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But in case of a charged conductor all parts of it, inside and outside, are at the same potential, the sphere is therefore all at the potential V of its center.

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or the capacity in electrostatic units of an isolated sphere surrounded by air is numerically equal to its radius.

If the medium surrounding the sphere has specific inductive capacity K, its capacity becomes (§573)

C=Kr.

584. Capacity of a Condenser Made of Two Concentric Spheres. Suppose we have a condenser such as shown in figure 329, consisting of two concentric metal spheres with air between them. Let r, be the outer radius of the inner sphere and T2 be the inner radius of the outer sphere. If a charge +Q is given to the inner sphere, an induced charge -Q will be found on the outer sphere. If the outer sphere is connected to earth it comes to zero potential and all charge disappears from its outer surface.

FIG. 329.-Spherical condenser.

The potential at O the center of the small sphere is therefore

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since the charge +Q is at a distance r1 from the center, and the charge-Q is at a distance r, from the center.

But the potential everywhere inside of a closed conductor is the same as at its surface. Hence the potential of the inner

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and since the outer sphere is at zero potential, V is the difference of potential between the two.

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If the medium between the spheres has a specific inductive capacity K, the capacity of the condenser will be

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If the spheres are close together we may write rr2 = r2 and r2-r1 =d where r is the mean radius of the spheres and dis the thickness of the space between them.

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but 4r2 is the area of surface of a sphere of radius r; therefore

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In this form the formula can be used for any condensers where the two surfaces are close together, as in a Leyden jar or in a condenser made of two flat parallel plates.

Problems.

1. How much work must be done to carry a unit positive charge from

a point 1 meter distant from a charge +100 to a point 2 cm. from it?

2. What is the potential at a point half-way between two equal spherical conductors having charges +100 and -100, respectively?

3. What is the potential at one corner of a rectangle which measures 40 x 30 cm. when there is a charge -300 at the diagonally opposite corner and +120 at each of the adjacent ones?

4. A spherical conductor 10 cm. in diameter has a charge of +200 units and a small body having an equal plus charge is situated 1 meter from the center of the sphere. What is the potential of the sphere?

5. How much work would be done in moving the small charged body of the preceding question up to 50 cm. from the center of the sphere?

ELECTRIC DISCHARGE.

585. Electric Discharge through Air at Ordinary Pressures.Three forms of discharge are recognized through air at ordinary pressures, the electric spark or disruptive discharge, brush discharge, and glow discharge.

In the ordinary spark discharge there is a flash of light accompanied by heat and sound and the medium is mechanically rent. The energy that was in the strained dielectric is dissipated in these various ways.

The discharge must not be thought of as "jumping across" from one body to the other, it cannot be said to leap from positive pole to negative or from negative to positive, but takes place simultaneously at every point along the path of discharge. Imagine a piece of rope AB held in position by a set of elastic bands which are attached to nails on each side of it, as shown in figure 330. If we try to pull the rope toward B the elastic

FIG. 330.

bands are stretched and resist; but if enough force is exerted they will break, and the breaking will begin not necessarily at one end or the other, but wherever the weakest one is found. But when breaking occurs all parts of the rope move forward at once. This illustrates very crudely what probably takes place in disruptive discharge; the strained medium begins to break down at the weakest point, wherever that may be, but the electric discharge takes place simultaneously at all points along the line of discharge.

The brush discharge is seen when in a darkened room the hand is brought near the positive conductor of a highly active electrical machine. If it is not held near enough for the spark discharge a luminous brush, like a little tree with branches of light ramifying from a short stem, extends out toward the hand from some point on the positively charged conductor. It seems to be caused by an almost continuous succession of extremely small discharges.

Sometimes in the dark when an electric machine is highly

excited, but when the conductors are separated too far for sparks to pass, a faint velvety glow of violet light known as the glow discharge is seen on the knob of the negative conductor.

586. Oscillatory Discharge. When a spring is bent and let fly it oscillates back and forth, coming to rest when its energy is finally spent in heat, sound, and air waves. So when a charged Leyden jar is discharged through a circuit of small resistance. the energy of the charge cannot be dissipated in the first rush and consequently there is a back-and-forth rush of current from one coating to the other until the energy is finally spent in sound, heat, light, and electric waves. This is known as the oscillatory

FIG. 331.-Oscillatory discharge.

discharge. If there is sufficient resistance in the discharge circuit there is no oscillation, just as a pendulum hung in molasses will sink to its lowest position without oscillation.

The oscillatory discharge was examined by Feddersen in 1863 by means of a rapidly rotating mirror. Seen in this way, each discharge showed as a group of sparks at regular intervals and rapidly dying out, as shown in figure 331; see also §762.

587. Mechanical and Heating Effects of Disruptive Discharge. When the discharge takes place through a sheet of glass it is pulverized at the point of discharge. Pasteboard is perforated by the discharge, the edges of the hole being raised in a burr on each side as if by the sudden expansion and bursting out of the contained air or moisture. When trees are struck by lightning they are apt to be splintered, large slivers being flung violently out sidewise, perhaps due to sudden vaporization of moisture. When a living tree is struck the discharge usually takes the sap layer and frequently follows the grain. A glass tube having a fine bore filled with water and with a wire thrust a short distance in each end may be burst by the discharge of a Leyden jar.

The electric discharge is accompanied by heat. Ether and bisulphide of carbon are readily ignited by it. Buildings con

taining inflammable material are occasionally set on fire when struck by lightning. A mixture of one volume of oxygen with two volumes of hydrogen explodes with violence if even a minute electric spark passes in it.

A little gunpowder placed between the ends of two wires. through which a discharge is sent will usually be scattered unless the discharge is retarded by causing it to pass through a wet string or other poor conductor, in which case the powder may be ignited.

Narrow strips of gold foil, one or two millimeters in width, gummed to a sheet of paper so that they form a conducting strip, may be deflagrated or volatilized by the discharge of a Leyden battery. The purple stain which is left is wider than the gold-foil strips and is streaked at right angles to its length as though the volatilized metal had been driven violently out from the path of discharge.

588. Lightning. The resemblance between lightning flashes and electric sparks was early noticed. Franklin, in 1752, performed the celebrated experiment of obtaining electric charges by means of a kite as a thunder storm was approaching. The kite was provided with metal points and the linen kite cord was a fairly good conductor when wet. To the lower end of the kite cord was fastened a metal key to which a silk cord was attached which was held in the hand and acted as an insulator. Sparks were obtained from the key and Leyden jars were charged, and the familiar phenomena of electric charges were observed.

The cause of the great difference of potential between clouds and earth which is observed in thunderstorms is unknown. Some have held that it is due to evaporation, some to condensation, and others to friction between air and water particles.

Experiment does not show that evaporation is always accompanied by electrification, nor is it satisfactorily explained how condensation by which larger drops are produced can raise the potential of a whole region.

The so-called globe lightning, described by different observers as a ball of fire slowly moving along and then suddenly exploding with terrific violence, has never been imitated by any electrical discharges obtained in the laboratory and is so different from the ordinary phenomena of discharge that many physicists con

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