Chapter I: Front Matter (1)

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Transcriber's notes:

(1) Numbers following letters (without space) like C2 were originally
printed in subscript. Letter subscripts are preceded by an
underscore, like C_n.

(2) Characters following a carat (^) were printed in superscript.

(3) Side-notes were relocated to function as titles of their respective
paragraphs.

(4) Macrons and breves above letters and dots below letters were not
inserted.

(5) [root] stands for the root symbol; [int] for integral; [alpha],
[beta], etc. for greek letters.

(6) The following typographical errors have been corrected:

ARTICLE ELECTROTHERAPEUTICS: "This is the normal reaction of the
nerve to faradism." 'normal' amended from 'mormal'.

ARTICLE ELECTROTHERAPEUTICS: "If the current be suddenly reversed,
so that what was the anode becomes the kathode, a stronger
contraction is obtained than by simply making and breaking the
current." 'stronger' amended from 'stonger'.

ARTICLE ELECTROTHERAPEUTICS: "Thus RD is present in infantile
paralysis, acute neuritis, &c., but absent in progressive muscular
atrophy where the wasting of nerve and muscle takes place extremely
slowly." 'progressive' amended from 'progessive'.

ARTICLE ELECTROTHERAPEUTICS: "In the medical application of these
facts it must be remembered that when an ion is introduced into the
body by electrolysis, it is probably forced into the actual
cellular constituents of the body ..." 'probably' amended from
'propably'.

ARTICLE ELGINSHIRE: "Findhorn has been twice visited by
calamities." 'visited' amended from 'vsited'.

ARTICLE ELIOT, GEORGE: "Romola, which is what is called an
historical novel, owes its vitality not to the portraits of
Savonarola or of the heroine ..." 'its' amended from 'it'.

ARTICLE ELZEVIR: "... a chronological list and detailed description
of all works printed by them, their various typographical marks
..." 'works' amended from 'words'.

ARTICLE EMBALMING : "... it being held by the Egyptians a
detestable thing to commit any violence or inflict a wound on the
body." 'detestable' amended from 'destestable'.

ARTICLE EMBRYOLOGY: "In this manner the germinal disc has become
converted into the blastoderm ..." 'become' amended from' beecome'.

ARTICLE EMBRYOLOGY: "For instance, in Vertebrata this tissue gives
rise to nervous tissue, blood-vessels, renal tubules ..." 'rise'
amended from 'tise'.

ARTICLE EMIN PASHA: "This last step followed upon his receipt of a
letter from Nubar Pasha, informing him that it was impossible for
the Egyptian government to send him help ..." 'from' amended from
'fom'.

ARTICLE EMPIRE: "S. Riezler, Die literarischen Widersacher der
Papste zur Zeit Ludwigs des Baiers (1874); ..." 'zur' amended from
'sur'.

ARTICLE EMPLOYERS' LIABILITY: "But it expressly applies to seamen."
'expressly' amended from 'expressely'.

ARTICLE EMPLOYERS' LIABILITY: "Compensation becomes payable after
the expiration of sixty days from the date of the accident."
'becomes' amended from 'bcomes'.

ARTICLE ENCAUSTIC PAINTING: "In this ley he boiled the wax, cut
into little bits, for half an hour, after which he removed it from
the fire and allowed it to cool." 'little' amended from 'litle'.

ARTICLE ENERGETICS: "... which throw much light on the phenomena of
actual systems not far removed from these ideal limits." 'from'
amended from 'trom'.

ARTICLE ENERGETICS: "It should be noted, however, that this
argument applies only to fluid phases, for in the case of
deposition of a solid ..." 'phases' amended from 'phrases'.

ARTICLE ENGEL, ERNST: "... and Zeitschrift des Statistischen
Bureaus des Konigreichs Sachsen." 'Statistischen' amended from
'Statistichen'.

ENCYCLOPAEDIA BRITANNICA

A DICTIONARY OF ARTS, SCIENCES, LITERATURE
AND GENERAL INFORMATION

ELEVENTH EDITION

VOLUME IX, SLICE III

Electrostatics to Engis

ARTICLES IN THIS SLICE:

