Electrostatics

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′/Kd² 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 ½QV 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 4πσ, where σ 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. § 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 4π. 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.  
  
---  
Fig. 1.  
  
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.

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 Potential of a sphere. 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/x² 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

W = ∫x∞ Qx−2dx = Q/x

(1).

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 σdS/r, where dS is an element of surface, σ 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 Capacity of a sphere. 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.

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 σ the uniform Capacity of a thin rod. surface electric density. Then consider a thin annulus of the wire of width dx; the charge on it is equal to 2πrσ/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

V = 2 ∫ l/2 0 | 2πrσ | dx = 4πrσ { loge(½l + √r² + ¼l²) − loger}.  
---|---|---  
√(r² + x²)  
  
If, then, r is small compared with l, we have V = 4πrσloge l/r. But the charge is Q = 2πrσ, and therefore the capacity of the thin wire is given by

C = ½ loge l/r

(2).

A more difficult case is presented by the ellipsoid5. 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 Potential of an ellipsoid. 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 interior6. 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 = μ. 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 σ is the surface density, δ the thickness of the shell at any point, and ρ the assumed volume density of the matter of the shell, we have σ = Aδρ. 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 4πabcρμ; therefore Q = A4πabcρμ and δ = μp, where p is the length of the perpendicular let fall from the centre of the ellipsoid on the tangent plane. Hence

σ = Qp / 4πabc

(3).

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 Capacity of an ellipsoid. 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 x²/a² + y²/b² + z²/c² = 1 to the tangent plane at x, y, z. Then it can be shown that 1/p² = x²/a4 \+ y²/b4 \+ z²/c4 (see Frost’s Solid Geometry, p. 172). Hence the density σ is given by

σ = | Q |   | 1 | .  
---|---|---|---|---  
4πabc | √(x² / a4 \+ y² / b4 \+ z² / c4)  
  
and the potential at the centre of the ellipsoid, and therefore its potential as a whole is given by the expression,

V = ∫ | σdS | = | Q | ∫ | dS  
---|---|---|---|---|---  
r | 4πabc | r √(x² / a4 \+ y² / b4 \+ z² / c4)  
  
(4).

Accordingly the capacity C of the ellipsoid is given by the equation

1 | = | 1 | ∫ | dS  
---|---|---|---|---  
C | 4πabc | √(x² + y² + z²) √(x² / a4 \+ y² / b4 \+ z² / c4)  
  
(5).

It has been shown by Professor Chrystal that the above integral may also be presented in the form,7

1 | = ½ ∫∞0 | dλ  
---|---|---  
C | √{(a² + λ) (b² + λ) (c² + λ)}  
  
(6).

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

dS = 2πyds = 2πydx / ( | dx | ) = 2πydx / ( | py | ) = | 2πb² | dx.  
---|---|---|---|---|---|---  
ds | b | p  
  
Hence, since σ = Qp / 4πab², σdS = 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  
---|---|---|---|---  
C | 2a | √(x² + y²)  
  
(7).

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 | logε ( | 1 + e | )  
---|---|---|---|---|---  
C1 | 2ae | 1 − e  
  
(8).

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

1 | = | sin−1ae  
---|---|---  
C² | ae  
  
(9).

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 / logε 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/π. This last result shows that the capacity of a thin disk is 2/π = 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.

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 Capacity of two concentric spheres. 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.

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 Capacity of two coaxial cylinders. 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 = 2πR1lσ, where σ is the surface density, and by Coulomb’s law σ = E1/4π, where E1 = A/R1 is the force at the surface of the inner cylinder.

Accordingly Q = 2πR1lA / 4πR1 = 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.

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 Capacity of two parallel planes. 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 σ is the surface density, E = 4πσ, and σ = Q/S. Hence we have

(V1 − V2) d = 4πQ / S or C = Q / (V1 − V2) = S / 4πd

(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 “Edge effect.” 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:—

C = | πr² | + | r | { d logε | 16πr (d + t) | \+ t logε | d + t | }  
---|---|---|---|---|---|---|---|---  
4πd | 4πd | εd² | t  
  
(14).

In the above formula ε 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.

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 Guard plates. 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 / 4πd. 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.

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 Systems of condensers. capacities, then Σ(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 / Σ(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.

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 Relative determinations. 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

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

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.  
  
---  
Fig. 2.  
  
