with unit positive electricity and carry it along any path to some point near the sphere. All the way we experience a repelling force, small at first, but rapidly increasing as we approach to the final position. In overcoming this repelling force we must impress mechanical energy on the pith ball, and the energy thus stored can again be recovered in letting the pith ball recede and perform mechanical work by overcoming some opposing force, so regulated that it balances at any point the repelling force of electricity.

We need not concern ourselves with the mechanism by which such a process could be carried out, since the whole experiment is only hypothetical and merely intended to illustrate principles. Our unit charge then is a carrier of energy, or a means of storing energy; and the amount of energy stored will depend on the charge on the sphere, the medium in which the approach takes place, and the distance from the centre of the sphere at which the approach is arrested. Thus to every point of space surrounding the sphere corresponds a definite amount of energy. The nearer the point is to the surface of the sphere, the greater is the amount of energy required to bring unit charge to that point. By moving the pith ball nearer to the sphere we must expend energy, that is, store it; by allowing it to move farther away we obtain energy, that is, we diminish the amount stored. If we move the pith ball round the sphere, taking care to keep at the same distance, we neither expend nor receive energy. In this case the movement takes place everywhere at right angles to the direction of the force, and consequently no work can be done. Our pith ball is only potent to give up energy if allowed to recede from the sphere in obedience to its repelling force, and the measure of this "potency," or, as we may shortly term it, the "potential," is a measure of the total energy which the pith ball yields if allowed to move from the point in question to a point so far away that the force has dwindled to zero—in mathematical language, to a point infinitely distant. The potential has a definite value for every point of the space surrounding the charged sphere. We may thus define it: The potential at any point of space is the energy required to bring unit positive charge from infinite distance to that point.

We have yet to find a mathematical expression for the potential. To do so we shall assume the approach to the sphere to take place in a straight line. It is quite permissible to restrict the movement to this condition, for if the shape of the path made any difference to the energy expended on approach and recovered on recession, we should be able to construct a perpetual motion machine, as may be easily seen from the following consideration: Imagine that a path of approach could be found which required a smaller expenditure of energy than can be recovered if the pith ball is constrained to follow on its outward journey some other path, then we could by a suitable sequence of the two motions create energy. We know that the creation of energy is impossible, and we must therefore conclude that all paths are equivalent as far as the potential is concerned. We are thus justified to take that shape of path which lends itself most easily to a mathematical investigation, and that is the straight line.

Let us now subdivide the straight line, along which the movement takes place, into a very large number of little bits, each in itself so small that we may neglect any change of the repelling force within the two ends of this little bit. The force varies by a small amount from bit to bit, but within the limits of one bit or small step on the journey we consider it constant. Such a conception is quite permissible if we take the steps or elements of the path small enough. The energy corresponding to each elemental part of the journey is the product of the length of the element divided by the square of the distance to the centre of the sphere, and multiplied by the charge on it. To each step thus corresponds a little bit of the total energy, and by adding up all these little bits of energy we get the potential. It would be very laborious to actually map out the whole of the journey in this way and make the innumerable calculations here indicated. Fortunately there is no necessity for all this arithmetical work. By the application of a mathematical method known as the infinitesimal calculus we are able to arrive at the result in a very simple way by one operation. The result is

$V={\frac {Q}{D}}$

where $V$ is the potential in dyne-centimetres, $Q$ the charge on the sphere in electrostatic units, and $D$ the distance of the point from the centre of the sphere in centimetres at which the approaching motion has terminated.