ELECTROSTATICS ELTVILLE
ELECTROTHERAPEUTICS ELTZ
ELECTROTYPING ELVAS
ELECTRUM, ELECTRON ELVEY, SIR GEORGE JOB
ELEGIT ELVIRA, SYNOD OF
ELEGY EL WAD
ELEMENT ELWOOD
ELEMI ELY, RICHARD THEODORE
ELEPHANT ELY
ELEPHANTA ISLE ELYOT, SIR THOMAS
ELEPHANTIASIS ELYRIA
ELEPHANT'S-FOOT ELYSIUM
ELETS ELZE, KARL
ELEUSIS ELZEVIR
ELEUTHERIUS EMANATION
ELEUTHEROPOLIS EMANUEL I.
ELEVATORS EMBALMING
ELF EMBANKMENT
ELGAR, SIR EDWARD EMBARGO
ELGIN (Illinois, U.S.A.) EMBASSY
ELGIN (Scotland) EMBER DAYS and EMBER WEEKS
ELGIN AND KINCARDINE, EARLS OF EMBEZZLEMENT
ELGINSHIRE EMBLEM
ELGON EMBLEMENTS
ELI EMBOSSING
ELIAS EMBRACERY
ELIAS, JOHN EMBRASURE
ELIAS LEVITA EMBROIDERY
ELIE EMBRUN
ELIE DE BEAUMONT, JEAN BAPTISTE EMBRYOLOGY
ELIJAH EMDEN
ELIJAH WILNA EMERALD
ELIOT, CHARLES WILLIAM EMERIC-DAVID, TOUSSAINT-BERNARD
ELIOT, GEORGE EMERITUS
ELIOT, SIR JOHN EMERSON, RALPH WALDO
ELIOT, JOHN EMERSON, WILLIAM
ELIS (district of Greece) EMERY
ELIS (city of Greece) EMETICS
ELIS, PHILOSOPHICAL SCHOOL OF EMEU
ELISAVETGRAD EMIGRATION
ELISAVETPOL (Russian government) EMILIA
ELISAVETPOL (Russian town) EMINENCE
ELISHA EMINENT DOMAIN
ELISHA BEN ABUYAH EMINESCU, MICHAIL
ELIXIR EMIN PASHA
ELIZABETH (queen of England) EMLYN, THOMAS
ELIZABETH [PETROVNA] EMMANUEL
ELIZABETH [AMELIE EUGENIE] EMMANUEL PHILIBERT
ELIZABETH (Frederick V. consort) EMMAUS
ELIZABETH [PAULINE OTTILIE] EMMENDINGEN
ELIZABETH (Charles I. daughter) EMMERICH
ELIZABETH (Marie Helene) EMMET, ROBERT
ELIZABETH, SAINT EMMET, THOMAS ADDIS
ELIZABETH (New Jersey, U.S.A.) EMMETT, DANIEL DECATUR
ELIZABETHAN STYLE EMMITSBURG
ELIZABETH CITY EMMIUS, UBBO
ELK EMMONS, EBENEZER
ELKHART EMMONS, NATHANAEL
ELKINGTON, GEORGE RICHARDS EMPEDOCLES
ELLA EMPEROR
ELLAND EMPHYSEMA
ELLENBOROUGH, EDWARD LAW (judge) EMPIRE
ELLENBOROUGH, EDWARD LAW (Earl) EMPIRICISM
ELLERY, WILLIAM EMPLOYERS' LIABILITY & COMPENSATION
ELLESMERE, FRANCIS EGERTON EMPOLI
ELLESMERE (town in England) EMPORIA
ELLICE (LAGOON) ISLANDS EMPORIUM
ELLICHPUR EMPSON, SIR RICHARD
ELLIOTSON, JOHN EMPYEMA
ELLIOTT, EBENEZER EMPYREAN
ELLIPSE EMS (river of Germany)
ELLIPSOID EMS (town of Germany)
ELLIPTICITY EMSER, JEROME
ELLIS, ALEXANDER JOHN ENAMEL
ELLIS, GEORGE ENCAENIA
ELLIS, SIR HENRY ENCAUSTIC PAINTING
ELLIS, ROBINSON ENCEINTE
ELLIS, WILLIAM ENCINA, JUAN DEL
ELLISTON, ROBERT WILLIAM ENCKE, JOHANN FRANZ
ELLORA ENCLAVE
ELLORE ENCOIGNURE
ELLSWORTH, OLIVER ENCYCLICAL
ELLSWORTH ENCYCLOPAEDIA
ELLWANGEN ENDECOTT, JOHN
ELLWOOD, THOMAS ENDIVE
ELM ENDOEUS
ELMACIN, GEORGE ENDOGAMY
ELMALI ENDOR
ELMES, HARVEY LONSDALE ENDOSPORA
ELMES, JAMES ENDYMION
ELMHAM, THOMAS ENERGETICS
ELMINA ENERGICI
ELMIRA ENERGY
ELMSHORN ENFANTIN, BARTHELEMY PROSPER
ELMSLEY, PETER ENFIDAVILLE
ELNE ENFIELD (Connecticut, U.S.A.)
EL OBEID ENFIELD (England)
ELOI, SAINT ENFILADE
ELONGATION ENGADINE
EL PASO ENGAGED COLUMN
ELPHINSTONE, MOUNTSTUART ENGEL, ERNST
ELPHINSTONE, WILLIAM ENGEL, JOHANN JAKOB
EL RENO ENGELBERG
ELSFLETH ENGELBRECHTSDATTER, DORTHE
ELSINORE ENGELHARDT, JOHANN GEORG VEIT
ELSSLER, FANNY ENGHIEN, LOUIS ANTOINE HENRI
ELSTER (rivers of Germany) ENGHIEN
ELSTER (spa of Germany) ENGINE
ELSWICK ENGINEERING
EL TEB ENGINEERS, MILITARY
ELTON, CHARLES ISAAC ENGIS

ELECTROSTATICS, the name given to that department of electrical science in which the phenomena of electricity at rest are considered. Besides their ordinary condition all bodies are capable of being thrown into a physical state in which they are said to be electrified or charged with electricity. When in this condition they become sources of electric force, and the space round them in which this force is manifested is called an "electric field" (see ELECTRICITY). Electrified bodies exert mechanical forces on each other, creating or tending to create motion, and also induce electric charges on neighbouring surfaces.

The reader possessed of no previous knowledge of electrical phenomena will best appreciate the meaning of the terms employed by the aid of a few simple experiments. For this purpose the following apparatus should be provided:--(1) two small metal tea-trays and some clean dry tumblers, the latter preferably varnished with shellac varnish made with alcohol free from water; (2) two sheets of ebonite rather larger than the tea-trays; (3) a rod of sealing-wax or ebonite and a glass tube, also some pieces of silk and flannel; (4) a few small gilt pith balls suspended by dry silk threads; (5) a gold-leaf electroscope, and, if possible, a simple form of quadrant electrometer (see ELECTROSCOPE and ELECTROMETER); (6) some brass balls mounted on the ends of ebonite penholders, and a few tin canisters. With the aid of this apparatus, the principal facts of electrostatics can be experimentally verified, as follows:--

_Experiment I._--Place one tea-tray bottom side uppermost upon three warm tumblers as legs. Rub the sheet of ebonite vigorously with warm flannel and lay it rubbed side downwards on the top of the tray. Touch the tray with the finger for an instant, and lift up the ebonite without letting the hand touch the tray a second time. The tray is then found to be electrified. If a suspended gilt pith ball is held near it, the ball will first be attracted and then repelled. If small fragments of paper are scattered on the tray and then the other tray held in the hand over them, they will fly up and down rapidly. If the knuckle is approached to the electrified tray, a small spark will be seen, and afterwards the tray will be found to be discharged or unelectrified. If the electrified tray is touched with the sealing-wax or ebonite rod, it will not be discharged, but if touched with a metal wire, the hand, or a damp thread, it is discharged at once. This shows that some bodies are _conductors_ and others _non-conductors_ or _insulators_ of electricity, and that bodies can be electrified by friction and impart their electric charge to other bodies. A charged conductor supported on a non-conductor retains its charge. It is then said to be insulated.