In the absolute determination of capacity we have to measure the ratio of the charge of a condenser to its plate potential difference. One of the best methods for doing this is to charge the condenser by the known voltage of a battery, and then Absolute determinations. discharge it through a galvanometer and repeat this process rapidly and successively. If a condenser of capacity C is charged to potential V, and discharged n times per second through a galvanometer, this series of intermittent discharges is equivalent to a current nCV. Hence if the galvanometer is calibrated by a potentiometer (q.v.) we can determine the value of this current in amperes, and knowing the value of n and V thus determine C. Various forms of commutator have been devised for effecting this charge and discharge rapidly by J.J. Thomson, R.T. Glazebrook, J.A. Fleming and W.C. Clinton and others.11 One form consists of a tuning-fork electrically maintained in vibration of known period, which closes an electric contact at every vibration and sets another electromagnet in operation, which reverses a switch and moves over one terminal of the condenser from a battery to a galvanometer contact. In another form, a revolving contact is used driven by an electric motor, which consists of an insulating disk having on its surface slips of metal and three wire brushes a, b, c (see fig. 2) pressing against them. The metal slips are so placed that, as the disk revolves, the middle brush, connected to one terminal of the condenser C, is alternately put in conductive connexion with first one and then the other outside brush, which are joined respectively to the battery B and galvanometer G terminals. From the speed of this motor the number of commutations per second can be determined. The above method is especially useful for the determinations of very small capacities of the order of 100 electrostatic units or so and upwards.

Dielectric constant.—Since all electric charge consists in a state of strain or polarization of the dielectric, it is evident that the physical state and chemical composition of the insulator must be of great importance in determining electrical phenomena. Cavendish and subsequently Faraday discovered this fact, and the latter gave the name “specific inductive capacity,” or “dielectric constant,” to that quality of an insulator which determines the charge taken by a conductor embedded in it when charged to a given potential. The simplest method of determining it numerically is, therefore, that adopted by Faraday.12 He constructed two equal condensers, each consisting of a metal ball enclosed in a hollow metal sphere, and he provided also certain hemispherical shells of shellac, sulphur, glass, resin, &c., which he could so place in one condenser between the ball and enclosing sphere that it formed a condenser with solid dielectric. He then determined the ratio of the capacities of the two condensers, one with air and the other with the solid dielectric. This gave the dielectric constant K of the material. Taking the dielectric constant of air as unity he obtained the following values, for shellac K = 2.0, glass K = 1.76, and sulphur K = 2.24.

Table I.—Dielectric Constants (K) of Solids (K for Air = 1).

Substance. | K. | Authority.  
---|---|---  
Glass, double extra dense flint, density 4.5 | 9.896 | J. Hopkinson  
Glass, light flint, density 3.2 | 6.72 |  ”  
Glass, hard crown, density 2.485 | 6.61 |  ”  
Sulphur | 2.24 | M. Faraday  
2.88 | Coullner  
3.84 | L. Boltzmann  
4.0 | P.J. Curie  
2.94 | P.R. Blondlot  
Ebonite | 2.05 | Rosetti  
3.15 | Boltzmann  
2.21 | Schiller  
2.86 | Elsas  
India-rubber, pure brown | 2.12 | Schiller  
India-rubber, vulcanized, grey | 2.69 |  ”  
Gutta-percha | 2.462 | J.E. H. Gordon  
Paraffin | 1.977 | Gibson and Barclay  
2.32 | Boltzmann  
2.29 | J. Hopkinson  
1.99 | Gordon  
Shellac | 2.95 | Wällner  
2.74 | Gordon  
3.04 | A.A. Winkelmann  
Mica | 6.64 | I. Klemenčič  
8.00 | P.J. Curie  
7.98 | E.M.L. Bouty  
5.97 | Elsas  
Quartz— |   |    
along optic axis | 4.55 | P.J. Curie  
perp. to optic axis | 4.49 | P.J. Curie  
Ice at −23° | 78.0 | Bouty  
  
Since Faraday’s time, by improved methods, but depending essentially upon the same principles, an enormous number of determinations of the dielectric constants of various insulators, solid, liquid and gaseous, have been made (see tables I., II., III. and IV.). There are very considerable differences between the values assigned by different observers, sometimes no doubt due to differences in method, but in most cases unquestionably depending on variations in the quality of the specimens examined. The value of the dielectric constant is greatly affected by the temperature and the frequency of the applied electric force.

Table II.—Dielectric Constant (K) of Liquids.

Liquid. | K. | Authority.  
---|---|---  
Water at 17° C. | 80.88 | F. Heerwagen  
”  ”  25° C. | 75.7 | E.B. Rosa  
”  ”  25.3° C. | 78.87 | Franke  
Olive oil |  3.16 | Hopkinson  
Castor oil |  4.78 |  ”  
Turpentine |  2.15 | P.A. Silow  
” |  2.23 | Hopkinson  
Petroleum |  2.072 | Silow  
” |  2.07 | Hopkinson  
Ethyl alcohol at 25° C. | 25.7 | Rosa  
Ethyl ether |  4.57 | Doule  
”  ” |  4.8 | Bouty  
Acetic acid |  9.7 | Franke  
  
Table III.—Dielectric Constant of some Bodies at a very low Temperature (−185° C.) (Fleming and Dewar).

