Fig. 6. Fig. 6 shows the field of force due to a charge I at A, and a like charge 4 at B. CHO Fig. 7. Fig. 7 shows the field of force between two parallel planes. At the edges its lines of force curve out. Some pass from the back of one plane to the back of the other. § 9a. TUBES OF FORCE. Imagine a small plane surface placed in an electric field perpendicular to the lines of force at that point. We may consider all the lines of force which pass through the boundary of the small plane as forming a tube of force which starts from the surface of the positive charge and ends at the negative. Since the sides of the tube are lines of force, the cross-section of a tube of force at any point must be proportional to the electric intensity at that point. We may imagine each tube as inclosing one unit of positive electrification at the origin and a unit of negative electrification at the end. These tubes are in a state of tension along the lines of force and a state of pressure at right angles to the line. Faraday introduced this method of representing the force in the electric field, and solved, by means of it, many of the problems in electrification without the aid of mathematical analyses. § 10. CONDUctors and NoN-CONDUCTORS FUR THER DEFINED. It is evident from the way in ELECTRICITY which the electric field is produced (88) that the lines of electric force cannot exist in a conductor; therefore a conductor may be further defined as a substance which cannot sustain electrostatic stress, and which permits the flow of electricity through it when such stress is applied. On the other hand, when electrostatic stress is applied to a dielectric, something of the nature of a displacement of electricity takes place all along the line, causing, as it were, an equal amount to appear on the opposite surface. This displacement is proportional to the stress applied, and disappears when the stress is removed. § 11. A CHARGE, OR CHARGED CONDUCTOR. It is well to keep in mind all that is included in the term charged conductor; namely, one of two equal and opposite charges separated by a dielectric in a state of stress, or as it is called, an electric field. There is no such thing as a single charge; its equal and opposite is somewhere on the earth or surrounding conductors. The intensity of the field at any point varies with the distance between the conductors, their shape, and the dielectric which separates them. When we speak of a single isolated charge, it is understood that the opposite is at a great distance from it. In the case of the two spheres (§ 9, Fig. 3), if one of them be removed to a great distance, the field of force about the other can be represented by Fig. 8; the lines of force are radial and equally distributed about the surface. Fig. 8. 812. ELECTRIFICATION BY INDUCTION, OR INFLUENCE. It is found that when an insulated conductor is introduced into the field about a charged conductor it becomes electrified with equal quantities of positive and negative electrification; further, if it then be connected with earth, the charge, which is like that on the original charged body, disappears, while that which is opposite in sign, remains. The conductor is then said to be charged by induction; but as the term induction has other uses, the term influence is sometimes preferred. 509 | divided, since they cannot exist in a conductor. We have lines ending on the side of C nearest A, and an equal number beginning on the side opposite to A, hence the side opposite A is charged positively, and the side near A is charged negatively, and, moreover, the charges are equal. Now, if the body C moves toward A until it touches A, the lines between A and C no longer exist, and some of the lines which were before on A are new on C, or, as we say, the body C has been charged by conduction; i. e., has taken from A a part of its charge. from A a part of its charge. But if the body C is taken, instead, toward B, the equal and opposite charge, and touches it, C shares with B the lines which are on it, or, what amounts to the same thing, we may connect C with B by means of a conducting wire; the surface of C then becomes electrically a part of the surface of B, and, being the part nearest A, will receive most of the lines of force. If, now, the connection between B and C is broken, we have the charge which was originally on B divided between B and C, the part on C depending upon its nearness to A when the contact was broken; hence we see that a charge given by induction is a charge by conduction, as in the first case, except that a part of the equal and opposite charge is given up to the conductor, C. 1 2 § 12а THE ELECTROPHORUS. The electrophorus is an instrument for the production of electrification by induction, or influence. It consists essentially of a resinous cake, A, a conducting lid, B, with its insulating handle, C (Fig. 9a). The cake is electrified on its upper surface by beating it with a cat's skin (or other convenient substance), which electrifies it negatively, as above, at (1), (Fig. 9a). The charge on the cat's skin finds it way to the earth, or neighboring conductor. The lid, when placed near the cake, is electrically between the charge on the cake and the equal and opposite charge; hence it intercepts most of the lines, and is charged both positively and negatively, as shown at (2). The lid is now connected with the earth by touching it with the finger; hence the surface of the lid becomes a part of the earth's surface, and the equal and opposite charge is now nearly all on the lid; if 3 4 Fig. 9a. the connection between the lid and the earth is broken, the lid may be removed and its charge used to do work in charging other conductors, etc. In actual use the lid is placed on the surface, but as the cake is a non-conductor, and the lid touches it only at a few points, it is electrically "near" the cake. It is evident that the process can be repeated as often as we choose, but the electrification is produced only by the expenditure of energy. Influence or static machines, as they are called, are but mechanical devices for performing automatically the processes described in connection with the electrophorus, thus producing a constant supply of electrification at the expense of the mechanical work necessary to operate the machine against electrical attraction. 