The reader should note that the conception of a person actually carrying a body containing unit charge in his hand, and approaching it to the sphere, is merely introduced as illustrating a mathematical relation between certain quantities, and must in no ways be taken literally. The formula only says that the potential is an attribute of the particular point $A$ in space distant $D$ cm. from the centre of the active mass $Q$. In another point, $A_{1}$ distant $D_{1}$ cm., the potential will have a different value, say $V_{1}$ If $A$ is nearer to the active centre than $A_{1}$, then $V$ will be greater than $V_{1}$, and we may therefore speak of a potential difference $V-V_{1}$ existing between the points $A$ and $A_{1}$. Or in symbols

$V-V_{1}=Q\left({\frac {1}{D}}-{\frac {1}{D_{1}}}\right)$

Since $D$ is smaller than $D_{1}$, the potential difference is positive. We must expend energy in bringing the unit positive charge, and in fact any positive charge, from $A_{1}$ to $A$. Conversely, the energy thus stored can be recovered if we allow the unit charge to recede from $A$ to $A_{1}$, which it will do under the repelling force from the active mass $Q$. Positive electricity, then, tends to move from the point of higher to that of lower potential.

This is self-evident; but how does the matter stand if the sphere is charged with negative electricity? We have then not repulsion, but attraction of the unit charge. The force has changed sign. The potentials at $A$ and $A_{1}$ are both negative, but that at $A$ is more negative than that at $A_{1}$. Now by referring both to the same datum line we may also say the potential at $A_{1}$ is positive as compared to that at $A$. To make this matter clear, let me illustrate by substituting height for potential: On a tableland 2000 ft. above sea-level there is a mountain 500 ft. high. At the foot of the mountain a shaft is sunk 500 ft. deep. Referring vertical distances to the level of the plain we say the level of the mountain-top is +500 ft. and the level of the bottom of the mine-shaft is —500 ft.; but if we refer all heights to the sea-level we would give the mountain-top as +2500 ft. and the bottom of the shaft as +1500 ft. Both levels (potentials) are positive, but the mountain is more positive than the mine.

Let us now revert to our positively charged sphere. At infinite distance the potential is zero, and as we approach the sphere it becomes positive and grows in value inversely as the distance diminishes. Its greatest possible value is at the least possible distance, which is on its surface. The maximum value, the potential of the sphere, is on its own surface, and is numerically given by

$V={\frac {Q}{R}}$

where $R$ denotes the radius of the sphere in cm. For a negatively charged sphere the potential at infinite distance is also zero, and on its surface it is

$V=-{\frac {Q}{R}}$

A unit charge free to move will therefore fly from infinity towards the sphere and right on to it. If the unit charge is carried on some conductor having ponderable mass, this conductor would strike the surface of the sphere with a certain velocity. It is easy to determine this, since we know the total energy (namely, the potential difference between infinity and the distance $R$), which, during the flight of this projectile, has been stored in it in the shape of kinetic energy. By a well-known law of mechanics the kinetic energy stored in a projectile is given by the product of half its mass and the square of the velocity. Since the mass and energy are known, the velocity can be calculated.

Let us apply, by way of illustration, this principle of equivalence between potential and kinetic energy to the calculation of the velocity with which a meteorite strikes our earth. The potential of gravity of the earth on a point on its surface is

$V={\frac {fM}{R}}$

where $M$ is the mass of the earth and $R$ its radius. The energy stored in a meteorite of mass $m$ is therefore

$E={\frac {fMm}{R}}$

which may also be written in the form

$E={\frac {fMm}{R^{2}}}R$

But ${\frac {fMm}{R^{2}}}$ is nothing else than the weight of the mass $m$, and we thus find that the kinetic energy of the meteorite is the product of its weight multiplied by the radius of the earth. Adopting the engineer's unit of energy as the metre-kilogram, and the mass unit as that mass which weighs 9.81 kg., we must take the radius of the earth in metres, and shall get the velocity $v$ in metres per second. Since the mass of our meteorite is supposed to be unity, we have the equation

$E=9.81R$

from which we find ${\frac {1}{2}}v^{2}=9.81\times 636000$

$v=11150$

The meteorite will strike the earth with a velocity of 11.15 kilometres a second; in reality a little less, because of the resistance of the air.