Substance. | K  
at 15° C. | K  
at −185°C.  
---|---|---  
Water | 80 | 2.4 to 2.9  
Formic acid | 62 | 2.41  
Glycerine | 56 | 3.2  
Methyl alcohol | 34 | 3.13  
Nitrobenzene | 32 | 2.6  
Ethyl alcohol | 25 | 3.1  
Acetone | 21.85 | 2.62  
Ethyl nitrate | 17.7 | 2.73  
Amyl alcohol | 16 | 2.14  
Aniline |  7.5 | 2.92  
Castor oil |  4.78 | 2.19  
Ethyl ether |  4.25 | 2.31  
  
The above determinations at low temperature were made with either a steady or a slowly alternating electric force applied a hundred times a second. They show that the dielectric constant of a liquid generally undergoes great reduction in value when the liquid is frozen and reduced to a low temperature.13

The dielectric constants of gases have been determined by L. Boltzmann and I. Klemenčič as follows:—

Table IV.—Dielectric Constants (K) of Gases at 15° C. and 760 mm. Vacuum = 1.

Gas. | Dielectric  
Constant  
K. | √K. | Optical  
Refractive  
Index.  
μ.  
---|---|---|---  
Air | 1.000590 | 1.000295 | 1.000293  
Hydrogen | 1.000264 | 1.000132 | 1.000139  
Carbon dioxide | 1.000946 | 1.000475 | 1.000454  
Carbon monoxide  | 1.000690 | 1.000345 | 1.000335  
Nitrous oxide | 1.000994 | 1.000497 | 1.000516  
Ethylene | 1.001312 | 1.000656 | 1.000720  
Marsh gas (methane) | 1.000944 | 1.000478 | 1.000442  
Carbon bisulphide | 1.002900 | 1.001450 | 1.001478  
Sulphur dioxide | 1.00954 | 1.004770 | 1.000703  
Ether | 1.00744 | 1.003720 | 1.00154  
Ethyl chloride | 1.01552 | 1.007760 | 1.001174  
Ethyl bromide | 1.01546 | 1.007730 | 1.00122  
  
In general the dielectric constant is reduced with decrease of temperature towards a certain limiting value it would attain at the absolute zero. This variation, however, is not always linear. In some cases there is a very sudden drop at or below a certain temperature to a much lower value, and above and below the point the temperature variation is small. There is also a large difference in most cases between the value for a steadily applied electric force and a rapidly reversed or intermittent force—in the last case a decrease with increase of frequency. Maxwell (Elec. and Magn. vol. ii. § 788) showed that the square root of the dielectric constant should be the same number as the refractive index for waves of the same frequency (see Electric Waves). There are very few substances, however, for which the optical refractive index has the same value as K for steady or slowly varying electric force, on account of the great variation of the value of K with frequency.

There is a close analogy between the variation of dielectric constant of an insulator with electric force frequency and that of the rigidity or stiffness of an elastic body with the frequency of applied mechanical stress. Thus pitch is a soft and yielding body under steady stress, but a bar of pitch if struck gives a musical note, which shows that it vibrates and is therefore stiff or elastic for high frequency stress.

Residual Charges in Dielectrics.—In close connexion with this lies the phenomenon of residual charge in dielectrics.14 If a glass Leyden jar is charged and then discharged and allowed to stand awhile, a second discharge can be obtained from it, and in like manner a third, and so on. The reappearance of the residual charge is promoted by tapping the glass. It has been shown that this behaviour of dielectrics can be imitated by a mechanical model consisting of a series of perforated pistons placed in a tube of oil with spiral springs between each piston.15 If the pistons are depressed and then released, and then the upper piston fixed awhile, a second discharge can be obtained from it, and the mechanical stress-strain diagram of the model is closely similar to the discharge curve of a dielectric. R.H.A. Kohlrausch called attention to the close analogy between residual charge and the elastic recovery of strained bodies such as twisted wire or glass threads. If a charged condenser is suddenly discharged and then insulated, the reappearance of a potential difference between its coatings is analogous to the reappearance of a torque in the case of a glass fibre which has been twisted, released suddenly, and then gripped again at the ends.

For further information on the qualities of dielectrics the reader is referred to the following sources:—J. Hopkinson, “On the Residual Charge of the Leyden Jar,” Phil. Trans., 1876, 166 [ii.], p. 489, where it is shown that tapping the glass of a Leyden jar permits the reappearance of the residual charge; “On the Residual Charge of the Leyden Jar,” ib. 167 [ii.], p. 599, containing many valuable observations on the residual charge of Leyden jars; W.E. Ayrton and J. Perry, “A Preliminary Account of the Reduction of Observations on Strained Material, Leyden Jars and Voltameters,” Proc. Roy. Soc., 1880, 30, p. 411, showing experiments on residual charge of condensers and a comparison between the behaviour of dielectrics and glass fibres under torsion. In connexion with this paper the reader may also be referred to one by L. Boltzmann, “Zur Theorie der elastischen Nachwirkung,” Wien. Acad. Sitz.-Ber., 1874, 70.

