The three-dimensional analog to Fig. The gradient of a scalar function may now be defined as follows: The gradient of a scalar function g is a vector whose magnitude is the maximum directional derivative at the point being considered and whose direction is the direction of the maximum directional derivative at the point. The most common symbols for the gradient are V and grad; of these we will most often use the latter. This is immediately evident from the geometry of Fig, In a more coi the same.
For our purposes we may consider three kinds of inte- grals: line, surface, and volume, according to the nature of the differential appearing in the integral. The integrand may be either vector or a -scalar; however, certain combinations of integrands and differentials give rise to uninteresting integrals.
Those of interest here are the Line integral of a vector, the surface integral of a vector, and the volume integrals of both vectors and scalars. Tf F is a vector, the line integral of F is written. The definition of the lire integral follows closely the Riemaun definition of the definite integral. The segment of C between a and b is divided into a large number of small increments Al,; for each increment an interior point is chosen and the value of F at that point found.
The sealar product of each increment with the corresponding value of F is found and the sum of these computed. The line integral is then defined as the limit of this sum as the number of increments becomes infinite in such a way that each increment goes to zero. For this reason one often encounters line inte- grals around undesignated closed paths, e. If any ambiguity is possible, it is wise to specify the contour. The basic approach to the evaluation of line integrals is to obtain a one-parameter description of the curve and then use this description to express the line integral as the sum of three ordinary one-dimensional integrals.
In all but the simplest cases this is a long and tedious procedure; fortunately, however, it is seldom necessary to evaluate the integrals in this fashion. As will be seen later, it is often possible to convert the line integral into a more tractable surface integral or to show that it does not depend on the path between the endpoints.
In the latter case a simple path may be chosen to simplify the integration. If F is again a vector, the surface integral of F is written f, F-nda, where S is the surface over which the integration is to be performed, da is an infinitesimal area on S and n is a unit normal to da.
There is a two- fold ambiguity in the choice of n, which is resolved by taking n to be the outward drawn normal if S is a closed surface. If S is not closed and is finite then it has a boundary, and the sense of the normal is important ony with respect to the arbitrary positive sense of traversing the boundary.
This is illustrated in Fig. Comments exactly parallel to those made for the line integral can be made for the surface integral. The definition of the surface integral is made in a way exactly comparable to that of the line integral. The detailed formulation is left as an exercise. These integrals are sufficiently familiar to require no further comment. Boundary Fig.
Relation of normal n to a surface, and the direction of traversal of the boundary. Another important operator, which is essentially a derivative, is the divergence operator. The divergence of vector F, written div F, is defined as follows: The divergence of a vector is the limit of ils surface integral per unit volume as the volume enclosed by the surface goes to zero. The above defini- tion has several virtues: it is independent of any special choice of coor- dinate system, and it may be used to find the explicit form of the divergence operator in any particular coordinate.
In rectangular coordinates the volume element Az Ay Az provides a convenient basis for finding the explicit form of the divergence. The volume enclosed by the coordinate intervals Ar, A9, Ag is chosen as the volume of integra- tion. This volume is r? If the fluid is incompressible, the surface invesral measures the total source of fluid enclosed by the surface.
The above definition of the diver- gence then indioates that it may be interpreted as the limit of the source strength per unit volume, or the source density of an incompressible fluid.
The divergence theorem is now obtained by letting the number of cells go to infinity in such a way that the volume of each cell goes to zero. We shall have frequent occasion to ex- ploit this theorem, both in the development of the theoretical aspects of electricity and magnetism and for the very practical purpose of evaluating integrals.
The third interesting vector differential operator is the curl. The curl of a vector, written curl F, is defined as follows: The curl of a vector is the limit of the ratio of the integral of its cross product with the outward drawn normal, over a closed surface, to the volume en- closed by the surface as the volume goes tw zero. Otherwise the defini- tions are the same. This definition is convenient for finding the explicit form of the curl in various coordinate systems; however, for other purposes a different but equivalent definition is more useful.
This alternative definition is: The component of curl F in the direction of the unit vector a is the limit of a line integral per unit area, as the enclosed area goes to zero, this area being perpendicular to a.
This equivalence can be shown without the use of the special volume used here; however, so doing sacrifices much of the simplicity of the proof given above. In rectangular coordinates the volume Az Ay Az is convenient. For the 2-component of the curl only the faces perpendicular to the y- and z-axes contribute. The y- and z-components may be found in exactly the'same way. The proof of this theorem is quite analogous to the proof of the divergence theorem.
The surface S is divided into a large number of cells. The surface of the ith cell is called AS; and the curve bounding it is C;. This theorem, like the divergence theorem, is useful both in the development of electromagnetic theory and in the evaluation of integrals. The operations of taking the gradient, divergence, or curl of appropriate kinds of fields may be repeated. For example, it makes sense to take the divergence of the gradient of a scalar field. Some of these repeated operations give zero for any well-behaved field.
One is of sufficient importance to have a special name; the others can be expressed in terms of simpler operations. An important double operation is the divergence of the gradient of a scalar field. This combined operator is known as the Laplacian operator and is usually written V? The curl of the gradient of any scalar field is zero. This statement is most easily verified by writing it out in rectangular coordinates. The divergence of any curl is also zero.
In any coordinate system other than rectangular the Laplacian -of a veotor is defined by Eq. There are many possible combinations of differential operators and products; those of most interest are tabulated in Table These identities may be readily verified in rectangular coordinates, which is sufficient to assure their validity in any coordinate system.
