Zkna

ZKNA = 47rZe and a + - a (KF„) + a = 47rp are ax ay az merely two different mathematical ways of expressing the fact that the number of unit tubes of electric force which come out of a charge e is equal to 4ire.

If V denotes any vector with components VZ, then avx — + a + a VZ is called the divergence of the vector, and is ax az denoted by div. V. It is equal to the number of unit tubes of the vector which start from unit volume. Hence we may write div. KF = 4irp.

The electric intensity F close to the surface of a charged con ductor where the specific inductive capacity is K may be found in the same way as in vacuo. Consider a cylinder of cross section a with its sides perpendicular to the charged surface. Apply Gauss' theorem to the volume en closed by this cylinder and two cross sec tions of it, one just outside and one just inside the surface of the conductor. The tensity inside is zero, and outside it is per pendicular to the surface, so that we have KFa = 47ras,where s is the surface density of charge. Thus F= 47rs/K, as was to be expected. The force on the charged sur face is Fs/2, as in vacuo, and this is equal to or to The intensity F, due to a charge e in a medium of specific inductive capacity K, is got by applying Gauss' theorem to a sphere of radius r with the charge e at its centre. This gives = 47re, or F = as was to be expected. Gauss' theorem of course makes KFa along a unit tube of force, since this was assumed in obtaining the generalized theorem.

Refraction of Tubes of Force.

Consider the electric field at the boundary between two insulators, one of specific inductive capacity and the other of specific inductive capacity K2. In fig. 12 let AB be the boundary, and let CD be a tube of force passing across it. Let and be the products of the intensity in and the cross section of the tube for each of the two media. Also, let e1 and 02 be the angles between the tube and the normal to the boundary. The potential must be the same in both media at the surface, so that we have We have also = Also the projection of on the boundary is equal to that of so that = These equations give Thus it appears that the tubes of force are refracted at a boundary between two different insulators.

The capacity of a condenser, consisting of two parallel metal plates at a distance d apart, with a uniform slab of an insulator of specific inductive capacity K and thickness t less than d, can be calculated easily. Consider a unit tube of force going from one plate to the other. KFa is constant along this tube, so that, since the tube is straight and a constant, the intensity will be K times smaller in the slab than in the air. If F is the electric intensity in the air, the potential difference between the plates is equal to F(d — t) +Ft/K. Also F = 47rs, so that P = 47rs(d — t) +47rst/K; the capacity per unit area is therefore Energy of Charged Conductors.—Consider a system of in sulated conductors, and let the charges on them be etc., and their potentials P3, etc. The work required to charge the system may be calculated as follows:—Suppose that we start with all the conductors uncharged, and then charge them all together so that each charge is the same fraction f of its final value. Thus the charges are etc., and during the charging process f increases from o to r. The potentials will also be etc., for the electric intensities clearly will be proportional to the charges. The work done, when f is increased from f to f+df is the total work is therefore Thus the electrostatic energy of a system of charged conductors is equal to one half the sum of the products of the charges on and the potentials of the conductors. In the case of a con denser consisting of two conductors with charges E and —E and potentials and P2, the energy is therefore 2E(P1—P2).

Energy in the Electrostatic Field.

The electrostatic energy can be regarded • as distributed throughout the electric field, between the conductors. To see this, consider a unit tube of force, starting from a conductor at potential and ending on another conductor at potential P2. The charges on the ends of the tube are 1/47r and —1/4i-, so that the energy associated with the tube is —P2)/8r. Suppose the tube divided into a great many short elements, and let l be the length and a the cross section of an element. The potential difference between the ends of the tube is then given by = 2F1, where F is the electric intensity in the tube. The energy is therefore (IFl)/87r. But KFa all along the tube, so that we can multiply the values of F for each element by KFa at the element, and so get for the energy The volume of an element of the tube is la, so that, if each element contained en ergy per unit volume equal to the total amount of energy would be that ac tually present. It is generally supposed that the electrostatic energy is distributed throughout the field so that its density is 7r.