Distribution of Electricity on Conductors.—We now proceed to consider in more detail the laws which govern the distribution of electricity at rest upon conductors. It has been shown above that the potential due to a charge of q units placed on a very small sphere, commonly called a point-charge, at any distance x is q/x. The mathematical importance of this function called the potential is that it is a scalar quantity, and the potential at any point due to any number of point charges q1, q2, q3, &c., distributed in any manner, is the sum of them separately, or

q1/x1 \+ q2/x2 \+ q3/x3 \+ &c. = Σ (q/x) = V

(17),

where x1, x2, x3, &c., are the distances of the respective point charges from the point in question at which the total potential is required. The resultant electric force E at that point is then obtained by differentiating V, since E = −dV / dx, and E is in the direction in which V diminishes fastest. In any case, therefore, in which we can sum up the elementary potentials at any point we can calculate the resultant electric force at the same point.

We may describe, through all the points in an electric field which have the same potential, surfaces called equipotential surfaces, and these will be everywhere perpendicular or orthogonal to the lines of electric force. Let us assume the field divided up into tubes of electric force as already explained, and these cut normally by equipotential surfaces. We can then establish some important properties of these tubes and surfaces. At each point in the field the electric force can have but one resultant value. Hence the equipotential surfaces cannot cut each other. Let us suppose any other surface described in the electric field so as to cut the closely compacted tubes. At each point on this surface the resultant force has a certain value, and a certain direction inclined at an angle θ to the normal to the selected surface at that point. Let dS be an element of the surface. Then the quantity E cos θdS is the product of the normal component of the force and an element of the surface, and if this is summed up all over the surface we have the total electric flux or induction through the surface, or the surface integral of the normal force mathematically expressed by ∫E cos θdS, provided that the dielectric constant of the medium is unity.  
  
---  
  
We have then a very important theorem as follows:—If any closed surface be described in an electric field which wholly encloses or wholly excludes electrified bodies, then the total flux through this surface is equal to 4π\- times the total quantity of electricity within it.16 This is commonly called Stokes’s theorem. The proof is as follows:—Consider any point-charge E of electricity included in any surface S, S, S (see fig. 3), and describe through it as centre a cone of small solid angle dω cutting out of the enclosing surface in two small areas dS and dS′ at distances x and x′. Then the electric force due to the point charge q at distance x is q/x, and the resolved part normal to the element of surface dS is q cosθ / x². The normal section of the cone at that point is equal to dS cosθ, and the solid angle dω is equal to dS cosθ / x². Hence the flux through dS is qdω. Accordingly, since the total solid angle round a point is 4π, it follows that the total flux through the closed surface due to the single point charge q is 4πq, and what is true for one point charge is true for any collection forming a total charge Q of any form. Hence the total electric flux due to a charge Q through an enclosing surface is 4πQ, and therefore is zero through one enclosing no electricity.

Stokes’s theorem becomes an obvious truism if applied to an incompressible fluid. Let a source of fluid be a point from which an incompressible fluid is emitted in all directions. Close to the source the stream lines will be radial lines. Let a very small sphere be described round the source, and let the strength of the source be defined as the total flow per second through the surface of this small sphere. Then if we have any number of sources enclosed by any surface, the total flow per second through this surface is equal to the total strengths of all the sources. If, however, we defined the strength of the source by the statement that the strength divided by the square of the distance gives the velocity of the liquid at that point, then the total flux through any enclosing surface would be 4π times the strengths of all the sources enclosed. To every proposition in electrostatics there is thus a corresponding one in the hydrokinetic theory of incompressible liquids.

Let us apply the above theorem to the case of a small parallel-epipedon or rectangular prism having sides dx, dy, dz respectively, its centre having co-ordinates (x, y, z). Its angular points have then co-ordinates (x ± ½dx, y ± ½dy, z ± ½dz). Let this rectangular prism be supposed to be wholly filled up with electricity of density ρ; then the total quantity in it is ρ dx dy dz. Consider the two faces perpendicular to the x-axis. Let V be the potential at the centre of the prism, then the normal forces on the two faces of area dy·dx are respectively

− ( | dV | \+ ½ | d²V | dx) and ( | dV | − ½ | d²V | dx),  
---|---|---|---|---|---|---|---|---  
dx | dx² | dx | dx²  
  
and similar expressions for the normal forces to the other pairs of faces dx·dy, dz·dx. Hence, multiplying these normal forces by the areas of the corresponding faces, we have the total flux parallel to the x-axis given by −(d²V / dx²) dx dy dz, and similar expressions for the other sides. Hence the total flux is

− ( | d²V | + | d²V | + | d²V | ) dx dy dz,  
---|---|---|---|---|---|---  
dx² | dy² | dz²  
  
and by the previous theorem this must be equal to 4πρdx dy dz.