This conciudes our brief discussion of vector analysis, In the interests of brevity, many well-known results have been relegated to the exercises. Ne attempt has been made to achieve a high degree of rigor; the approach has been utilitarian, What we wili need we have developed; everything alse has heen omitted.
Show that the lines AB and CD are parollel and find the ratio of their lengths. Verify that Eq. It should be noted that r and have different meanings here than in Eqs. In spherical coordinates r is the magnitude of the radius vector from the origin and is the polar angle.
In cylindrical coordi- nates, r is tho perpendicular distance from the cylinder axis and 9 is the azimuthal angle about this axis.
From the definition of the divergence, obtain an expression for div F in cylindrical coordinates. Find the divergence of the vector A x? Also find the curl. Note: u is any veotor. Prove identities I-6 and in Table If r is the magnitude of the vector from the origin to the point z, y, 2 , and f r is an arbitrary function of r, prove that a r dr Verify Eq. Prove identities I and I in Table Hint: Use the diver- gence theorem and one or more identities from the first half of Table The first observation of the electrification of ob- jects by rubbing is lost in antiquity; however, it is common experience that rubbing 9 hard rubber comb on a piece of wool endows the rubber with the ability to pick up small pieces of paper.
As a result of rubbing the two objects together strictly speaking, as a result of bringing them into close contact , both the rubber and the wool acquire a new property; they are charged. This experiment serves to introduce. But charge, itself, is not created during this process; the total charge, or the sum of the charges on the two bodies, is still the same as-before elec- trification.
In the light of modern physics we know that microscopic charged particles, specifically electrons, are transferred from the wool to the rubber, leaving the wool positively charged and the rubber comb negatively charged.
Charge is a fundamental and characteristic property of the elementary particles which make up matter. In fact, all matter is composed ultimately of protons, neutrons, and electrons, and two of these particles bear charges. But even though on a microscopic scale matter is composed of a large number of charged particles, the powerful electrical forces associated with these particles are fairly well hidden in a macroscopic observation.
The reason is that there are two kinds of charge, positive and negative, and an ordinary piece of matter contains approximately equal amounts of each kind. When we say that an object is charged, we mean that it has an excess charge, either an excess of electrons negative or an excess of protons positive. In this and the following chapters, charge will usually be denoted by the symbol g.
Since charge is a fundamental property of the ultimate particles making up matter, the total charge of a closed system cannot change. From the macroscopic point of view charges may be regrouped and combined in different ways; nevertheless, we may state that net charge is conserved in a closed system. Towards the end of the eighteenth century tech- niques in experimental science achieved sufficient sophistication to make fdssible refined observations of the forces between electric charges.
The last two statements, with the first as preamble, are known as Coulomb's law in honor of Chatles Augustin de Coulomb , who was one of the leading eighteenth century students of electricity.
In Eq. If the force on gz is to be found, it is only necessary to change every subscript 1 to 2 and every 2 to4. Understanding this notation is important, since in future work it will provide a tech- nique for keeping track of field and source variables. To the best of our knowledge, Coulomb's law also applies to the interactions of elementary particles such as protons and electrons.
Equation is an experimental law; nevertheless, there is both theoretical and experimental evidence to indicate that the inverse square law is exact, ie. The same experiment: was performed earlier by Kelvin and by Maxwell. Maxwell established the exponent of 2 to within one part in 20, There is considerable advantage tu having the results of calculations come out in the serae units as those which are used in the laboratory; hence we hall use the r A mks or Giorgi system of units in the present volume.
In Appendix I the otinitions of the coulomb, the ampere, the permeability, and permittivity f free space aro slated to one another and to the velocity of light in a logical way; since a logical formulation of these definitions requires a Imowledge cf magnetic phenomena and of electromagnetic wave prc agation, it is not appropriate to pursue them now.
In Appendix IT othe systems of electrical units, in particular the gaussian system, ure disensced. This is, of course, just the superposition principle for forces, which says that the total force acting on a body is the vector sum of the individual forces which act on it. A simple extension of the ideas of N interacting point charges is the interaction of a point charge with a continuous charge distribution. We deliberately choose this configuration to avoid certain difficulties which may be encountered when the interaction of two continuous charge dis- tributions is considered.
Before proceeding further the meaning of a con- tinuous distribution of charge should be examined. In other words, if any charge were examined in great detail, its magnitude would be found to be an integral multiple of the magnitude of the electronic charge. The smallness of the basic unit means that macroscopic charges are in- variably composed of a very large number of electronic charges; this in turn means that in a macroscopic charge distribution any small element of volume contains a large number of electrons.
One may then describe a charge distribution in terms. Care must be used, however, in applying this kind of description to atomic problems, since in these cases only a small number of electrons is involved, and the process of taking the limit is meaningless.
It may appear at first sight that if point r falls inside the charge distribution, the first integral of should diverge. It is clear that the force on g as given by Eq. This observation leads ys, tp introduce a vector field which is independent of g, namely, the force per unit charge. The electric field at a point is defined as the limit of the following ratio: the force on a test charge placed at the point, to the charge of the test charge, the limit being taken as the magnitude of the test charge goes to zero.
The customary symbol for the electric field is E. If, for example, positive charge is distributed on the surface of a conductor a conductor is a material in which charge is free to move , then bringing a test charge into the vicinity of the conductor will cause the charge on the conductor to redistribute itself.