Force on Insulators in an

static Field.—There is a force on lators in a non-uniform electric field. As a simple illustration of this, consider the case of a slab of an insulator of thickness t, partly between two equal parallel metal plates at a distance d apart, as shown in fig. 13 ; AB and CD are the plates, and EF the slab. Let the breadth of the slab be b, and the length of it between the plates x. The capacity of C between the plates is + 47rd { Kd — where A is the area of each plate. The electrical energy W is 2PE, where P is the potential difference between the plates, and E the charge on one plate or, since E =PC, it is Hence, if we suppose P kept constant, dW = 2 dx' P2 dC do To keep the potential difference con stant requires work, PdE or to be done, so that if f is the force on the slab then we have, by the conservation of energy, section of this article dealing with the electron theory. The in ductive capacity of a substance is sometimes called its dielectric constant. The following table gives the specific inductive capacities of several substances, that of a vacuum being taken as equal to unity: Glass . . . 5-10 Water . . . .8o Hard rubber . . . 3.1 Ice . . • . 3 Paraffin wax . . . 2.1 Alcohol . . .26 Mica . . . . 5.6-6•o Air . . . i •0006 Sulphur . . . . 4 Hydrogen . . . 1-00026 Pyroelectricity.—It was discovered in 1700 that a tourmaline crystal put in hot ashes attracts the ashes. Aepinus in 1756 found the effect to be due to electrical charges on the ends of the crystal, which are positive on one end and negative on the other. A little later Canton showed that the charges are produced by a change of temperature, and that the charges due to cooling are opposite to those due to raising the temperature. Many other crystals have since been found to possess the same property which is called pyroelectricity. Hauy, about 1800, found that only hemihedral crystals with inclined faces develop pyro electricity. Lord Kelvin in 186o worked out a theory of pyro electricity which is generally accepted. Kelvin supposed that the molecules of pyroelectric crystals contain equal positive and negative charges which do not neutralize each other, so that the molecules form electric doublets. The electric moment of a doublet, consisting of a charge e and a charge —c at a distance d apart, is equal to de. In the crystal the molecules are arranged in a regular way, and Kelvin supposed that all the electric doublets lie in parallel directions, so that a crystal containing N mole cules, each having an electric moment M, has a total moment NM. The moment per unit volume is called the polarization, or the intensity of electrification, so that, if it is denoted by P then P = NM/ V, where V is the volume of the crystal.

The electric moment of a rod of length 1 and cross section a, polarized along its length, is equal to Pla, and is the same as that due to charges +Pa at one end and —Pa at the other end. If a crystal is kept at a constant temperature for a long time, the ends attract charges sufficient to reduce the electric mo ment to zero, so that the crystal becomes electrically neutral. It is still polarized, but has charges —Pa and +Pa, on its ends, which have a moment equal and opposite to that due to the polarization. The neutralization of the polarization can be brought about immediately by passing the crystal through the conducting gases rising from a flame.

If the temperature of a crystal is raised, then its volume V increases, so diminishing P. Also, the molecular moment M may depend on the temperature. The charge Pa on the end of a crystal is equal to NMa/V, or to NM/l, so that, if we suppose that l = +aT), where T is the temperature and a the co efficient of linear expansion along 1, and that M = where (3 is a constant, then we have, for the charge Pa, or E, or, approximately, E = -- (j3 — a) T) . Thus, if the tem perature of a neutral crystal is changed from to it will acquire apparent charges, on its ends, given by For a tourmaline crystal, Voigt found that the polarization P at 24° C is equal to in electrostatic units, and that it changes by +1.2 for one degree rise of temperature.

Piezoelectricity.

J. and P. Curie in 1880 discovered that pyroelectric crystals become electrified when subjected to pres sure or tension along the direction of the polarization. Tension produces the same effect as raising the temperature, and pressure the same effect as lowering it. Since E = NM/l, we might expect increasing 1 by tension to have the same effect as increasing it by raising the temperature, provided M also changes to the same extent in both cases. It appears that, in many cases,the effect of tension is nearly the same as that of a change of temperature which gives an equal change of length. Lippmann in 1881 pointed out that, since straining the crystal alters its polarization, we should expect an electric field to strain it, i.e., to alter its size and shape. This was shown to be the case by J. and P. Curie. If a rapidly alternating electric field is applied to a pyroelectric crystal, the crystal is made to vibrate, and, if the frequency agrees with the natural frequency of the crystal, resonance occurs and the amplitude of oscillation becomes relatively large.

In the piezo effect, the quantity of electricity set free is pro portioned to the pressure, so that the effect may be used to measure pressures. The effect is suitable for measuring rapidly changing pressures, since the movements of the crystal under pressure are so small that troublesome inertia effects, which take place when springs are used, are eliminated. In 1919 Sir. J. J. Thomson suggested the use of piezo crystals for recording ex plosive pressures, and many experiments have been carried out on these lines, a cathode ray oscillograph to measuring the rate of liberation of electricity. (See Keys "A Piezoelectric Method of Measuring Explosion Pressures," Phil. Mag. 1921.) Electrets.—Eguchi in 1925 discovered that if certain mixtures of waxes are allowed to solidify in a strong electric field, they become electrically polarized. A mixture of Carnauba wax 45%, white resin 45% and white beeswax 1o% is suitable. A slab of this mixture, formed in an electric field perpendicular to its surfaces, maintains a positive charge on one surface and a negative charge on the other. The charges do not diminish ap preciably in several years. Such a polarized slab of wax is the electrical analogue of a permanent magnet and is called an electret. It is supposed that the wax molecules have electrical moments, and the field lines them up when the wax is liquid, and after it has solidified they remain all pointing in the same direc tion, even when the field is removed. The polarization is neutral ized by charges attracted from surrounding bodies, as with pyroelectric crystals, but the neutralization does not become exact because the polarization very slowly diminishes.