Hence

d²V | + | d²V | + | d²V | \+ 4πρ = 0  
---|---|---|---|---|---  
dx² | dy² | dz²  
  
(18).

This celebrated equation was first given by S.D. Poisson, although previously demonstrated by Laplace for the case when ρ = 0. It defines the condition which must be fulfilled by the potential at any and every point in an electric field, through which ρ is finite and the electric force continuous. It may be looked upon as an equation to determine ρ when V is given or vice versa. An exactly similar expression holds good in hydrokinetics, provided that for the electric potential we substitute velocity potential, and for the electric force the velocity of the liquid.

The Poisson equation cannot, however, be applied in the above form to a region which is partly within and partly without an electrified conductor, because then the electric force undergoes a sudden change in value from zero to a finite value, in passing outwards through the bounding surface of the conductor. We can, however, obtain another equation called the “surface characteristic equation” as follows:—Suppose a very small area dS described on a conductor having a surface density of electrification σ. Then let a small, very short cylinder be described of which dS is a section, and the generating lines are normal to the surface. Let V1 and V2 be the potentials at points just outside and inside the surface dS, and let n1 and n2 be the normals to the surface dS drawn outwards and inwards; then −dV1 / dn1 and −dV2 / dn2 are the normal components of the force over the ends of the imaginary small cylinder. But the force perpendicular to the curved surface of this cylinder is everywhere zero. Hence the total flux through the surface considered is −{(dV1 / dn1) + (dV2 / dn2)} dS, and this by a previous theorem must be equal to 4πσdS, or the total included electric quantity. Hence we have the surface characteristic equation,17

(dV1 / dn1) + (dV2 / dn2) + 4πσ = 0

(19).

Let us apply these theorems to a portion of a tube of electric force. Let the part selected not include any charged surface. Then since the generating lines of the tube are lines of force, the component of the electric force perpendicular to the curved surface of the tube is everywhere zero. But the electric force is normal to the ends of the tube. Hence if dS and dS′ are the areas of the ends, and +E and -E′ the oppositely directed electric forces at the ends of the tube, the surface integral of normal force on the flux over the tube is

EdS − E′dS′

(20),

and this by the theorem already given is equal to zero, since the tube includes no electricity. Hence the characteristic quality of a tube of electric force is that its section is everywhere inversely as the electric force at that point. A tube so chosen that EdS for one section has a value unity, is called a unit tube, since the product of force and section is then everywhere unity for the same tube.

In the next place apply the surface characteristic equation to any point on a charged conductor at which the surface density is σ. The electric force outward from that point is −dV/dn, where dn is a distance measured along the outwardly drawn normal, and the force within the surface is zero. Hence we have

−dV/dn = 4.0πσ or σ = −(¼π) dV/dn = E/4π.

The above is a statement of Coulomb’s law, that the electric force at the surface of a conductor is proportional to the surface density of the charge at that point and equal to 4π times the density.18

If we define the positive direction along a tube of electric force as the direction in which a small body charged with positive electricity would tend to move, we can summarize the above facts in a simple form by saying that, if we have any closed surface described in any manner in an electric field, the excess of the number of unit tubes which leave the surface over those which enter it is equal to 4π-times the algebraic sum of all the electricity included within the surface.

Every tube of electric force must therefore begin and end on electrified surfaces of opposite sign, and the quantities of positive and negative electricity on its two ends are equal, since the force E just outside an electrified surface is normal to it and equal to σ/4π, where σ is the surface density; and since we have just proved that for the ends of a tube of force EdS = E′dS′, it follows that σdS = σ′dS′, or Q = Q′, where Q and Q′ are the quantities of electricity on the ends of the tube of force. Accordingly, since every tube sent out from a charged conductor must end somewhere on another charge of opposite sign, it follows that the two electricities always exist in equal quantity, and that it is impossible to create any quantity of one kind without creating an equal quantity of the opposite sign.  
  
---  
Fig. 4.  
  
We have next to consider the energy storage which takes place when electric charge is created, i.e. when the dielectric is strained or polarized. Since the potential of a conductor is defined to be the work required to move a unit of positive electricity from the surface of the earth or from an infinite distance from all electricity to the surface of the conductor, it follows that the work done in putting a small charge dq into a conductor at a potential v is v dq. Let us then suppose that a conductor originally at zero potential has its potential raised by administering to it small successive doses of electricity dq. The first raises its potential to v, the second to v′ and so on, and the nth to V. Take any horizontal line and divide it into small elements of length each representing dq, and draw vertical lines representing the potentials v, v′, &c., and after each dose. Since the potential rises proportionately to the quantity in the conductor, the ends of these ordinates will lie on a straight line and define a triangle whose base line is a length equal to the total quantity Q and height a length equal to the final potential V. The element of work done in introducing the quantity of electricity dq at a potential v is represented by the element of area of this triangle (see fig. 4), and hence the work done in charging the conductor with quantity Q to final potential V is ½QV, or since Q = CV, where C is its capacity, the work done is represented by ½CV² or by ½Q² / C.