_Experiment II._--Arrange two tea-trays, each on dry tumblers as before. Rub the sheet of ebonite with flannel, lay it face downwards on one tray, touch that tray with the finger for a moment and lift up the ebonite sheet, rub it again, and lay it face downwards on the second tray and leave it there. Then take two suspended gilt pith balls and touch them (a) both against one tray; they will be found to repel each other; (b) touch one against one tray and the other against the other tray, and they will be found to attract each other. This proves the existence of two kinds of electricity, called _positive_ and _negative_. The first tea-tray is positively electrified, and the second negatively. If an insulated brass ball is touched against the first tray and then against the knob or plate of the electroscope, the gold leaves will diverge. If the ball is discharged and touched against the other tray, and then afterwards against the previously charged electroscope, the leaves will collapse. This shows that the two electricities neutralize each other's effect when imparted equally to the same conductor.

_Experiment III._--Let one tray be insulated as before, and the electrified sheet of ebonite held over it, but not allowed to touch the tray. If the ebonite is withdrawn without touching the tray, the latter will be found to be unelectrified. If whilst holding the ebonite sheet over the tray the latter is also touched with an insulated brass ball, then this ball when removed and tested with the electroscope will be found to be negatively electrified. The sign of the electrification imparted to the electroscope when so charged--that is, whether positive or negative--can be determined by rubbing the sealing-wax rod with flannel and the glass rod with silk, and approaching them gently to the electroscope one at a time. The sealing-wax so treated is electrified negatively or _resinously_, and the glass with positive or _vitreous_ electricity. Hence if the electrified sealing-wax rod makes the leaves collapse, the electroscopic charge is positive, but if the glass rod does the same, the electroscopic charge is negative. Again, if, whilst holding the electrified ebonite over the tray, we touch the latter for a moment and then withdraw the ebonite sheet, the tray will be found to be positively electrified. The electrified ebonite is said to act by "electrostatic induction" on the tray, and creates on it two induced charges, one of positive and the other of negative electricity. The last goes to earth when the tray is touched, and the first remains when the tray is insulated and the ebonite withdrawn.

_Experiment IV._--Place a tin canister on a warm tumbler and connect it by a wire with the gold-leaf electroscope. Charge positively a brass ball held on an ebonite stem, and introduce it, without touching, into the canister. The leaves of the electroscope will diverge with positive electricity. Withdraw the ball and the leaves will collapse. Replace the ball again and touch the outside of the canister; the leaves will collapse. If then the ball be withdrawn, the leaves will diverge a second time with negative electrification. If, before withdrawing the ball, after touching the outside of the canister for a moment the ball is touched against the inside of the canister, then on withdrawing it the ball and canister are found to be discharged. This experiment proves that when a charged body acts by induction on an insulated conductor it causes an electrical separation to take place; electricity of opposite sign is drawn to the side nearest the inducing body, and that of like sign is repelled to the remote side, and these quantities are equal in amount.

_Seat of the Electric Charge._--So far we have spoken of electric charge as if it resided on the conductors which are electrified. The work of Benjamin Franklin, Henry Cavendish, Michael Faraday and J. Clerk Maxwell demonstrated, however, that all electric charge or electrification of conductors consists simply in the establishment of a physical state in the surrounding insulator or dielectric, which state is variously called _electric strain_, _electric displacement_ or _electric polarization_. Under the action of the same or identical electric forces the intensity of this state in various insulators is determined by a quality of them called their _dielectric constant_, _specific inductive capacity_ or _inductivity_. In the next place we must notice that electrification is a measurable magnitude and in electrostatics is estimated in terms of a unit called the _electrostatic unit_ of electric quantity. In the absolute C.G.S. system this unit quantity is defined as follows:--If we consider a very small electrified spherical conductor, experiment shows that it exerts a repulsive force upon another similar and similarly electrified body. Cavendish and C.A. Coulomb proved that this mechanical force varies inversely as the square of the distance between the centres of the spheres. The unit of mechanical force in the "centimetre, gramme, second" (C.G.S.) system of units is the _dyne_, which is approximately equal to 1/981 part of the weight of one gramme. A very small sphere is said then to possess a charge of one electrostatic unit of quantity, when it repels another similar and similarly electrified body with a force of one dyne, the centres being at a distance of one centimetre, provided that the spheres are _in vacuo_ or immersed in some insulator, the dielectric constant of which is taken as unity. If the two small conducting spheres are placed with centres at a distance d centimetres, and immersed in an insulator of dielectric constant K, and carry charges of Q and Q' electrostatic units respectively, measured as above described, then the mechanical force between them is equal to QQ'/Kd^2 dynes. For constant charges and distances the mechanical force is inversely as the dielectric constant.

_Electric Force._--If a small conducting body is charged with Q electrostatic units of electricity, and placed in any electric field at a point where the electric force has a value E, it will be subject to a mechanical force equal to QE dynes, tending to move it in the direction of the resultant electric force. This provides us with a definition of a unit of electric force, for it is the strength of an electric field at that point where a small conductor carrying a unit charge is acted upon by unit mechanical force, assuming the dielectric constant of the surrounding medium to be unity. To avoid unnecessary complications we shall assume this latter condition in all the following discussion, which is equivalent simply to assuming that all our electrical measurements are made in air or _in vacuo_.

