Import ant pro perty of tubes of force Charge mea sured by tubes of force. 20 tube between the surfaces our fundamental equation (9). We thus get, since there is no normal component perpen dicular to the generating lines of the tube, RdS-K d$ = 0, .... (12), provided the tube does not cut through electrified matter between the two surfaces. Here R and R denote the resul tant force at dS and </S, which are supposed so small that the force may be considered uniform all over each of them. It appears then that the product of Ike resultant force into the area of the normal section of a tube of force is constant for the same tube so long as it does not cut through electrified matter ; or what amounts to the same, tJie resultant force at any point of a tube of force varies inversely as tlie normal section of the tube at that point. If we divide up any level surface into a series of small elements, such that the product HdS is constant for each element and equal to unity, and draw tubes of force through each small element, then the electric induction through any finite area of the surface is equal to the number of tubes of force which pass through that area ; for if n be that number, we have, summing over the whole of the area 2RJS = n ..... (13), the left hand side of which is the electric induction through the finite area. It is clear, from the constancy of the pro duct RJS for each tube of force, that if this is true for one level surface it will be true for every other cut by the tubes of force. It is evident that the proposition is true for any surface, whether a level surface or not, as may be seen by projecting the area considered by lines of force on a level surface, and applying to the cylinder thus formed the surface integral of electric induction, it being remarked as obvious that the same number of tubes of force pass through the area as through the projection. This enables us to state the proposition involved in equation (9) in the following manner : The excess of the number of tubes of forces which leave a closed surface over the number which enter is equal to 4?r times the algebraical sum of all the electricity within the surface. (y.B. The positive direction of a line of force is that direction in which a unit of + electricity would tend to move along it.) This proposition enables us to measure the charge of a body by means of the lines 1 of force. We have only to draw a surface inclosing the body, and very near to it, and count the lines of force entering and leaving the surface. If the number of the latter, diminished by the number of the former, be divided by 4?r, the result is the charge on the body. If we apply (13) to a portion of an equipotential surface so small that R may be considered uniform over the whole of it, we may write Result- or in words : The resultant force at any point is equal to ant force tf ie numoer o f H ties O f f orce p er un fc O y arm O f i eve i sw -f ace sured by a * ^ ia * P^ nt j meaning thereby the number of lines of lines of force which would pass through a unit of area of level sur- force. face if the force were uniform throughout, and equal to its value at the point considered. We are now able to express by means of the lines of force the resultant force at any point of the field, and the charge in any element of space. The electrical language thus constructed was invented by Faraday, who continually used it in his electrical researches. In the hands of Sir William Thomson, and particularly of Professor Clerk Max well, this language has become capable of representing, not 1 Here we drop the distinction between line and tube of force. We shall hereafter suppose the lines of force to be always drawn so as to form unit tuLes, and shall speak of these tubes as lines of force, thereby following the usual custom. [ELECTKOSTATICAL THEORY. only qualitatively, but also quantitatively, with mathematical accuracy, the state of the electric field. It has the additional advantages of being well fitted for the use of the practical electrician, and of lending itself very readily to graphical representation. It will be convenient, before passing to electrical applica tions, to state here another general property of the potential which follows from our fundamental proposition. The potential cannot have a maximum or minimum value Maxi- at a point where there is no electricity. mum or For if a maximum value were possible, we could draw round the mimniuin point a surface at every point of which the potential was decreasing P c outwards ; consequently at every point of this surface the normal im P s- component of the resultant force in the outward direction would be . l e positive, and a positive number of lines of force would leave the m surface. But this is impossible, since, by our hypothesis, there is no s P ace - electricity within. Similarly there could be no minimum value. From this it follows at once that if the potential have the Case of same value at every point of the boundary of a space in ivhich s P ace there is no electrified body, then the potential is constant ^ ounile( i throughout that space, awl equal to the value at the boundary. Jbfa.^ For if the potential at any point within had any value greater or less than the value at the boundary, this would be a case of maximum or minimum potential at a point in free space, which we have seen to be impossible. In order that there may be electrical equilibrium in a perfect conductor, it is necessary that the resultant electric force should be zero at every point of its substance. For if it were not so at any point the positive electricity there would move in the direction of the resultant force and the negative electricity in the opposite direction, which is incon sistent with our supposition of equilibrium. This condition must be satisfied at any point of the conductor, however near the surface. At the surface there must be no tangential component of resultant force, otherwise electricity would move along the surface. In other words, the resultant force at the surface must be normal ; its magnitude is not other wise restricted ; 2 for by our hypothesis electricity cannot penetrate into the non-conducting medium. These conditions are clearly sufficient. We may sum them up in the following single statement : If the electricity in any conductor is in equilibrium, the Coiidi- potential must have the same value at every point in its tion of 7 . electrical tubstance. eouili- For if the potential be constant, its differential coefficients jj r j um% are zero, so that inside the conductor the resultant force vanishes. Also the surface of the conductor is a level sur face, and therefore the resultant force is everywhere normal to it. This constant value of the potential we shall hence forth speak of as the potential of the conductor. Since the potential is constant at every point in the Elec- substance of a charged conductor, we have at every point tr V 2 V = 0, and hence by the equation of Poisson p = ; that ^^ is, there is no electricity in the substance of the conductor. sur f ace . We thus get, as a theoretical conclusion from our hypothesis, the result already suggested by experiment, that electricity resides wholly on the surface of conductors. If we apply the surface characteristic equation to any point of the surface of a conductor, we get ~ T, dv = 4 T ( 15 )> which gives the surface density in terms of the resultant _ R force and reciprocally. 4* We may put this into the language of the lines of force by saying that the charge on any portion of the surface of a conductor is equal to the number of lines of force issuing from it divided by 47r. Since the surface of a conductor in electric equilibrium 2 Of course in practice there is an upper limit, at which disruptive
discharge occurs.