dielectric and is called its specific inductive capacity or dielectric

constant.

The derivation of this formula is given in §584.

Caution. The student is warned against thinking that the capacity of a condenser is the greatest charge which it can hold. The maximum possible charge of a condenser depends upon its insulation and the strength of the dielectric between its coatings to resist disruptive discharge. One condenser may be charged to many times its capacity. before it discharges, while another may break down or discharge before it is charged to one-tenth of its capacity.

Some Specific Inductive Capacities, or Dielectric Constants.

Hard rubber..
Glass.
Mica.

8.0

2.5 Paraffin..... 2.0 .6. to 8. Turpentine .. 2.2 Petroleum 3.1

Air (normal pressure). 1.00059
Carbon dioxide..
Hydrogen.

B

1.00090

1.00028

574. Hydraulic Analogy.-It is instructive to consider the following hydraulic model of a condenser. A metal box is divided into two parts A and B by a partition of thick sheet rubber. Each side is provided with a tubular opening controlled by a stopcock, the whole is then filled with water and immersed in a pond. While the stopcocks are open the pressure is the same in A and B and the rubber is not strained. It is like a Leyden jar uninsulated and discharged. Now attach a pump to B and force water in while the stopcock of A is left open or, what amounts to the same thing, connect the pump both to A and B so that it pumps water out of A and into B. The rubber will be strained as shown in the figure, the side A will be at the pressure of the pond which while the other side is at a higher pressure p. This difference in pressure p between the two sides is due to the strain of the rubber. If A and B are now connected by a pipe and the stopcocks are opened there will be a flow from one side to the other as the rubber springs back into the unstrained condition.

FIG. 318.-Hydraulic model of Leyden jar.

may be called zero,

So when a Leyden jar is discharged electricity may be thought of as forced from one coating to the other by the springing back of the displaced electricity in the dielectric.

If the rubber diaphragm were thicker more difference in pressure would be required to force in a given charge. So in a Leyden jar, the thicker the dielectric the greater the difference of potential between its coatings when it has a given charge.

If the diaphragm were made of a substance that was more yielding

than rubber, it would correspond to a dielectric of greater specific inductive capacity; for a given pressure would then force in a greater charge.

Also suppose the stopcock A is closed and the pump connected to B, pressure will be produced in B and perhaps a slight amount of water forced in due to the elastic yielding of the box itself, but the rubber diaphragm will not be appreciably strained and the pressure will be the same on both sides. This is the case of trying to charge an insulated jar. The stiff and but slightly yielding walls of the box represent the insulating dielectric that surrounds the Leyden jar and extends to the walls of the room, while the rubber diaphragm represents the thin glass dielectric between the coatings of the jar.

Remembering that the dielectric surrounding the jar is slightly yielding will enable the student to explain the succession of small sparks obtained from the insulated jar as described in §570.

575. Energy of Charge.-When we begin to charge a Leyden jar or condenser the two coatings are at the same potential and therefore no work is required to transfer the first little portion of charge from one coating to the other. But as the charging goes on the difference of potential between the two coatings increases and more work is required to produce a given increase in charge.

Suppose the final potential to which the jar is charged is V, and suppose that in charging it Q units of electricity are transferred from one coating to the other, giving one a charge + Q and the other a charge Q. If during this transfer the difference of potential between the coatings were to remain constant and equal to V the work done in charging would be QV ergs. But since the difference of potential is zero at the start and increases in proportion to the charge, the average potential during charging is V and the work actually done in charging is QV, which is therefore the energy of the charge.

The case is analogous to the filling of a cylindrical water tower, the pressure is zero when the tower is empty, and increases as the water rises until the final pressure p is reached. The work done is therefore pv where v is the total volume of water pumped in.

The energy of the condenser exists as electrical strain in the dielectric.

576. Dissected Leyden Jar.-That the energy of the charge is in the dielectric and not in the conducting surfaces is shown by the following experiment.

Take a Leyden jar, such as is shown in figure 319, in which the metal coatings can be removed from the glass. Charge it strongly and first remove one coating while the other is insulated, and then remove the other also. They are found to have only slight charges, but when they are again fitted upon the glass a vigorous discharge may be obtained.

577. Leyden Battery. The Leyden jars in the combination shown in figure 320 have their inner coatings connected together and are mounted in a box lined with tinfoil by which their outer coatings are also joined. Such an arrangement is known as a Leyden battery, the jars are also said to be connected in parallel or multiple, and the combination is equivalent to a single large jar having a capacity equal to the sum of the capacities of the separate jars.

578. Leyden Jars Connected in Cascade or Series.-In each of the two arrangements shown in figure 321 four jars on insulating stands are connected in such a way that if the discharge were to burst through the glass of the jars it would have to pierce all four jars to pass from one end to the other, as four layers of glass intervene between the terminal conductors. In such a case the jars are said to be joined in cascade or in series. The diagram (Fig. 322) shows the state of electrification, the regions between the charged plates representing the layers of glass.

Four similar jars joined in this way are like a single jar having a dielectric four times as thick, and the capacity of the combination is one-fourth that of a single jar.

This case is well illustrated by the hydrostatic analogue (Fig. 323) in which four models such as are described in §574 are connected in series. Clearly when water is pumped in at A and out at B the rubber diaphragms are all strained and an equal quantity of water is displaced from each into

the next succeeding, thus representing the equality of the charges in each. The pressures represent the potentials. Evidently the pressure P1 is greater than po, and p, is the greatest of all, and to force in a given quantity of water four times as much pressure must be used as to force it into a single one of the cells.

The chief practical advantage of the cascade arrangement is that it has great dielectric strength and sparks do not easily burst through the glass; for this reason the small jars used on induction electrical machines are usually connected two in series, one being connected to one pole of the machine and one to the other, while their outer coatings are joined by a wire.

Problems.

1. A Leyden jar 14 cm. in diameter and made of glass 3 mm. thick is coated on the bottom and sides up to a height of 20 cm. What is its capacity and what charge is required to bring it to potential 30? Take dielectric constant of glass = 6.

2. Two Leyden jars, one of capacity 300 and charged to potential 20, the other brought to potential 30 by a charge of 7200 units, are connected together in parallel, positive coating being connected to positive and negative to negative. Find the resulting potential and the charge in each jar after being connected.

3. If in the preceding problem the positive coating of each jar is connected to the negative of the other, what will be the resulting difference in potential and charge in each jar?

4. A jar of capacity 1000 is charged to potential 50; find the heat developed in gram-calories when it is discharged through a long fine wire.

5. Three jars each of capacity 500 are charged each to potential 12. If they are now joined in series, find the energy of the charge. 6. When the three Leyden jars of the preceding problem are joined in parallel, find the energy of the charge.

CALCULATION OF POTENTIAL AND CAPACITY.

579. Potential at a Point.-Up to this point we have thought of electrical potential simply as a certain condition which determines the flow of electricity; and we have shown that the difference of potential between two conductors may be measured by the work done in transferring unit charge from one to the other ($556).

But potential is not a property of conductors only. When a little charge is brought up to any point whatever in space, work must in general be expended in bringing it to that point, on account of the attractions or repulsions of neighboring charges; and this work, per unit charge, is used as the measure of the potential at that point.

Definition: The potential at any point is measured by the work done against electrostatic forces in bringing a unit positive charge up to that point from an infinite distance.

This work may be calculated as follows:

Suppose there is a charge of q units of electricity at A, (Fig. 324) and it is required to find the work done in carrying