Owing to the confusion introduced by the employment of the term force, Maxwell and other writers sometimes use the words _electromotive intensity_ instead of electric force. The reader should, however, notice that what is generally called electric force is the analogue in electricity of the so-called acceleration of gravity in mechanics, whilst electrification or quantity of electricity is analogous to mass. If a mass of M grammes be placed in the earth's field at a place where the acceleration of gravity has a value g centimetres per second, then the mechanical force acting on it and pulling it downwards is Mg dynes. In the same manner, if an electrified body carries a positive charge Q electrostatic units and is placed in an electric field at a place where the electric force or electromotive intensity has a value E units, it is urged in the direction of the electric force with a mechanical force equal to QE dynes. We must, however, assume that the charge Q is so small that it does not sensibly disturb the original electric field, and that the dielectric constant of the insulator is unity.

Faraday introduced the important and useful conception of _lines_ and _tubes_ of electric force. If we consider a very small conductor charged with a unit of positive electricity to be placed in an electric field, it will move or tend to move under the action of the electric force in a certain direction. The path described by it when removed from the action of gravity and all other physical forces is called a line of electric force. We may otherwise define it by saying that a line of electric force is a line so drawn in a field of electric force that its direction coincides at every point with the resultant electric force at that point. Let _any_ line drawn in an electric field be divided up into small elements of length. We can take the sum of all the products of the length of each element by the resolved part of the electric force in its direction. This sum, or integral, is called the "line integral of electric force" or the _electromotive force_ (E.M.F.) along this line. In some cases the value of this electromotive force between two points or conductors is independent of the precise path selected, and it is then called the _potential difference_ (P.D.) of the two points or conductors. We may define the term potential difference otherwise by saying that it is the work done in carrying a small conductor charged with one unit of electricity from one point to the other in a direction opposite to that in which it would move under the electric forces if left to itself.

_Electric Potential._--Suppose then that we have a conductor charged with electricity; we may imagine its surface to be divided up into small unequal areas, each of which carries a unit charge of electricity. If we consider lines of electric force to be drawn from the boundaries of these areas, they will cut up the space round the conductor into tubular surfaces called tubes of electric force, and each tube will spring from an area of the conductor carrying a unit electric charge. Hence the charge on the conductor can be measured by the number of unit electric tubes springing from it. In the next place we may consider the charged body to be surrounded by a number of closed surfaces, such that the potential difference between any point on one surface and the earth is the same. These surfaces are called "equipotential" or "level surfaces," and we may so locate them that the potential difference between two adjacent surfaces is one unit of potential; that is, it requires one absolute unit of work (1 erg) to move a small body charged with one unit of electricity from one surface to the next. These enclosing surfaces, therefore, cut up the space into shells of potential, and divide up the tubes of force into electric cells. The surface of a charged conductor is an equipotential surface, because when the electric charge is in equilibrium there is no tendency for electricity to move from one part to the other.

We arbitrarily call the potential of the earth zero, since all potential difference is relative and there is no absolute potential any more than absolute level. We call the difference of potential between a charged conductor and the earth the potential of the conductor. Hence when a body is charged positively its potential is raised above that of the earth, and when negatively it is lowered beneath that of the earth. Potential in a certain sense is to electricity as difference of level is to liquids or difference of temperature to heat. It must be noted, however, that potential is a mere mathematical concept, and has no objective existence like difference of level, nor is it capable per se of producing physical changes in bodies, such as those which are brought about by rise of temperature, apart from any question of difference of temperature. There is, however, this similarity between them. Electricity tends to flow from places of high to places of low potential, water to flow down hill, and heat to move from places of high to places of low temperature. Returning to the case of the charged body with the space around it cut up into electric cells by the tubes of force and shells of potential, it is obvious that the number of these cells is represented by the product QV, where Q is the charge and V the potential of the body in electrostatic units. An electrified conductor is a store of energy, and from the definition of potential it is clear that the work done in increasing the charge q of a conductor whose potential is v by a small amount dq, is vdq, and since this added charge increases in turn the potential, it is easy to prove that the work done in charging a conductor with Q units to a potential V units is 1/2 QV units of work. Accordingly the number of electric cells into which the space round is cut up is equal to twice the energy stored up, or each cell contains half a unit of energy. This harmonizes with the fact that the real seat of the energy of electrification is the dielectric or insulator surrounding the charged conductor.[1]

We have next to notice three important facts in electrostatics and some consequences flowing therefrom.

(i) _Electrical Equilibrium and Potential._--If there be any number of charged conductors in a field, the electrification on them being in equilibrium or at rest, the surface of each conductor is an equipotential surface. For since electricity tends to move between points or conductors at different potentials, if the electricity is at rest on them the potential must be everywhere the same. It follows from this that the electric force at the surface of the conductor has no component along the surface, in other words, the electric force at the bounding surface of the conductor and insulator is everywhere at right angles to it.

By the _surface density_ of electrification on a conductor is meant the charge per unit of area, or the number of tubes of electric force which spring from unit area of its surface. Coulomb proved experimentally that the electric force just outside a conductor at any point is proportional to the electric density at that point. It can be shown that the resultant electric force normal to the surface at a point just outside a conductor is equal to 4[pi][sigma], where [sigma] is the surface density at that point. This is usually called Coulomb's Law.[2]

(ii) _Seat of Charge._--The charge on an electrified conductor is wholly on the surface, and there is no electric force in the interior of a closed electrified conducting surface which does not contain any other electrified bodies. Faraday proved this experimentally (see _Experimental Researches_, series xi. S 1173) by constructing a large chamber or box of paper covered with tinfoil or thin metal. This was insulated and highly electrified. In the interior no trace of electric charge could be found when tested by electroscopes or other means. Cavendish proved it by enclosing a metal sphere in two hemispheres of thin metal held on insulating supports. If the sphere is charged and then the jacketing hemispheres fitted on it and removed, the sphere is found to be perfectly discharged.[3] Numerous other demonstrations of this fact were given by Faraday. The thinnest possible spherical shell of metal, such as a sphere of insulator coated with gold-leaf, behaves as a conductor for static charge just as if it were a sphere of solid metal. The fact that there is no electric force in the interior of such a closed electrified shell is one of the most certainly ascertained facts in the science of electrostatics, and it enables us to demonstrate at once that particles of electricity attract and repel each other with a force which is inversely as the square of their distance.