This digression has been inserted to show the application of the potential theory to a purely mechanical problem. Let us now return to the electrical aspect of this theory. We have a large sphere, charged with $Q$ units of positive electricity, and suspended in the middle of a large room. The potential difference between any point of the wall and the surface of the sphere is numerically equal to the energy required to bring a unit of positive electricity from the wall to the sphere. The wall of the room being in contact with the earth, both must be considered as at the same potential, and if we arbitrarily fix this as zero (which is evidently permissible since we deal with potential differences and may take our datum line where we like), then the energy expended is the absolute potential of the sphere. We may also now drop the conception of an immensely large room, and assume the sphere suspended in a room of any size, or even in the open. This does not mean that it will in all cases be equally easy to give the sphere the same charge $Q$ irrespective of the surroundings; but it means that for the same charge $Q$ the potential on the surface will be the same whatever the surroundings may be. Thus we may imagine the sphere charged in the room to the potential

$V={\frac {Q}{R}}$

If now we knock down the walls, or carry the sphere into another room or into the open, there will be no change in its potential provided that we can avoid loss of charge by dispersion. Now how are we to give the charge $Q$ to our sphere? We cannot pick up positive units of electricity from the ground as if they were pebbles and carry them by hand to the sphere; we must proceed in a different way. Let us then take a frictional machine and connect its negative wire to the ground, and the positive to the sphere. If the machine is worked, it will push negative electricity into the ground and positive on to the sphere. In other words, a charging current will flow along that wire, and more and more electricity will accumulate on the sphere the longer the machine is at work. There is, however, a limit; beyond which the process of charging cannot go. At first, it is easy enough to push electricity on to the sphere, because there is only a little quantity there which repels the influx of new units, but as the charge proceeds the quantity accumulated grows, the potential grows, and it requires more and more energy to bring every single unit on to the sphere. Finally, a point is reached when the pushing force of the machine, or, as we term it technically, its electromotive force, is just able to balance the repelling force of the charge accumulated, but not able to add a single unit. Thus a state of equilibrium is reached, and the charging process has come to an end. If we want to charge still a little more electricity on the sphere, we must increase the electromotive force, or e.m.f. of the machine, by working it quicker; this will again raise the potential, but a point must eventually be reached when e.m.f. and potential again balance.

We have thus, as the limiting condition of the process of charging, equality between potential and e.m.f. It may be objected that this statement cannot have any physical meaning, because we are comparing two things which by their nature are different. Potential is of the nature of energy, whereas e.m.f. is only one of the factors which make up energy. When we pay our electric light bill, we pay, really, for energy, and not for current by itself; nor do we pay for e.m.f. by itself; nor for the product of the two. What we pay for is the product of three things, namely, current, e.m.f. and time. The electricity metre, which says how much we have to pay, takes account of all three factors, and gives the energy as the product of amperes, volts, seconds, or, as a mere matter of convenience, it gives it in kilovoltampere hours, the unit legalised by Act of Parliament, and known as the "Board of Trade Unit" of electrical energy. As a matter of strict logic it is fore not permissible to equate potential and e.m.f., but it becomes permissible as a numerical proposition the moment we adopt such a unit for the current, that the product of unit current and unit time equals unit charge in the same system as that adopted in expressing the charge on the sphere. Thus a current of $i$ such units, flowing for $t$ seconds, corresponds to a charge of $q$ units. If the electromotive force required to push these $q$ units on to the sphere is denoted by $e$ units, then the energy expended is

$e\times i\times t=eq$

On the other hand, we know from the definition of the potential that the energy required to bring $q$ units from the wall of the room to the sphere requires the energy $Vq$; and hence it is evident that $e$ and $V$ are numerically equal. By adopting the system of units here explained, we are therefore justified in considering e.m.f. and potential as numerically equal, and can write

