The latter formula applies to any two parallel disks of surface S, whether circular or otherwise, provided they are large as compared with the distance b between them. To calculate down to microfarads the numbers given by any of the above must be divided by 900,000.

305. Energy of Discharge of Leyden Jar or Condenser. It follows from the definition of potential, given in Art. 263, that in bringing up one + unit of electricity to the potential V, the work done is V ergs. This assumes, however, that the total potential V is not thereby raised, and on this assumption the work done in bringing up Q units would be QV ergs. If, however, the potential is nothing to begin with, and is raised to V by the charge Q, the average potential during the operation is only V; hence the total work done in bringing up the charge Q from zero potential to potential V is QV ergs. Now, according to the principle of the conservation of energy, the work done in charging a jar or condenser with electricity is equal to the work which could be done by that quantity of electricity when the jar is discharged. Hence QV represents also the energy of the discharge. Since Q VK, it follows that we may write QV in the form. That is to say, if a condenser of capacity K K

=

is charged by having a charge Q imparted to it, the energy of the charge is proportional directly to the square of the quantity, and inversely to the capacity of the condenser. 306. Symbol for Condenser. Electricians use as symbols for condensers in diagrams of electric circuits those given in Fig. 160. The origin of these symbols is the alternate layers of tinfoils. The symbol on the right

Fig. 160.

suggests six layers of foil, of which the first, third, and

* If Q is given in coulombs and V in volts, the work will be expressed not in ergs but in joules (Art. 354).

fifth are joined together, and the second, fourth,. and sixth are also joined together.

307. Capacities joined in Parallel. To join two condensers together in parallel the positive foils of one are joined to the positive foils of the other, and their negative foils are also joined together. In Fig. 161 the two condensers K, and K, are joined in parallel. They will thus act simply like one large condenser of capacity = K1+ K. Any charge flowing in on the + side will divide between the two in proportion to their capacities.

If two equal Leyden jars are charged to the same potential, and then their inside and outside coatings are respectively joined, their united charge will be the same as that of a jar of equal thickness, but having twice the amount of surface.

+

K2

+

Fig. 161.

If a charged Leyden jar is placed similarly in communication with an uncharged jar of equal capacity, the charge will be shared equally between the two jars, and the passage of electricity from one to the other will be evidenced by the production of a spark when the respective coatings are put into communication. Here, however, half the energy of the charge is lost in the operation of sharing the charge, for each jar will have only Q for its charge and V for its potential; hence the energy of the charge of each, being half the product of charge and potential, will only be one quarter of the original energy. The spark which passes in the operation of dividing the charge is, indeed, evidence of the loss of energy; it is about half as powerful as the spark would have been if the first jar had been simply discharged, and it is just twice as powerful as the small sparks yielded finally by the discharge of each jar after the charge has been shared between them.

The energy of a charge of the jar manifests itself, as

stated above, by the production of a spark at discharge; the sound, light, and heat produced being the equivalent of the energy stored up. If discharge is effected slowly through a long thin wire of high resistance the air spark may be feeble, but the wire may be perceptibly heated. A wet string being a feeble conductor affords a slow and almost silent discharge; here probably the electrolytic conduction of the moisture is accompanied by an action resembling that of secondary batteries (Lesson 492) tending to prolong the duration of the discharge.

308. Capacities joined in Series. If two condensers are joined in series they will act as a condenser having a lesser capacity than either of them separately. Their joint capacity in series will be the reciprocal of the sum of the reciprocals of their capacities separately.

Proof. Let two condensers K1 and K2 be set in series (Fig. 162) between two points across which there is a difference of

potential V. This difference of potential will be divided between the two inversely in proportion to their capacities, seeing that the quantities of electricity that are displaced into and out of their respective coatings are necessarily equal. Or, if Q be this quantity, and Kg the effective or joint capacity of the two together, to find the latter, we have:

K,

Fig. 162.

K2

Example. If two condensers, respectively 3 and 2 microfarads, are joined in series, they will act as a single condenser of capacity = 1 / (+1)=1 microfarads.

309. Charge of Jars arranged in Cascade. — Franklin suggested that a series of jars might be arranged, the outer coating of one being connected with the inner one of the next, the outer coating of the last being connected to earth. The object of this arrangement was that the second jar might be charged with the electricity repelled from the outer coating of the first, the third from that of the second, and so on. This "cascade" arrangement, however, is of no advantage, the sum of the charges accumulated in the series being only equal to that of one single jar if used alone. For if the inner coating of the first jar be raised to V, that of the outer coating of the last jar remaining at zero in contact with earth, the difference of potential between the outer and inner coat1 ing of any one jar will be only V, where n is number of

n

jars. And as the charge in each jar is equal to its capacity K, multiplied by its potential, the charge in each will only be KV, and in the whole n jars the total

n

charge will be nKV, or KV, or equals the charge of one

n

jar of capacity K raised to the same potential V.

LESSON XXIV. - Phenomena of Discharge

310. Conductive Discharge. An electrified conductor may be discharged in at least three different ways, depending on the medium through which the discharge is effected, and varying with the circumstances of the discharge. If the discharge takes place by the passage of a continuous current, as when electricity flows through a thin wire connecting the knobs of an influence machine, or joining the positive pole of a battery to the negative

pole, the operation is termed a "conductive discharge. Under some circumstances a conductive discharge takes the nature of an oscillation to and fro (Art. 515).

311. Disruptive Discharge. It has been shown how influence across a non-conducting medium is always accompanied by a mechanical stress upon the medium; the tension along the electric lines of force increasing as the square of the intensity of the electric field. If this stress is very great the non-conducting medium will suddenly give way and a spark will burst across it. Such a discharge is called a "disruptive" discharge.

A very simple experiment will set the matter in a clear light. Suppose a metal ball charged with + electrification to be hung by a silk string above a metal plate lying on the ground. If we lower down the suspended ball a spark will pass between it and the plate when they come very near together, and the ball will then be found to have lost all its previous charge. It was charged with a certain quantity of electricity; and as it had, when suspended out of the range of other conductors, a certain capacity (numerically equal to its radius in centimetres), the electricity on it would be at a certain potential (namely = Q/K), and the charge would be distributed uniformly all over it. The plate lying on the earth would be all the while at zero potential. But when the suspended ball was lowered down towards the plate the previous state of things was altered. In the presence of the charge of the ball the potential of the plate would rise, were it not that, by influence, just enough negative electrification appears on it to keep its potential still the same as that of the earth. The tension in the electric field will draw the charge of the ball downwards, and alter the distribution of the charge, the surfacedensity becoming greater at the under surface of the ball

*The student must remember that, by the definition of potential in Art. 263, the potential at a point is the sum of all the separate quantities of electricity near it, divided each by its distance from the point.