We may give in the first place an elementary proof of the converse proposition by the aid of a simple lemma:--

_Lemma._--If particles of matter attract one another according to the law of the inverse square the attraction of all sections of a cone for a particle at the vertex is the same. _Definition._--The solid angle subtended by any surface at a point is measured by the quotient of its apparent surface by the square of its distance from that point. Hence the total solid angle round any point is 4[pi]. The solid angles subtended by all normal sections of a cone at the vertex are therefore equal, and since the attractions of these sections on a particle at the vertex are proportional to their distances from the vertex, they are numerically equal to one another and to the solid angle of the cone.

Let us then suppose a spherical shell O to be electrified. Select any point P in the interior and let a line drawn through it sweep out a small double cone (see fig. 1). Each cone cuts out an area on the surface equally inclined to the cone axis. The electric density on the sphere being uniform, the quantities of electricity on these areas are proportional to the areas, and if the electric force varies inversely as the square of the distance, the forces exerted by these two surface charges at the point in question are proportional to the solid angle of the little cone. Hence the forces due to the two areas at opposite ends of the chord are equal and opposed.

Hence we see that if the whole surface of the sphere is divided into pairs of elements by cones described through any interior point, the resultant force at that point must consist of the sum of pairs of equal and opposite forces, and is therefore zero. For the proof of the converse proposition we must refer the reader to the _Electrical Researches of the Hon. Henry Cavendish_, p. 419, or to Maxwell's _Treatise on Electricity and Magnetism_, 2nd ed., vol. i. p. 76, where Maxwell gives an elegant proof that if the force in the interior of a closed conductor is zero, the law of the force must be that of the inverse square of the distance.[4] From this fact it follows that we can shield any conductor entirely from external influence by other charged conductors by enclosing it in a metal case. It is not even necessary that this envelope should be of solid metal; a cage made of fine metal wire gauze which permits objects in its interior to be seen will yet be a perfect electrical screen for them. Electroscopes and electrometers, therefore, standing in proximity to electrified bodies can be perfectly shielded from influence by enclosing them in cylinders of metal gauze.

Even if a charged and insulated conductor, such as an open canister or deep cup, is not perfectly closed, it will be found that a proof-plane consisting of a small disk of gilt paper carried at the end of a rod of gum-lac will not bring away any charge if applied to the deep inside portions. In fact it is curious to note how large an opening may be made in a vessel which yet remains for all electrical purposes "a closed conductor." Maxwell (_Elementary Treatise_, &c., p. 15) ingeniously applied this fact to the insulation of conductors. If we desire to insulate a metal ball to make it hold a charge of electricity, it is usual to do so by attaching it to a handle or stem of glass or ebonite. In this case the electric charge exists at the point where the stem is attached, and there leakage by creeping takes place. If, however, we employ a hollow sphere and let the stem pass through a hole in the side larger than itself, and attach the end to the interior of the sphere, then leakage cannot take place.

Another corollary of the fact that there is no electric force in the interior of a charged conductor is that the potential in the interior is constant and equal to that at the surface. For by the definition of potential it follows that the electric force in any direction at any point is measured by the space rate of change of potential in that direction or E = [+-]dV/dx. Hence if the force is zero the potential V must be constant.

(iii.) _Association of Positive and Negative Electricities._--The third leading fact in electrostatics is that positive and negative electricity are always created in equal quantities, and that for every charge, say, of positive electricity on one conductor there must exist on some other bodies an equal total charge of negative electricity. Faraday expressed this fact by saying that no absolute electric charge could be given to matter. If we consider the charge of a conductor to be measured by the number of tubes of electric force which proceed from it, then, since each tube must end on some other conductor, the above statement is equivalent to saying that the charges at each end of a tube of electric force are equal.

The facts may, however, best be understood and demonstrated by considering an experiment due to Faraday, commonly called the ice pail experiment, because he employed for it a pewter ice pail (_Exp. Res._ vol. ii. p. 279, or _Phil. Mag._ 1843, 22). On the plate of a gold-leaf electroscope place a metal canister having a loose lid. Let a metal ball be suspended by a silk thread, and the canister lid so fixed to the thread that when the lid is in place the ball hangs in the centre of the canister. Let the ball and lid be removed by the silk, and let a charge, say, of positive electricity (+Q) be given to the ball. Let the canister be touched with the finger to discharge it perfectly. Then let the ball be lowered into the canister. It will be found that as it does so the gold-leaves of the electroscope diverge, but collapse again if the ball is withdrawn. If the ball is lowered until the lid is in place, the leaves take a steady deflection. Next let the canister be touched with the finger, the leaves collapse, but diverge again when the ball is withdrawn. A test will show that in this last case the canister is left negatively electrified. If before the ball is withdrawn, after touching the outside of the canister with the finger, the ball is tilted over to make it touch the inside of the canister, then on withdrawing it the canister and ball are found to be perfectly discharged. The explanation is as follows: the charge (+Q) of positive electricity on the ball creates by induction an equal charge (-Q) on the inside of the canister when placed in it, and repels to the exterior surface of the canister an equal charge (+Q). On touching the canister this last charge goes to earth. Hence when the ball is touched against the inside of the canister before withdrawing it a second time, the fact that the system is found subsequently to be completely discharged proves that the charge -Q induced on the inside of the canister must be exactly equal to the charge +Q on the ball, and also that the inducing action of the charge +Q on the ball created equal quantities of electricity of opposite sign, one drawn to the inside and the other repelled to the outside of the canister.