$e={\frac {Q}{R}}$or$Q=eR$

The charge that can be accumulated on a sphere is the product of its radius and the e.m.f. developed by the electric machine. It has been pointed out that the conception of a very large room, in which the sphere is suspended, is not necessary to our arguments. We may reduce the room to any extent, and still the definition of potential, or, as we now see, that of e.m.f., holds good. It is the energy required to carry unit positive charge from the wall to the sphere. Since there is no restriction to the size of the room, other than there must not be actual contact between wall and sphere, let the room shrink until it has become merely a spherical shell surrounding the sphere closely, the distance being a very small length $\delta$. Let this be a mere clearance space, so small in comparison with the radius of the sphere that the repelling force of the charge $Q$ on our unit has sensibly the same value at any point within this very narrow space. The repelling force is

${\frac {Q}{R^{2}}}$

and since the product of force and distance traversed is energy, we find

$e=Q{\frac {\delta }{R^{2}}}$

and

$Q=e{\frac {R^{2}}{\delta }}$

instead of $eR$ as found previously, when the sphere was in a large room or in the open. Comparing now the two cases, namely, the sphere in the open and the sphere closely surrounded by a metallic envelope, it will be seen that to get the same charge on the spheres is not equally easy. The sphere hanging free requires the application of a much larger e.m.f. than the sphere within an envelope, or, to put it another way, the sphere with an envelope will, under the application of the same e.m.f., acquire a much greater charge than the sphere hanging free in space. The capacity of the sphere for taking a charge has been increased. This reasoning leads us to the conception of capacity as a property of the configuration of metallic bodies. We define capacity as the ratio of charge divided by e.m.f. Using the symbol $C$ for capacity, the definition mathematically expressed is

$C={\frac {Q}{e}}$ and $Q=eC$

Since we found previously that $Q=eR$, it follows that the capacity of a sphere in the open is given by the length of its radius expressed in cm. For the sphere with its envelope the capacity is

$C={\frac {R^{2}}{\delta }}$

The ratio of the square of a length and a length is again a length, so that we have in both cases the capacity expressed as so many cm.

In deducing the conception of capacity we assumed that the conductor has a spherical shape, but obviously if, instead of suspending a sphere and charging it, we had suspended a metallic body of any shape and forced electricity on to it by the frictional machine, it would have acquired some charge proportional to the e.m.f. applied. The body of irregular shape also has capacity, only we may not always be able to calculate it exactly beforehand. It can, however, always be found experimentally. For this purpose we apply a known e.m.f. to charge the body and then discharge it through a special kind of measuring instrument. The instrument indicates the quantity of charge which has passed through it; and from the two measurements, namely, e.m.f. and quantity, we can determine the capacity. For certain shapes the determination of capacity, by mathematical reasoning, is quite easy. One case, namely that of the sphere, either free or in a shell, we have already treated. The case of concentric cylinders, or parallel cylinders, or a cylinder and a parallel plane is also easily treated, but it would exceed the limits of this book to enter into such details, which have more immediate interest for the cable engineer or the telegraphist. The case of two parallel plates may, however, be here given, because the derivation of a mathematical expression for the capacity is exceedingly simple. We found that the capacity of concentric spheres is given by the expression

$C={\frac {R^{2}}{\delta }}$

If we multiply nominator and denominator with $4\pi$ we do not alter the equation, so that we also may write

$C={\frac {4\pi R^{2}}{4\pi \delta }}$

$4\pi R^{2}$ is nothing else than the surface of the sphere, so that we also have

$C={\frac {S}{4\pi \delta }}$

The capacity is therefore given by the surface divided by $4\pi \delta$. The radius does no longer appear in our formula. If we assume the radius to be infinitely large, any part $S$ of the surface becomes a plane, and we thus have for the capacity of two parallel plane surfaces of $S$ square cm., distant $\delta$ cm., the expression

$C={\frac {S}{4\pi \delta }}$

which is again a length.