If σ is the surface density and dS an element of surface, then ∫σdS is the whole charge, and hence ½ ∫ VσdS is the expression for the energy of charge of a conductor.

We can deduce a remarkable expression for the energy stored up in an electric field containing electrified bodies as follows:19 Let V denote the potential at any point in the field. Consider the integral

W = | 1 | ∫ ∫ ∫ {( | dV | ) | ² | \+ ( | dV | ) | ² | \+ ( | dV | ) | ² | } dx dy dz.  
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---  
8π | dx |   | dy |   | dz  
  
(21)

where the integration extends throughout the whole space unoccupied by conductors. We have by partial integration

∫ ∫ ∫ ( | dV | ) | ² | dx dy dz = ∫ ∫ V | dV | dy dz − ∫ ∫ ∫ V | d²V | dx dy dz,  
---|---|---|---|---|---|---|---|---  
dx |   | dx | dx²  
  
and two similar equations in y and z. Hence

1 | ∫ ∫ ∫ {( | dV | ) | ² | \+ ( | dV | ) | ² | \+ ( | dV | ) | ² | } dx dy dz =  
---|---|---|---|---|---|---|---|---|---|---|---|---|---  
8π | dx |   | dy |   | dz  
1 | ∫ ∫ V | dV | dS − | 1 | ∫ ∫ ∫ V∇V dx dy dz  
---|---|---|---|---|---  
8π | dn | 8π  
  
(22)

where dV/dn means differentiation along the normal, and ∇ stands for the operator d²/dx² + d²/dy² + d²/dz². Let E be the resultant electric force at any point in the field. Then bearing in mind that σ = (1/4π) dV/dn, and ρ = −(1/4π) ∇V, we have finally

1 | ∫ ∫ ∫ E²dv = | 1 | ∫ ∫ Vσ dS + | 1 | ∫ ∫ ∫ Vρ dv.  
---|---|---|---|---|---  
8π | 2 | 2  
  
The first term on the right hand side expresses the energy of the surface electrification of the conductors in the field, and the second the energy of volume density (if any). Accordingly the term on the left hand side gives us the whole energy in the field.

Suppose that the dielectric has a constant K, then we must multiply both sides by K and the expression for the energy per unit of volume of the field is equivalent to ½DE where D is the displacement or polarization in the dielectric.

Furthermore it can be shown by the application of the calculus of variations that the condition for a minimum value of the function W, is that ∇V = 0. Hence that distribution of potential which is necessary to satisfy Laplace’s equation is also one which makes the potential energy a minimum and therefore the energy stable. Thus the actual distribution of electricity on the conductor in the field is not merely a stable distribution, it is the only possible stable distribution.  
  
---  
Fig. 5.  
  
Method of Electrical Images.—A very powerful method of attacking problems in electrical distribution was first made known by Lord Kelvin in 1845 and is described as the method of electrical images.20 By older mathematical methods it had only been possible to predict in a few simple cases the distribution of electricity at rest on conductors of various forms. The notion of an electrical image may be easily grasped by the following illustration: Let there be at A (see fig. 5) a point-charge of positive electricity +q and an infinite conducting plate PO, shown in section, connected to earth and therefore at zero potential. Then the charge at A together with the induced surface charge on the plate makes a certain field of electric force on the left of the plate PO, which is a zero equipotential surface. If we remove the plate, and yet by any means can keep the identical surface occupied by it a plane of zero potential, the boundary conditions will remain the same, and therefore the field of force to the left of PO will remain unaltered. This can be done by placing at B an equal negative point-charge −q in the place which would be occupied by the optical image of A if PO were a mirror, that is, let −q be placed at B, so that the distance BO is equal to the distance AO, whilst AOB is at right angles to PO. Then the potential at any point P in this ideal plane PO is equal to q/AP − q/BP = O, whilst the resultant force at P due to the two point charges is 2qAO/AP³, and is parallel to AB or normal to PO. Hence if we remove the charge −q at B and distribute electricity over the surface PO with a surface density σ, according to the Coulomb-Poisson law, σ = qAO / 2πAP³, the field of force to the left of PD will fulfil the required boundary conditions, and hence will be the law of distribution of the induced electricity in the case of the actual plate. The point-charge −q at B is called the “electrical image” of the point-charge +q at A.