_Electrical Capacity._--We must next consider the quality of a conductor called its electrical capacity. The potential of a conductor has already been defined as the mechanical work which must be done to bring up a very small body charged with a unit of positive electricity from the earth's surface or other boundary taken as the place of zero potential to the surface of this conductor in question. The mathematical expression for this potential can in some cases be calculated or predetermined.

Potential of a sphere.

Thus, consider a sphere uniformly charged with Q units of positive
electricity. It is a fundamental theorem in attractions that a thin
spherical shell of matter which attracts according to the law of the
inverse square acts on all external points as if it were concentrated
at its centre. Hence a sphere having a charge Q repels a unit charge
placed at a distance x from its centre with a force Q/x^2 dynes, and
therefore the work W in ergs expended in bringing the unit up to that
point from an infinite distance is given by the integral
_
/x
W = | Qx^-2 dx = Q/x (1).
_/[oo]

Capacity of a sphere.

Hence the potential at the surface of the sphere, and therefore the
potential of the sphere, is Q/R, where R is the radius of the sphere
in centimetres. The quantity of electricity which must be given to the
sphere to raise it to unit potential is therefore R electrostatic
units. The capacity of a conductor is defined to be the charge
required to raise its potential to unity, all other charged conductors
being at an infinite distance. This capacity is then a function of the
geometrical dimensions of the conductor, and can be mathematically
determined in certain cases. Since the potential of a small charge of
electricity dQ at a distance r is equal to dQ/r, and since the
potential of all parts of a conductor is the same in those cases in
which the distribution of surface density of electrification is
uniform or symmetrical with respect to some point or axis in the
conductor, we can calculate the potential by simply summing up terms
like [sigma]dS/r, where dS is an element of surface, [sigma] the
surface density of electricity on it, and r the distance from the
symmetrical centre. The capacity is then obtained as the quotient of
the whole charge by this potential. Thus the distribution of
electricity on a sphere in free space must be uniform, and all parts
of the charge are at an equal distance R from the centre. Accordingly
the potential _at_ the centre is Q/R. But this must be the potential
_of_ the sphere, since all parts are at the same potential V. Since
the capacity C is the ratio of charge to potential, the capacity of
the sphere in free space is Q/V = R, or is numerically the same as its
radius reckoned in centimetres.

Capacity of a thin rod.

We can thus easily calculate the capacity of a long thin wire like a
telegraph wire far removed from the earth, as follows: Let 2r be the
diameter of the wire, l its length, and [sigma] the uniform surface
electric density. Then consider a thin annulus of the wire of width
dx; the charge on it is equal to 2[pi]r[sigma]/dx units, and the
potential V at a point on the axis at a distance x from the annulus
due to this elementary charge is

_l/2
/ 2[pi]r[sigma]
V = 2 | -----------------dx = 4[pi]r[sigma] {log_e (1/2 l + [root][r^2 + 1/4 l^2]) - log_e ^r}.
_/ [root](r^2 + x^2)
0

If, then, r is small compared with l, we have V = 4[pi]r[sigma]log_e
l/r. But the charge is Q = 2[pi]r[sigma], and therefore the capacity
of the thin wire is given by

C = 1/2 log_e l/r (2).

Potential of an ellipsoid.

A more difficult case is presented by the ellipsoid[5]. We have first
to determine the mode in which electricity distributes itself on a
conducting ellipsoid in free space. It must be such a distribution
that the potential in the interior will be constant, since the
electric force must be zero. It is a well-known theorem in attractions
that if a shell is made of gravitative matter whose inner and outer
surfaces are similar ellipsoids, it exercises no attraction on a
particle of matter in its interior[6]. Consider then an ellipsoidal
shell the axes of whose bounding surfaces are (a, b, c) and (a + da),
(b + db), (c + dc), where da/a = db/b = dc/c = [mu]. The potential of
such a shell at any internal point is constant, and the equipotential
surfaces for external space are ellipsoids confocal with the
ellipsoidal shell. Hence if we distribute electricity over an
ellipsoid, so that its density is everywhere proportional to the
thickness of a shell formed by describing round the ellipsoid a
similar and slightly larger one, that distribution will be in
equilibrium and will produce a constant potential throughout the
interior. Thus if [sigma] is the surface density, [delta] the
thickness of the shell at any point, and [rho] the assumed volume
density of the matter of the shell, we have [sigma] = A[delta][rho].
Then the quantity of electricity on any element of surface dS is A
times the mass of the corresponding element of the shell; and if Q is
the whole quantity of electricity on the ellipsoid, Q = A times the
whole mass of the shell. This mass is equal to 4[pi]abc[rho][mu];
therefore Q = A4[pi]abc[rho][mu] and [delta] = [mu]p, where p is the
length of the perpendicular let fall from the centre of the ellipsoid
on the tangent plane. Hence

[sigma] = Qp/4[pi]abc (3).

Capacity of an ellipsoid.

Accordingly for a given ellipsoid the surface density of free
distribution of electricity on it is everywhere proportional to the
length of the perpendicular let fall from the centre on the tangent
plane at that point. From this we can determine the capacity of the
ellipsoid as follows: Let p be the length of the perpendicular from
the centre of the ellipsoid, whose equation is x^2/a^2 + y^2/b^2 +
z^2/c^2 = 1 to the tangent plane at x, y, z. Then it can be shown that
1/p^2 = x^2/a^4 + y^2/b^4 + z^2/c^4 (see Frost's _Solid Geometry_, p.
172). Hence the density [sigma] is given by

Q 1
[sigma] = -------- -----------------------------------,
4[pi]abc [root](x^2/a^4 + y^2/b^4 + z^2/c^4)

and the potential at the centre of the ellipsoid, and therefore its
potential as a whole is given by the expression,
_ _
/ [sigma]dS Q / dS
V = | --------- = -------- | ------------------------------------ (4).
_/ r 4[pi]abc _/ r[root](x^2/a^4 + y^2/b^4 + z^2/c^4)

Accordingly the capacity C of the ellipsoid is given by the equation
_
1 1 / dS
-- = -------- | ---------------------------------------------------------- (5).
C 4[pi]abc _/ [root](x^2 + y^2 + z^2)[root](x^2/a^4 + y^2/b^4 + z^2/c^4)

It has been shown by Professor Chrystal that the above integral may
also be presented in the form,[7]
_
1 /[oo] d[lambda]
-- = 1/2 | -------------------------------------------------------- (6).
C _/0 [root]{(a^2 + [lambda])(b^2 + [lambda])(c^2 + [lambda])}

The above expressions for the capacity of an ellipsoid of three
unequal axes are in general elliptic integrals, but they can be
evaluated for the reduced cases when the ellipsoid is one of
revolution, and hence in the limit either takes the form of a long rod
or of a circular disk.

