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500. As of immense importance, in the theory not only of gravitation but of electricity, of magnetism, of fluid motion, of the conduction of heat, etc., we give here an investigation of the most important properties of the Potential.
501. This function was introduced for gravitation by Laplace, but the name was first given to it by Green, who may almost be said to have created the theory, as we now have it. Green's work was neglected till 1846, and before that time most of its important theorems had been re-discovered by Gauss, Chasles, Sturm, and Thomson.
In $ 245, the potential energy of a conservative system in any configuration was defined. When the forces concerned are forces acting, either really or apparently, at a distance, as attraction of gravitation, or attractions or repulsions of electric or magnetic origin, it is in general most convenient to choose, for the zero configuration, infinite distance between the bodies concerned. We have thus the following definition :
502. The mutual potential energy of two bodies in any relative position is the amount of work obtainable from their mutual repulsion, by allowing them to separate to an infinite distance asunder. When the bodies attract mutually, as for instance when no other force than gravitation is operative, their mutual potential energy, according to the convention for zero now adopted, is negative, or their exhaustion of potential energy is positive.
503. The Potential at any point, due to any attracting or repelling body, or distribution of matter, is the mutual potential energy between it and a unit of matter placed at that point. But in the case of gravitation, to avoid defining the potential as a negative quantity, it is convenient to change the sign. Thus the gravitation potential, at any point, due to any mass, is the quantity of work required to remove a unit of matter from that point to an infinite distance.
504. Hence, if V be the potential at any point P, and V, that at a proximate point Q, it evidently follows from the above definition that V-V, is the work required to remove an independent unit of matter from P to l; and it is useful to note that this is altogether independent of the form of the path chosen between these two points, as it gives us a preliminary idea of the power we acquire by the introduction of this mode of representation.
Suppose ( to be so near to P that the attractive forces exerted on unit of matter at these points, and therefore at any point in the line PQ, may be assumed to be equal and parallel. Then if F represent the resolved part of this force along PQ, F.PQ is the work required to transfer unit of matter from P to l. Hence
V- V or
that is, the attraction on unit of matter at P in any direction PQ,
is the rate at which the potential at P increases per unit of length
505. A surface, at every point of which the potential has the same value, and therefore called an Equipotential Surface, is such that the attraction is everywhere in the direction of its normal. For in no direction along the surface does the potential change in value, and therefore there is no force in any such direction. Hence if the attracted particle be placed on such a surface (supposed smooth and rigid), it will rest in any position, and the surface is therefore sometimes called a Surface of Equilibrium. We shall see later, that the force on a particle of a liquid at the free surface is always in the direction of the normal, hence the term Level Surface, which is often used for the other terms above.
506. If a series of equipotential surfaces be constructed for values of the potential increasing by equal small amounts, it is evident from $ 504 that the attraction at any point is inversely proportional to the normal distance between two successive surfaces close to that point; since the numerator of the expression for F is, in this case, constant.
507. A line drawn from any origin, so that at every point of its length its tangent is the direction of the attraction at that point, is called a Line of Force; and it obviously cuts at right angles every equipotential surface which it meets.
These three last sections are true whatever be the law of attraction ; in the next we are restricted to the law of the inverse square of the distance.
508. If, through every point of the boundary of an infinitely small portion of an equipotential surface, the corresponding lines of force be drawn, we shall evidently have a tubular surface of infinitely small section. The resultant force, being at every point tangential to the direction of the tube, is inversely as its normal transverse section.
This is an immediate consequence of a most important theorem, which will be proved later. The surface integral of the attraction exerted by any distribution of matter in the direction of the normal at every point of any closed surface is 49M; where M is the amount of matter within the surface, while the attraction is considered positive or negative according as it is inwards or outwards at any point of the surface.
For in the present case the force perpendicular to the tubular part of the surface vanishes, and we need consider the ends only. When none of the attracting mass is within the portion of the tube considered, we have at once
This is equivalent to the celebrated equation of Laplace.
When the attracting body is symmetrical about a point, the lines of force are obviously straight lines drawn from this point. Hence the tube is in this case a cone, and, by $ 486, w is proportional to the square of the distance from the vertex. Hence F is inversely as the square of the distance for points external to the attracting mass.
When the mass is symmetrically disposed about an axis in infinitely long cylindrical shells, the lines of force are evidently perpendicular to the axis. Hence the tube becomes a wedge, whose section is proportional to the distance from the axis, and the attraction is therefore inversely as the distance from the axis.
When the mass is arranged in infinite parallel planes, each of uniform density, the lines of force are obviously perpendicular to these planes; the tube becomes a cylinder; and, since its section is constant, the force is the same at all distances.