812b. The WIMSHURST INFLUENCE-MACHINE. Fig. 9b shows the Wimshurst machine; it is perhaps the simplest and most effective machine of this class, and an explanation of its action will m' 1 FTTTT ་ FTTTT m may be seen at once. Suppose a segment, a, on the back plate, to have a slight positive charge; as the first segment a passes a', a is acted on by induction, and receives a slight negative charge, the equal positive charge appearing on b, which is connected with a, at the moment, by the brushes and connector r. These charges are carried forward until they are opposite and d, which at this moment are connected by the brushes and connector, r', of the rear plate; c and d will then receive, by induction, positive and negative charges, and these, passing onward, act on the front segments, under a and b. The charges on all the sections will thus be built up by a reciprocal action, the front sections on the upper half carrying negative charges from left to right, and the back sections carrying positive charges from right to left. The sections on the lower part of the plate, are passing through a similar, but inverse, set of operations; hence the negative charges are carried by both plates to the collector, m, which is connected with the terminal N, and positive charges are carried to m' in connection with the other terminal, P. P and N become highly charged, the air between them ceases to resist the stress, and a discharge takes place across the gap. The terminals may be connected with any body which it is desired to charge; as, for example, a Leyden jar. The small initial charge necessary to start the action is probably acquired by the friction of the plates and air. It is evident that if P and N be supplied with electrification from another source, the machine would work backward as a motor. 13. UNIT-CHARGE. Unit-charge of electricity is defined as that charge such that two bodies having this charge repel each other with unit force (one dyne) when separated by unit distance. The dielectric is taken as air, and the dimensions of the bodies are assumed to be small, as compared with unit distance. S14. THE ELECTRIC DISCHARGE. We have seen that every electric charge is essentially of two equal and opposite charges separated by a dielectric which is in a state of Fig. 10. B stress. We have assumed that the cause of this state of the bodies and dielectric is, first, the unequal distribution of the electricity which the bodies naturally contain; and second, their separation. Let A and B (Fig. 10) represent the two equal and oppositely charged conductors; then, if A and B are brought in contact, the electrification disappears, and the bodies are said to be discharged. Instead of bringing them in contact, we connect them with a conducting wire, and the same effect is observed. The bodies A and B and the dielectric return to their normal condition, provided A and B are not the two substances which, when in contact, produced the unequal Fig. 9c. distribution; for in that case the return to the be understood by reference to Fig. 9c, where the normal condition would be opposed by the very disks are shown as cylinders, in order that both | thing which caused the unequal distribution. PN In ELECTRICITY case the stress in the dielectric becomes too great, it gives way, and acts for the moment as a conductor, thus allowing the two charges to reunite. This form of the discharge is usually accompanied by a sharp report and a spark, due to the heat generated in the dielectric along the path of the discharge. It is usually spoken of as the electric spark, or, better, a disruptive discharge, in order to distinguish it from the ordinary discharge through a conductor. 5. SURFACE DENSITY. If the electrification on the surface of a charged conductor is uniformly distributed over the surface, the amount of electrification per unit area is called the surface density. If the distribution is not uniform, then the surface density at any point is the amount that would be on unit area if it had a uniform distribution equal to that at the point. § 16. INTENSITY OF THE ELECTRIC FIELD. If a small body charged with unit positive electricity is placed at any point in an electric field, the force which it experiences is the measure of the electric intensity at the point. 17. FORCE BETWEEN CHARGED BODIES. Coulomb demonstrated, experimentally, by means of the torsion balance, that the force between two charged bodies, which are small as compared with the distance between them, is proportional to the product of the charges, and inversely proportional to the distance between them; hence it follows from the definition of unit charge that the force qq' between two small charged bodies in air is r-2 q and q' being the charges on the bodies, and r the distance between them. If q and q' are both positive, or both negative, the expression is positive, but if they are unlike in sign, it is negative; hence the positive sign denotes repulsion, and the negative attraction. 