We find a precisely analogous effect in optics which justifies the term “electrical image.” Suppose a room lit by a single candle. There is everywhere a certain illumination due to it. Place across the room a plane mirror. All the space behind the mirror will become dark, and all the space in front of the mirror will acquire an exalted illumination. Whatever this increased illumination may be, it can be precisely imitated by removing the mirror and placing a second lighted candle at the place occupied by the optical image of the first candle in the mirror, that is, as far behind the plane as the first candle was in front. So the potential distribution in the space due to the electric point-charge +q as A together with −q at B is the same as that due to +q at A and the negative induced charge erected on the infinite plane (earthed) metal sheet placed half-way between A and B.  
  
---  
Fig. 6.  
  
The same reasoning can be applied to determine the electrical image of a point-charge of positive electricity in a spherical surface, and therefore the distribution of induced electricity over a metal sphere connected to earth produced by a point-charge near it. Let +q be any positive point-charge placed at a point A outside a sphere (fig. 6) of radius r, and centre at C, and let P be any point on it. Let CA = d. Take a point B in CA such that CB·CA = r², or CB = r²/d. It is easy then to show that PA : PB = d : r. If then we put a negative point-charge −qr/d at B, it follows that the spherical surface will be a zero potential surface, for

q | − | rq | · | 1 | = 0  
---|---|---|---|---|---  
PA | d | PB  
  
(24).

Another equipotential surface is evidently a very small sphere described round A. The resultant force due to these two point-charges must then be in the direction CP, and its value E is the vector sum of the two forces along AP and BP due to the two point-charges. It is not difficult to show that

E = − (d² − r²) q / rAP³

(25),

in other words, the force at P is inversely as the cube of the distance from A. Suppose then we remove the negative point-charge, and let the sphere be supposed to become conductive and be connected to earth. If we make a distribution of negative electricity over it, which has a density σ varying according to the law

σ = −(d² − r²) q / 4πrAP³

(26),

that distribution, together with the point-charge +q at A, will make a distribution of electric force at all points outside the sphere exactly similar to that which would exist if the sphere were removed and a negative point charge −qr/d were placed at B. Hence this charge is the electrical image of the charge +q at A in the spherical surface.

We may generalize these statements in the following theorem, which is an important deduction from a wider theorem due to G. Green. Suppose that we have any distribution of electricity at rest over conductors, and that we know the potential at all points and consequently the level or equipotential surfaces. Take any equipotential surface enclosing the whole of the electricity, and suppose this to become an actual sheet of metal connected to the earth. It is then a zero potential surface, and every point outside is at zero potential as far as concerns the electric charge on the conductors inside. Then if U is the potential outside the surface due to this electric charge inside alone, and V that due to the opposite charge it induces on the inside of the metal surface, we must have U + V = 0 or U = −V at all points outside the earthed metal surface. Therefore, whatever may be the distribution of electric force produced by the charges inside taken alone, it can be exactly imitated for all space outside the metal surface if we suppose the inside charge removed and a distribution of electricity of the same sign made over the metal surface such that its density follows the law

σ = −(¼π) dU / dn

(27),

where dU/dn is the electric force at that point on the closed equipotential surface considered, due to the original charge alone.

Bibliography.—For further developments of the subject we must refer the reader to the numerous excellent treatises on electrostatics now available. The student will find it to be a great advantage to read through Faraday’s three volumes entitled Experimental Researches on Electricity, as soon as he has mastered some modern elementary book giving in compact form a general account of electrical phenomena. For this purpose he may select from the following books: J. Clerk Maxwell, Elementary Treatise on Electricity (Oxford, 1881); J.J. Thomson, Elements of the Mathematical Theory of Electricity and Magnetism (Cambridge, 1895); J.D. Everett, Electricity, founded on part iii. of Deschanel’s Natural Philosophy (London, 1901); G.C. Foster and A.W. Porter, Elementary Treatise on Electricity and Magnetism (London, 1903); S.P. Thompson, Elementary Lessons on Electricity and Magnetism (London, 1903)·

When these elementary books have been digested, the advanced student may proceed to study the following: J. Clerk Maxwell, A Treatise on Electricity and Magnetism (1st ed., Oxford, 1873; 2nd ed. by W.D. Niven, 1881; 3rd ed. by J.J. Thomson, 1892); Joubert and Mascart, Electricity and Magnetism, English translation by E. Atkinson (London, 1883); Watson and Burbury, The Mathematical Theory of Electricity and Magnetism (Oxford, 1885); A. Gray, A Treatise on Magnetism and Electricity (London, 1898). In the collected Scientific Papers of Lord Kelvin (3 vols., Cambridge, 1882), of James Clerk Maxwell (2 vols., Cambridge, 1890), and of Lord Rayleigh (4 vols., Cambridge, 1903), the advanced student will find the means for studying the historical development of electrical knowledge as it has been evolved from the minds of some of the master workers of the 19th century.

(J. A. F.)