Thus if the ellipsoid is one of revolution, and ds is an element of
arc which sweeps out the element of surface dS, we have

/dx\ /py\ 2[pi]b^2
dS = 2[pi]yds = 2[pi]ydx / ( -- ) = 2[pi]ydx / ( -- ) = -------- dx.
\ds/ \ b/ p

Hence, since [sigma] = Qp/4[pi]ab^2, [sigma]dS = Qdx/2a.

Accordingly the distribution of electricity is such that equal
parallel slices of the ellipsoid of revolution taken normal to the
axis of revolution carry equal charges on their curved surface.

The capacity C of the ellipsoid of revolution is therefore given by
the expression
_
1 1 / dx
-- = -- | ----------------- (7).
C 2a _/ [root](x^2 + y^2)

If the ellipsoid is one of revolution round the major axis a (prolate)
and of eccentricity e, then the above formula reduces to

1 1 /1 + e\
-- = --- log_[epsilon]( ----- ) (8).
C1 2ae \1 - e/

Whereas if it is an ellipsoid of revolution round the minor axis b
(oblate), we have

1 sin^-1 ae
--- = ---------- (9).
C^2 ae

In each case we have C = a when e = 0, and the ellipsoid thus becomes
a sphere.

In the extreme case when e = 1, the prolate ellipsoid becomes a long
thin rod, and then the capacity is given by

C1 = a/log_[epsilon] 2a/b (10),

which is identical with the formula (2) already obtained. In the other
extreme case the oblate spheroid becomes a circular disk when e = 1,
and then the capacity C2 = 2a/[pi]. This last result shows that the
capacity of a thin disk is 2/[pi] = 1/1.571 of that of a sphere of the
same radius. Cavendish (_Elec. Res._ pp. 137 and 347) determined in
1773 experimentally that the capacity of a sphere was 1.541 times that
of a disk of the same radius, a truly remarkable result for that date.

Three other cases of practical interest present themselves, viz. the
capacity of two concentric spheres, of two coaxial cylinders and of
two parallel planes.

Capacity of two concentric spheres.

Consider the case of two concentric spheres, a solid one enclosed in a
hollow one. Let R1 be the radius of the inner sphere, R2 the inside
radius of the outer sphere, and R2 the outside radius of the outer
spherical shell. Let a charge +Q be given to the inner sphere. Then
this produces a charge -Q on the inside of the enclosing spherical
shell, and a charge +Q on the outside of the shell. Hence the
potential V at the centre of the inner sphere is given by V =
Q/R1 - Q/R2 + Q/R3. If the outer shell is connected to the earth, the
charge +Q on it disappears, and we have the capacity C of the inner
sphere given by

C = 1/R1 - 1/R2 = (R2 - R1)/R1R2 (11).

Such a pair of concentric spheres constitute a condenser (see LEYDEN
JAR), and it is obvious that by making R2 nearly equal to R1, we may
enormously increase the capacity of the inner sphere. Hence the name
_condenser_.

Capacity of two coaxial cylinders.

The other case of importance is that of two coaxial cylinders. Let a
solid circular sectioned cylinder of radius R1 be enclosed in a
coaxial tube of inner radius R2. Then when the inner cylinder is at
potential V1 and the outer one kept at potential V2 the lines of
electric force between the cylinders are radial. Hence the electric
force E in the interspace varies inversely as the distance from the
axis. Accordingly the potential V at any point in the interspace is
given by
_
/
E = -dV/dR = A/R or V = -A | R^-1 dR, (12),
_/

where R is the distance of the point in the interspace from the axis,
and A is a constant. Hence V2 - V1 = -A log R2/R1. If we consider a
length l of the cylinder, the charge Q on the inner cylinder is Q =
2[pi]R1l[sigma], where [sigma] is the surface density, and by
Coulomb's law [sigma] = E1/4 [pi], where E1 = A/R1 is the force at the
surface of the inner cylinder.

Accordingly Q = 2[pi]R1lA/4[pi]R1 = Al/2. If then the outer cylinder
be at zero potential the potential V of the inner one is

V = A log (R2/R1), and its capacity C = l/2 log R2/R1.

This formula is important in connexion with the capacity of electric
cables, which consist of a cylindrical conductor (a wire) enclosed in
a conducting sheath. If the dielectric or separating insulator has a
constant K, then the capacity becomes K times as great.

Capacity of two parallel planes.

"Edge effect."

The capacity of two parallel planes can be calculated at once if we
neglect the distribution of the lines of force near the edges of the
plates, and assume that the only field is the uniform field between
the plates. Let V1 and V2 be the potentials of the plates, and let a
charge Q be given to one of them. If S is the surface of each plate,
and d their distance, then the electric force E in the space between
them is E = (V1-V2)/d. But if [sigma] is the surface density, E =
4[pi][sigma], and [sigma] = Q/S. Hence we have

(V1 - V2) d = 4[pi]Q/S or C = Q/(V1 - V2) = S/4[pi]d (13).