511 q a' q OA" ОА the induction Fa; the =qa, since ; that is to say, the por over the element a" is a". Let a be the angle -2° OA between the planes a' and a", which is equal to that between the normals I and In, then the induction q q normal to a' is a" cosa = a' OA OA a' a" cosa a' and a = ОА tion of the normal induction due to the element a" is equal to q multiplied by the area on the unit sphere subtended by the same solid angle; and the normal induction over the whole surface is equal to q, multiplied by the area subtended by the surface on the sphere of unit radius and center at O. The normal inductions on any finite § 18. TOTAL NORMAL INDUCTION OVER A SUR- portion of the surface will be the sum of the in、 FACE. If an imaginary surface be placed any-ductions on the elements into which the portion where in an electric field, we may suppose it to be of the surface is supposed to be divided, which is completely divided up into elementary surfaces q times the area cut from the sphere of unit so small that the electric intensity at any point radius and center at O, by a cone having the in an element may be regarded as constant. boundary of the surface as its base and the vertex at O. Case 1. Let O be inside a closed surface. The normal induction over the surface is q multiplied by the whole surface of the sphere having unit radius and center at O. The area of the unit sphere being 47, the total induction over the whole surface, due to the charge q inside of it, becomes 4 q. If the intensity at a point in each element be resolved along the outward-drawn normal to the surface at the point, and each element be multiplied by its normal intensity, the sum of all these products is defined as the total normal induction over the surface. $ 19. GAUSS'S THEOREM. A few of the more important cases of electrical intensity will be considered and solved by the aid of Gauss's theorem, which may be stated as follows: The total normal induction over any closed surface drawn in the electric field is equal to 4 times the total charge of electricity inside the surface. To demonstrate the theorem when the field is due to a single charged body, let O (Fig. 11) be the body, the dimensions of which are small as compared with the distance to the surface, and q the charge on the body. Let A, B, C, D be one of the elementary surfaces. Through O, with unit radius, describe a sphere, and denote, by a, the portion surface. Case 2. Let O be outside the closed An elementary cone, with vertex at O. q times the area cut from unit sphere by the same cone. The component of the induction on a' is in a direction away from the surface, while that on a is toward the surface; hence they will have opposite signs; and since the whole surface can be divided up into pairs of elementary surfaces, the surfaces of each pair neutralizing each other, the total normal induction for the closed surface will be o. point inside the shell, and f the electric intensity at the point. Through P draw a spherical surface with center at O. Since the charge is symmetrical about this surface, f, the normal intensity will be the same at all points on the sur face, and the total normal induction will be f (47 X OP), which, by Gauss's theorem, is 4 times the charge inside the spherical surface passing through P. There is no charge inside the surface; hence f(47 × OP3) = 0; therefore fo. There is then no electric intensity at a point inside the uniformly electrified shell. as describe a cylinder hav- If the electric field be due to any other distribution of electrification, we may regard it as due to several small charges, q,, q,, q,,... etc. If Nis the normal induction due to all the charges, q,, (c) Electric intensity at a point outside of an inq, q,, etc., and N,, N., N,,... etc., the in-finitely long circular cylinder uniformly charged. duction on an element of the surface by q, q1, ¶„ etc., then N=N1+N,+N2+... etc.; if a is the area of an element of the surface, Na= ΣN,a + Na + ΣN,a,. . . etc.; that is, the total normal induction over the surface due to the field is equal to the sum of the normal inductions due to the several small charges to which the field is supposed to be due. But since the normal in-planes at unit distance duction over a closed surface due to a charge in-fied cylinder is infinitely apart. Since the electriside it is 47 times the charge, and o, if the charge long and symmetrical is outside the surface it follows that the sum of about its axis, the inthe inductions over the surface due to the several side the charged cylinder will be along the radius tensity at any point outthrough the point, and will be the same for all points at the same distance from the charged sur charges is 4 times that part of the total charge which is inside the closed surface over which the normal induction is taken. § 20. CASES OF ELECTRIC INTENSITY. The following cases, solved by Gauss's theorem, are very important: (a) Electric intensity at a point outside a uniformly charged sphere. Let P (Fig. 13) be the point and O the center of the sphere. Through P draw a sphere with the center at O. Since the sphere is uniformly electrified, the direction of the force at P will be along O P; and it will have the same Fig. 13. value at any point on the sphere with radius OP. Call this value f. Since at all points on this surface the normal force is f, the total normal induction over the spherical surface is f multiplied by the surface of the sphere, which is f(4 X OP'); but by Gauss's theorem this is equal to 47 times the total charge inside the surface. Let q be the total q charge; then f X 4TOP =47q and f= Hence 2 OP2 face. Fig. 15. The intensity at any point on the plane ends of ends; hence the normal induction on these two the cylinder through P will be in the plane of the ends will be o, and the total normal induction will be that on the curved portion of the section only. Let f be the intensity at the point P, then the total normal induction is f(2xr), r being the distance of P from the axis, and the length of the section unity. Let q be the charge per unit length on the electrified cylinder; then by Gauss's theorem the total normal induction over the unit section is 47q; therefore f(2′′г) = 4′′q, and f=24; hence it is seen that the electric intensity 2q r Let A B (Fig. 16) be a portion of the plane, and P the point. It is evident that the intensity will |