Bodies constructed for holding an electric charge, that is, intended to condense electricity on their surfaces, are technically termed condensers. The first condenser used by physicists was the so-called "Leyden Jar" accidentally discovered by Musschenbroek (1692-1761), Professor of Physics in Leyden, Holland. In the eighteenth century electricity was considered a "fluid," and Musschenbroek attempted to collect some of this fluid in a glass filled with water. He held the glass in the hand, and electrified the water by a wire placed in the glass and projecting sufficiently far out so that he could touch the conductor of his frictional machine with the wire. When removing the glass and taking out the wire, he received an electric shock much more violent than he could obtain from his machine directly. In this case the water formed the inner conductor and the hand the outer shell, whilst the space between the two was filled by glass.

This form of condenser has become known under the name of Leyden Jar, and is used to this day by physicists. It consists of a glass jar coated on the inner and outer surface with tinfoil about half-way up. The uncoated part of the jar is varnished to minimise loss of charge along the surface of the glass. An improved form of Leyden jar has been designed by Mr. Mosicki, and is largely used in wireless telegraphy. The coating of tinfoil is replaced by silvering, and the shape of the glass vessel is designed with special reference to its ability to withstand very high e.m.f.'s. Whereas the ordinary jar of the physical laboratory can only be used with an e.m.f. of about 20,000 volts, the Mosicki condenser, as made for wireless telegraph stations, can be used up to an e.m.f. of 60,000 volts. Mosicki condensers are also used for the protection of electric power lines from atmospheric electricity, and from the effects of sudden electric disturbances. They act as a kind of electric buffer or elastic link, able to soften the blow which the line and machinery might otherwise receive with full force if there were, from any cause, a sudden increase in the charge on the system.

When the condenser is not subjected to a very high potential difference, the insulator separating its two surfaces or coatings need not be glass, but may be a cheaper material, such as paraffined paper. The object of using some lining between the plates is twofold. In the first place it would be technically very difficult to insure a very small intervening space without the risk that the plates come actually into contact. If the condenser is not required to have a large capacity, and especially if it is to be used as a standard of capacity for comparison with other condensers, then the intervening space between the plates may be left without filling material. Such condensers are called "air condensers." Where a condenser of larger capacity is required as a standard, then the filling-in material, the so-called "dielectric," may be mica. This, even in thin sheets, is electrically very strong, and is also an excellent insulator. It is thus possible to make the space between the plates very small, and by this means obtain a larger capacity with a given plate surface than with an air condenser. The other reason for using another material than air as a dielectric is that the material by itself has the property of increasing the capacity.

We have seen in Chapter I that the attractive force depends on the medium between the two charged surfaces. If this medium is air, the force is greatest; if it is an insulator such as oil, glass, mica or paper, it is $K$ times smaller. This means that to bring a unit of positive charge to our sphere through such a medium takes $K$ times less energy than to bring it through air. In other words, to obtain the same charge an e.m.f. $K$ times smaller is sufficient, or, if the e.m.f. is the same, the resulting charge will be $K$ times larger. Hence by using a dielectric other than air the capacity of our condenser is increased $K$ times. The following table gives the value of $K$ for some dielectric materials—

Values of the Specific Inductive Capacity K in the C.G.S. Electrostatic System

Material

Glass

Mica

Insulating Oil used in transformers

Paraffin Wax

India Rubber

Gutta Percha

Paper, as used for power cables

Paper, as used in telephone cables

Paper, paraffined, as used in condensers

Distilled Water

Pure Alcohol

K

2—10

5—6

2—1

2.3

2.2—2.8

2.5—4

2.6—3.5

2—2.5

7.2

76

26

In the old method of making condensers with paper as a dielectric the coatings or "electrodes" were sheets of tinfoil, but in the modern type of condenser developed by Mr. Mansbridge, of the British Post Office, so-called metallised paper is used interleaved with plain paper, both being paraffined. The effect of this improvement is that the bulk, weight and cost of paper condensers have been reduced to less than one tenth of what they were formerly. The capacity of any condenser is given in electrostatic c.g.s. units by the formula