In this calculation we neglect altogether the fact that electric force
distributed on curved lines exists outside the interspace between the
plates, and these lines in fact extend from the back of one plate to
that of the other. G.R. Kirchhoff (_Gesammelte Abhandl._ p. 112) has
given a full expression for the capacity C of two circular plates of
thickness t and radius r placed at any distance d apart in air from
which the edge effect can be calculated. Kirchhoff's expression is as
follows:--

[pi]r^2 r / 16[pi]r(d+t) d + t\
C = ------- + ------ ( d log_[epsilon] ------------ + t log_[epsilon] ----- ) (14).
4[pi]d 4[pi]d \ [epsilon]d^2 t /

In the above formula [epsilon] is the base of the Napierian
logarithms. The first term on the right-hand side of the equation is
the expression for the capacity, neglecting the curved edge
distribution of electric force, and the other terms take into account,
not only the uniform field between the plates, but also the
non-uniform field round the edges and beyond the plates.

Guard plates.

In practice we can avoid the difficulty due to irregular distribution
of electric force at the edges of the plate by the use of a guard
plate as first suggested by Lord Kelvin.[8] If a large plate has a
circular hole cut in it, and this is nearly filled up by a circular
plate lying in the same plane, and if we place another large plate
parallel to the first, then the electric field between this second
plate and the small circular plate is nearly uniform; and if S is the
area of the small plate and d its distance from the opposed plate, its
capacity may be calculated by the simple formula C = S/4[pi]d. The
outer larger plate in which the hole is cut is called the "guard
plate," and must be kept at the same potential as the smaller inner or
"trap-door plate." The same arrangement can be supplied to a pair of
coaxial cylinders. By placing metal plates on either side of a larger
sheet of dielectric or insulator we can construct a condenser of
relatively large capacity. The instrument known as a Leyden jar (q.v.)
consists of a glass bottle coated within and without for three parts
of the way up with tinfoil.

Systems of condensers.

If we have a number of such condensers we can combine them in
"parallel" or in "series." If all the plates on one side are connected
together and also those on the other, the condensers are joined in
parallel. If C1, C2, C3, &c., are the separate capacities, then
[Sigma](C) = C1 + C2 + C3 + &c., is the total capacity in parallel. If
the condensers are so joined that the inner coating of one is
connected to the outer coating of the next, they are said to be in
series. Since then they are all charged with the same quantity of
electricity, and the total over all potential difference V is the sum
of each of the individual potential differences V1, V2, V3, &c., we
have

Q = C1V1 = C2V2 = C3V3 = &c., and V = V1 + V2 + V3 + &c.

The resultant capacity is C = Q/V, and

C = 1/(1/C1 + 1/C2 + 1/C3 + &c.) = 1/[Sigma](1/C) (15).

These rules provide means for calculating the resultant capacity when
any number of condensers are joined up in any way.

If one condenser is charged, and then joined in parallel with another
uncharged condenser, the charge is divided between them in the ratio
of their capacities. For if C1 and C2 are the capacities and Q1 and Q2
are the charges after contact, then Q1/C1 and Q2/C2 are the potential
differences of the coatings and must be equal. Hence Q1/C1 = Q2/C2 or
Q1/Q2 = C1/C2. It is worth noting that if we have a charged sphere we
can perfectly discharge it by introducing it into the interior of
another hollow insulated conductor and making contact. The small
sphere then becomes part of the interior of the other and loses all
charge.

_Measurement of Capacity._--Numerous methods have been devised for the
measurement of the electrical capacity of conductors in those cases in
which it cannot be determined by calculation. Such a measurement may
be an _absolute_ determination or a _relative_ one. The dimensions of
a capacity in electrostatic measure is a length (see UNITS, PHYSICAL).
Thus the capacity of a sphere in electrostatic units (E.S.U.) is the
same as the number denoting its radius in centimetres. The unit of
electrostatic capacity is therefore that of a sphere of 1 cm.
radius.[9] This unit is too small for practical purposes, and hence a
unit of capacity 900,000 greater, called a microfarad, is generally
employed. Thus for instance the capacity in free space of a sphere 2
metres in diameter would be 100/900,000 = 1/9000 of a microfarad. The
electrical capacity of the whole earth considered as a sphere is about
800 microfarads. An absolute measurement of capacity means, therefore,
a determination in E.S. units made directly without reference to any
other condenser. On the other hand there are numerous methods by which
the capacities of condensers may be compared and a relative
measurement made in terms of some standard.

Relative determinations.

One well-known comparison method is that of C.V. de Sauty. The two
condensers to be compared are connected in the branches of a
Wheatstone's Bridge (q.v.) and the other two arms completed with
variable resistance boxes. These arms are then altered until on
raising or depressing the battery key there is no sudden deflection
either way of the galvanometer. If R1 and R2 are the arms' resistances
and C1 and C2 the condenser capacities, then when the bridge is
balanced we have R1 : R2 = C1 : C2.

Another comparison method much used in submarine cable work is the
method of mixtures, originally due to Lord Kelvin and usually called
Thomson and Gott's method. It depends on the principle that if two
condensers of capacity C1 and C2 are respectively charged to
potentials V1 and V2, and then joined in parallel with terminals of
opposite charge together, the resulting potential difference of the
two condensers will be V, such that

(C1V1 - C2V2)
V = ------------- (16);
(C + C)

and hence if V is zero we have C1 : C2 = V2 : V1.

The method is carried out by charging the two condensers to be
compared at the two sections of a high resistance joining the ends of
a battery which is divided into two parts by a movable contact.[10]
This contact is shifted until such a point is found by trial that the
two condensers charged at the different sections and then joined as
above described and tested on a galvanometer show no charge. Various
special keys have been invented for performing the electrical
operations expeditiously.

A simple method for condenser comparison is to charge the two
condensers to the same voltage by a battery and then discharge them
successively through a ballistic galvanometer (q.v.) and observe the
respective "throws" or deflections of the coil or needle. These are
proportional to the capacities. For the various precautions necessary
in conducting the above tests special treatises on electrical testing
must be consulted.

Absolute determinations.