field, whose charge we will, for simplicity, assume to be some whole number of units, to be divided into areas such that each is charged with a unit of positive electricity, and let each of these areas be taken as the base of a tube of force extending to the negative boundary. In this way the whole field will be exhaustively divided into tubes of force whose number gives the numerical value of the charge of the field. If we now suppose a series of equipotential surfaces to be drawn in the field, the potential' of each differing by unity from that of the next, every tube of force will be divided into a set of compartments or cells, the number of which gives the numerical value of the difference of potentials between the boundaries of the field. That is, the number of tubes of force gives the value of Q, and the number of cells into which each tube is divided by the equipotential surfaces gives the value of V – V'; consequently the whole number of cells gives the value of Q (V - V'), or half the number of cells into which the whole field is divided by tubes of force and equipotential surfaces drawn in the way described gives the electric energy, † Q (V – V'), of the field.

If we imagine one or both of the boundaries of the field' to be capable of yielding to the electric forces which tend to draw them together, the field would ultimately collapse, its energy being exhausted by the work done on the movable boundaries. During this process the difference of potentials between the boundaries would gradually fall from the initial value V - V' to nothing when the surfaces were in contact, the mean difference of potentials being (V -- V'), and the work done being the product of this mean value into the charge of the field.

57. Condensers.-When two electric conductors are so placed that their surfaces form the boundaries of a field of relatively great capacity, the arrangement is very frequently spoken of as a condenser. For example, we have seen that a conducting sphere of radius a centimetres, at a great distance from other conductors, forms one boundary of a field of capacity Ka; whereas, if the sphere be surrounded by a concentric conducting shell of radius b, ab Thus, if a = 10 cm., and

the resulting, field has capacity K b

a

b= 11 cm., the capacity in the second case is II times that in the first; or, a being still 10 cm., if b

=

10.1, the capacity of the field bounded by the two spheres becomes 1010 K, instead of 10 K.

58. Various Forms of Condensers.-One of the simplest forms of condenser is that known as Æpinus's condenser (Fig. 41); consisting of two equal insulated metal plates, supported so that they can be brought into contact with the opposite faces of a plate of glass larger than themselves, or simply placed so as to face each other at a short distance in air. In either case the capacity can be diminished by drawing the plates apart, or increased by moving them nearer together. To charge the condenser, it is only necessary to connect the two plates with opposite sides of an electric machine or other contrivance for producing a difference of potentials.

Another equivalent arrangement, except that the distance be

B

FIG. 41.

tween the conductors cannot be varied, is that known as Franklin's pane, consisting of two equal pieces of tin-foil pasted opposite each other on the two faces of a sheet of glass large enough to project a good way beyond them on all sides. For better insulation, the uncoated parts of the glass are usually covered with shellac varnish.

When condensers of very great capacity are required, but are not to be exposed to great differences of potential, they are often made by superposing alternately sheets of tin-foil and thin plates of mica, or sheets of paraffined paper. The first, third, fifth... sheets of tin-foil are connected' together, and form one boundary of a field of great area but very small thickness, and the second,

Another form of condenser that is very frequently employed in connection with high differences of potential is the Leyden jar (Fig. 42). This consists of a glass jar or bottle, coated internally and externally with tinfoil to within a moderate distance of the opening, the rest of the surface being varnished with shellac. A brass rod, terminated by a ball, connected with the inner coating, and passing out through the neck without touching it, enables electrical connection to be made with the inner coating when required. Sometimes the jar is closed at the mouth by a non

P

Р

M

FIG. 42.

conducting lid; in this case the rod is usually supported by being put tight through the lid, and makes contact with the inner coating by means of a flexible spring or a few inches of chain. Electrically, the Leyden jar is exactly equivalent to the coated pane, but it is more compact and convenient to handle. The tin-foil coatings may be replaced by conductors of any other material; the reason why tin-foil is commonly used being chiefly that its flexibility allows it to be readily adapted to the surface of the glass. Where very high insulation is required, it answers well to use strong sulphuric acid (oil of vitriol) to make contact with the inner surface of the glass (which, in this case, is not varnished), and to connect the acid with the other apparatus by a platinum wire dipping into it.

59. Quantity of Electricity in a Condenser.-The charge of a condenser is, of course, the charge of the electric field which is bounded by its conducting surfaces; it is therefore represented by

Q = C (V - V')

and depends partly on the condenser itself and partly on the difference of potentials established between its surfaces.

The formula C =

SK 4 пе

(§ 52) applies with sufficient accuracy to the capacity of any of the forms of condenser described. From this we see that, with a given insulating material, the capacity can

be increased by increasing the area S of the opposed conducting surfaces, for these determine the area of cross-section of the field, and also by decreasing the distance e between the surfaces, or, in other words, the length of the field. But if the charge is to be great, not only the capacity, but the difference of potentials, V- V', must have a considerable value. This is usually determined by the electric machine or other instrument used for producing the charge, and under fixed conditions in this respect may be taken as constant. The value of the difference of potentials that is to be employed puts a practical limit to the extent to which capacity can be increased by diminishing the thickness, e, of the dielectric. A small thickness and great difference of potentials implies a high

intensity in the field, and, as already mentioned (§ 20), when the stress in the field reaches a limit depending on the nature of the material, the dielectric gives way and is burst through by a dis

FIG. 44.

ruptive discharge. The maximum stress that a given material can support without allowing discharge to take place through it is called its electric strength. Little is known about the exact value of this property for different substances, but it is certainly many times greater for glass and various other solid dielectrics than it is for air. For this reason, in condensers intended to support great differences of potential, as well as to have a large capacity, glass is generally used as the dielectric, and the commonest form is that of the Leyden jar. To obtain sufficient surface without using jars of unwieldy size, several jars are often combined, as shown in Fig. 43, all the inner coatings being electrically connected by insulated metal rods, and the outer coatings by placing the jars on a tray, or in a box lined with tinfoil. The capacity of such a combination of jars, sometimes

called a Leyden battery, is equal to the sum of the capacities of the separate jars (comp. § 50).

When it is not desirable to expose a condenser to the full difference of potentials that may be available, it is sometimes advantageous to connect two or more jars in series, or in cascade as it is called, in the way shown in Fig. 44. The inner coating of the jar