* * *

1 See Maxwell, Elementary Treatise on Electricity (Oxford, 1881), p. 47.

2 See Maxwell, Treatise on Electricity and Magnetism (3rd ed., Oxford, 1892), vol. i. p. 80.

3 Maxwell, Ibid. vol. i. § 74a; also Electrical Researches of the Hon. Henry Cavendish, edited by J. Clerk Maxwell (Cambridge, 1879), p. 104.

4 Laplace (Mec. Cel. vol. i. ch. ii.) gave the first direct demonstration that no function of the distance except the inverse square can satisfy the condition that a uniform spherical shell exerts no force on a particle within it.

5 The solution of the problem of determining the distribution on an ellipsoid of a fluid the particles of which repel each other with a force inversely as the nth power of the distance was first given by George Green (see Ferrer’s edition of Green’s Collected Papers, p. 119, 1871).

6 See Thomson and Tait, Treatise on Natural Philosophy, § 519.

7 See article “Electricity,” Encyclopaedia Britannica (9th edition), vol. viii. p. 30. The reader is also referred to an article by Lord Kelvin (Reprint of Papers on Electrostatics and Magnetism, p. 178), entitled “Determination of the Distribution of Electricity on a Circular Segment of a Plane, or Spherical Conducting Surface under any given Influence,” where another equivalent expression is given for the capacity of an ellipsoid.

8 See Maxwell, Electricity and Magnetism, vol. i. pp. 284-305 (3rd ed., 1892).

9 It is an interesting fact that Cavendish measured capacity in “globular inches,” using as his unit the capacity of a metal ball, 1 in. in diameter. Hence multiplication of his values for capacities by 2.54 reduces them to E.S. units in the C.G.S. system. See Elec. Res. p. 347.

10 For fuller details of these methods of comparison of capacities see J.A. Fleming, A Handbook for the Electrical Laboratory and Testing Room, vol. ii. ch. ii. (London, 1903).

11 See Fleming, Handbook for the Electrical Laboratory, vol. ii. p. 130.

12 Faraday, Experimental Researches on Electricity, vol. i. § 1252. For a very complete set of tables of dielectric constants of solids, liquids and gases see A. Winkelmann, Handbuch der Physik, vol. iv. pp. 98-148 (Breslau, 1905); also see Landolt and Börnstein’s Tables of Physical Constants (Berlin, 1894).

13 See the following papers by J.A. Fleming and James Dewar on dielectric constants at low temperatures: “On the Dielectric Constant of Liquid Oxygen and Liquid Air,” Proc. Roy. Soc., 1897, 60, p. 360; “Note on the Dielectric Constant of Ice and Alcohol at very low Temperatures,” ib., 1897, 61, p. 2; “On the Dielectric Constants of Pure Ice, Glycerine, Nitrobenzol and Ethylene Dibromide at and above the Temperature of Liquid Air,” id. ib. p. 316; “On the Dielectric Constant of Certain Frozen Electrolytes at and above the Temperature of Liquid Air,” id. ib. p. 299—this paper describes the cone condenser and methods used; “Further Observations on the Dielectric Constants of Frozen Electrolytes at and above the Temperature of Liquid Air,” id. ib. p. 381; “The Dielectric Constants of Certain Organic Bodies at and below the Temperature of Liquid Air,” id. ib. p. 358; “On the Dielectric Constants of Metallic Oxides dissolved or suspended in Ice cooled to the Temperature of Liquid Air,” id. ib. p. 368.

14 See Faraday, Experimental Researches, vol. i. § 1245; R.H.A. Kohlrausch, Pogg. Ann., 1854, 91; see also Maxwell, Electricity and Magnetism, vol. i. § 327, who shows that a composite or stratified dielectric composed of layers of materials of different dielectric constants and resistivities would exhibit the property of residual charge.

15 Fleming and Ashton, “On a Model which imitates the behaviour of Dielectrics.” Phil. Mag., 1901 [6], 2, p. 228.

16 The beginner is often puzzled by the constant appearance of the factor 4π in electrical theorems. It arises from the manner in which the unit quantity of electricity is defined. The electric force due to a point-charge q at a distance r is defined to be q/r², and the total flux or induction through the sphere of radius r is therefore 4πq. If, however, the unit point charge were defined to be that which produces a unit of electric flux through a circumscribing spherical surface or the electric force at distance r defined to be ¼πr², many theorems would be enunciated in simpler forms.

17 See Maxwell, Electricity and Magnetism, vol. i. § 78b (2nd ed.).

18 Id. ib. vol. i. § 80. Coulomb proved the proportionality of electric surface force to density, but the above numerical relation E = 4πσ was first established by Poisson.

19 See Maxwell, Electricity and Magnetism, vol. i. § 99a (3rd ed., 1892), where the expression in question is deduced as a corollary of Green’s theorem.

20 See Lord Kelvin’s Papers on Electrostatics and Magnetism, p. 144.

* * *