If an infinitely small length l of the portion of the tube considered pass through matter of density p, and if w be the area of the section of the tube in this part, we have
509. In estimating work done against a force which varies inversely as the square of the distance from a fixed point, the mean force is to be reckoned as the geometrical mean between the forces at the beginning and end of the path : and, whạtever may be the path followed, the effective space is to be reckoned as the difference of distances from the attracting point. Thus the work done in any course is equal to the product of the difference of distances of the extremities from the attracting point, into the geometrical mean of the forces at these distances; or, if O be the attracting point, and m its force on a unit mass at unit distance, the work done in moving a particle, of unit mass, from any position P to any other position P', is
Opa OP2 ? OP OP
To prove this it is only necessary to remark, that for any infinitely small step of the motion, the effective space is clearly the difference of distances from the centre, and the working force may be taken as the force at either end, or of any intermediate value, the geometrical mean for instance: and the preceding expression applied to each infinitely small step shows that the same rule holds for the sum making up the whole work done through any finite range, and by any path.
Hence, by $ 503, it is obvious that the potential at P, of a mass m situated at O, is ; and thus that the potential of any mass at a
point P is to be found by adding the quotients of every portion of the mass, each divided by its distance from P.
510. Let S be any closed surface, and let O be a point, either external or internal, where a mass, m, of matter is collected. Let N be the component of the attraction of m in the direction of the normal drawn inwards from any point P, of S. Then, if do denotes an element of S, and SS integration over the whole of it,
SSN do = 4 Tm, or = 0, according as O is internal or external.
Case 1, O internal. Let OP, P,P,... be a straight line drawn in any direction from 0, cutting S in P, P2, P3, etc., and therefore passing out at P., in at P2, out again at P2, in again at Pų, and so
Let a conical surface. be described by lines through O, all infinitely near OP, P, ..., and let w be its solid angle (§ 482). The portions of SSN do corresponding to the elements cut from S by this cone will be clearly each equal in absolute magnitude to wm, but will be alternately positive and negative. Hence as there is an odd number of them, their sum is twm. And the sum of these, for all solid angles round O is (8 483) equal to 4 m; that is to say, SSNdo 47 m.
Case II, 0 external. Let OP,P2P,... be a line drawn from O passing across S, inwards at P1, outwards at P2, and so on. Drawing, as before, a conical surface of infinitely small solid angle, w, we have still wm for the absolute value of each of the portions of ST Ndo corresponding to the elements which it cuts from S; but their signs are alternately negative and positive: and therefore as their number is even, their sum is zero. Hence
SSNdo=0. From these results it follows immediately that if there be any continuous distribution of matter, partly within and partly without a closed surface S, and N and do be still used with the same signification, we have
SI Ndo = 40 M if M denote the whole amount of matter within S.
511. From this it follows that the potential cannot have a maximum or minimum value at a point in free space. For if it were so, a closed surface could be described about the point, and indefinitely near it, so that at every point of it the value of the potential would be less than, or greater than, that at the point; so that N would be negative or positive all over the surface, and therefore SsNdo would be finite, which is impossible, as the surface contains none of the attracting mass.
512. It is also evident that N must have positive values at some parts of this surface, and negative values at others, unless it is zero all over it. Hence in free space the potential, if not constant round any point, increases in some directions from it, and diminishes in
others; and therefore a material particle placed at a point of zero force under the action of any attracting bodies, and free from all constraint, is in unstable equilibrium, a result due to Earnshaw 1.
513. If the potential be constant over a closed surface which contains none of the attracting mass, it has the same constant value throughout the interior. For if not, it must have a maximum or minimum value somewhere within, which is impossible.
514. The mean potential over any spherical surface, due to matter entirely without it, is equal to the potential at its centre; a theorem apparently first given by Gauss. See also Cambridge Mathematical Journal, Feb, 1845 (vol. iv. p. 225). This proposition is merely an extension, to any masses, of the converse of the following statement, which is easily seen to follow from the results of $$ 479, 488 expressed in potentials instead of forces. The potential of an uniform spherical shell at an external point is the same as if its mass were condensed at the centre. At all internal points it has the same value as at the surface.
515. If the potential of any masses has a constant value, V, through any finite portion, K, of space, unoccupied by matter, it is equal to V through every part of space which can be reached in any way without passing through any of those masses : a very remarkable proposition, due to Gauss. For, if the potential differ from V in space contiguous to K, it must ($ 513) be greater in some parts and less in others.
From any point C within K, as centre, in the neighbourhood of a place where the potential is greater than V, describe a spherical surface not large enough to contain any part of any of the attracting masses, nor to include any of the space external to K except such as has potential greater than V. But this is impossible, since we have just seen (§ 514) that the mean potential over the spherical surface must be V. Hence the supposition that the potential is greater than V in some places and less in others, contiguous to K and not including masses, is false.
516. Similarly we see that in any case of symmetry round an axis, if the potential is constant through a certain finite distance, however short, along the axis, it is constant throughout the whole space that can be reached from this portion of the axis, without crossing any of the masses.
517. Let S be any finite portion of a surface, or complete closed surface, or infinite surface, and let E be any point on S. (a) It is possible to distribute matter over S so as to produce potential equal to F(E), any arbitrary function of the position of E, over the whole of S. (6) There is only one whole quantity of matter, and one distribution of it, which can satisfy this condition. For the proof of