$C=K{\frac {S}{4\pi \delta }}$

This unit is inconveniently small, and for practical work a much larger unit, namely, the "microfarad," has been adopted. As the name implies, the microfarad is the one millionth part of the farad, and the magnitude of the farad is given by the following definition: A condenser of one farad capacity, when charged under the e.m.f. of one volt, will store that quantity of electricity which is represented by the flow of one ampere during one second. The ratio of the electrostatic unit of capacity to the microfarad is $1$ to $9\times 10^{5}$, so that the capacity of a condenser expressed in microfarads is given by the formula

$M={\frac {K}{113}}{\frac {S}{\delta }}$

where $S$ is the surface of the dielectric in square metres, $\delta$ is the thickness of the dielectric in millimetres, and $K$ has the value given in the above table.

In developing the theory of the potential we started with the experiment of bringing unit electricity from the wall of the room to a point outside the charged sphere; and, as a limiting condition, to its surface. Beyond that we did not go. But what happens if we pass the surface and carry our unit through to the inside? A mathematical investigation shows that in this case no force at all is acting on the unit charge, and in fact on any body carrying a charge of any magnitude. Since in moving such a body about within the hollow sphere we experience no resisting force whatever, no energy is required to perform the motion, and consequently all points of the interior space must have the same potential. Any point of the inner surface of the sphere is a point in the interior, but since the surface has the potential ${\frac {Q}{R}}$ it follows that this is also the potential right through the cavity of the hollow sphere. This law that the potential at any point inside a conductor is the same as the potential on its surface, may also be proved without the aid of mathematics by the following reasoning: Imagine a conductor of any shape, and assume it at first to be solid right through. No free electricity could possibly remain in the substance of the metal, since the mutual repulsion of all the elementary charges would cause these to try to move apart as far as they can. As the carrier of these charges is metallic, that is to say, offers no resistance to the free displacement or flow of electricity, there is nothing to hinder the movement, and consequently the charge will all accumulate on the outside surface. There is, therefore, no charge in the body of the metal, and we may, without changing the electrical condition, take away the inside and leave only the merest shell, and still there can be no electrical effect produced inside the shell. We may charge such a shell with the strongest machine made, and yet in the inside not a trace of electricity can be detected.

This has first been proved experimentally by Faraday, who constructed for the purpose a large cage of wire gauze and went into it armed with the most delicate instrument for the detection of electric charges. The cage was placed on insulating supports and strongly electrified by a frictional machine. Not a trace of electrification could be detected in the interior or the inner surface of the wire gauze. The principle of the "Faraday Cage" has been applied as a protective device in various ways. Professor Artemieff, of the Moscow University, has constructed an electrical safety dress, which completely envelopes the wearer so that he is literally enclosed in a tight-fitting Faraday Cage. The dress is of metal gauze, and as long as the surface is continuous, the wearer is absolutely safe from shock. If a discharge flash from a high-tension apparatus should strike him, the charge flows through the dress to earth without doing any damage.

Another important application of the same principle is the protection of underground or submarine cables. The part of the cable that is below the ground or the sea is naturally protected against lightning strokes, but somewhere the end of the cable must be brought out to the surface of the earth and connected to some apparatus or machine. At that point both the end of the cable and the machinery are liable to be struck and must be protected. The best possible protection is to place the whole of the machinery and the apparatus connected to the cable into an iron house. It is not necessary that the walls of the house be entirely made of iron. In Milan there is such a house for the protection of the junction of the overhead power lines coming from the Alps, and joining by means of certain apparatus with the underground cable network that supplies the town with electricity. In appearance this house is not different from any of the other factory buildings of the neighbourhood; nevertheless it is a Faraday Cage. The roof has an iron lining, the stanchions are well bonded with it and with each other, and go down to the moist subsoil. Below the plastering of the walls is a heavy expanded metal lining all connected to the roof and stanchions, and the windows have iron frames also in good electrical connection with the metal walls. The building thus forms a metal shell, and affords complete protection to its contents against any electrical disturbance from outside.