Electromagnetic Theory: Volume 3
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Refresh and try again. Open Preview See a Problem? Details if other :. Thanks for telling us about the problem. Return to Book Page. Preview — Electromagnetic Theory Vol. Electromagnetic Theory Vol. Englishman OLIVER HEAVISIDE left school at 16 to teach himself electrical engineering, eventually becoming a renowned mathematician and one of the world's premiere authorities on electromagnetic theory and its applications for communication, including the telegraph and telephone. Here in three volumes are his collected writings on electromagnetic theory-Volume Englishman OLIVER HEAVISIDE left school at 16 to teach himself electrical engineering, eventually becoming a renowned mathematician and one of the world's premiere authorities on electromagnetic theory and its applications for communication, including the telegraph and telephone.
Here in three volumes are his collected writings on electromagnetic theory-Volume III was first published in This is a catalog of the bulk of his postulations, theorems, proofs, and common problems and solutions in electromagnetism, many of which had been published in article form. Part scientific history-including references to some contemporary criticisms, long since shown to be poorly based, of Heaviside's scholarship-and part guide to understanding a complex applied science, this work shows both the genius and the eccentricity of a man whose work includes precursory theories to Einstein, and revolutionary principles that today are the commonly assumed truths in the field of electrical engineering.
Get A Copy. Hardcover , Third Edition , pages. More Details Original Title. Electromagnetic Theory 3. Other Editions 4. Friend Reviews. To see what your friends thought of this book, please sign up. To ask other readers questions about Electromagnetic Theory Vol. Be the first to ask a question about Electromagnetic Theory Vol. Lists with This Book. Community Reviews. Showing Rating details.
All Languages. More filters. Sort order. Ben Davidson rated it it was amazing May 01, Cosimo Books added it Jan 12, Adam marked it as to-read Sep 07, Rageofanath marked it as to-read Jan 09, Kpieas84 marked it as to-read Dec 26, Peter Hofbauer marked it as to-read Jun 05, We can make the resemblance complete if we regard the element. In the hitter two identifications we arc using the rules for connecting rc. Then eqns. After division, by these become identical with. This model of a network of impedances is usually used in discussions of transmission lines, and because the final equations are identical with 1 1.
In this model the conductor. This differs from our original specification of two conductors. The discrepancy can be avoided if we regard the two conductors as separated by an infinite conducting plane at zero potential. The second line is then the image of the line of the first conductor in this plane. This plane is taken to be the earth conductor in the model. We now complete our discussion of the transmission line starting from theequations The mathematical discussion consists of finding a solution of eqns.
Because of the resistance in the conductors the signal suffers attenuation as it is transmitted down the line; the important practical case is that in which attenuation occurs without distortion. If the attenuation depends on the frequency, a given signal, which contains many different frequencies, will have these different components attenuated to different degrees when it reaches the end of the line, and so suffers distortion. Similarly distortion occurs if the different frequencies are transmitted with velocities which depend on the frequency. We shall see that by adjusting the values of the parameters i?
Such a line can have amplifiers inserted into it at suitable points along its length to over- come the attenuation, and a clear signal can then be transmitted over long distances. If the quantities a, arc independent of to, then all frequencies arc pro- p. In this case the line is distortionless. If we eliminate a from cqns. T- GR. The general solution of the eqns. Thus far we have not considered the effect of the ends of the line.
There are three particular results: 1. In general, the solution of problems concerning transmission lines requires ideas and techniques very closely similar to those required for w'aves on strings, both finite and infinite in length. In a irnnsmission line each of the elements AiA-, A complex alternating current. Vo is fed in at the terminal Ai,, and. Sliow that if the terminal yl,. A, Jj liy VLliy L. By this means Kirchholfs fitM l. Hence this transmission line will transmit only signals with this frequency without diminishing the amplitude, i.
A uniform cable has constant resistance R, capacity C and inductancei per unit length. Show that it induces a periodic potential at distance. At t 0 the end a- 0 is earthed. Show that at subsequent time t the potential at. State Maxwell's equations for an isotropic conducting medium of conductivity tr, pcrmc.
Electromagnetic Theory (3 Volumes) (v. 3)
Show that It, li and I! A plane electromagnetic wave, whose wavelength is! Obtain from. Maxwell's equations for a homogeneous medium of cUclectric constant. If tlic medium is highly conducting, show tliai the E waves and 11 waves are out of phase bv approximately. State the other conditions which must hold.
Determine the current in the sheet necessary to maintain the oscillation. A general dielectric medium is divided into two regions, denoted 1 and 2, by a metal sheet 5 which carries a current of surface density i. The sheet S is in the shape of an infinite cylinder of radius a, whose axis lies along the z-axis; the space inside and outside S is empty.
Kinp, r nnd 6. Determine the airrent in the sheet ncccssarv- to maintain this c'cilbiion. Prmc that in an cleciromapnctic field independent of the coordinate y, where the axes Ox, Oy, Or are rectangular cartesian, and i. J, k arc unit vectors along them, the vectors E es. Show that Maxwell's equations for anisotropic homogeneous non-conducting charge- free medium can be satisfied by t.
Find the reflected wave. It is now time to take up the general question of invariance. A scientific theory is, generally speaking, invariant under some group of transforma- tions. Tltc reader may be reminded that by a group is meant a set of quan- tities Ixitwecn which a binary operation is defined. This binarj' operation, often regarded as a product, for transformations consists of applying two traii'-formaiions in succession.
The binary operation is associative and for the existence of a group there must be an identity element and a reciprocal of every element.
In the ease of transformations the identity element is tlie identity transformation and the reciprocal element is the inverse transfor- mation. These ideas arc already familiar in the case of the orthogonal group in three dimensions i. Indeed the whole invariance under this group is automatically built into a theory' as soon as it is c,xprcssed in vectorial form. Accordingly the theory' of electromagnetism must be invariant under some group which has the orthogonal group in three dimensions as a subgroup.
Exactly what group this is wc shall return to shortly. Before that, let us consider exactly what it means to. For example, in the case of electromagnetic theory the numbers might be the six components of the electric field and the magnetic flux density, E and B. On the other band, if they are merely numbers represent- ing in part the properties of the coordinate system, it must make a difference whether one proceeds from the first to the third system directly or via some intermediate stage, and in this ease the conditions ul — v will not be ful- filled, Tlic condition just expressed on the transformations of the sets of numbers is known technically as requiring the transformations to be a representation of the original group and the argument which we have just given which is the general form of the principle of relativity can be put by saying that any physically significant set of numbers must transform under a representation of the group of transformations of the theory'.
It is instructive to look first at the example of the orthogonal group in three dimensions. Here we know of one example of a representation in the components of an ordin- ary' vector. However, not all linear transfor- mations of this kind arc permitted,. In this case, then, we see that the components of the vectors transform in the same way as the unit vectors themselves, but this is a coincidence.
What has basically been done here is to derive a representation of the group by regarding a vector as a displacement; in other words, one chooses something which, from its geometrical or physical interpretation, is known to be independent of the coordinate system and so to transform under a representation of the group, and uses its representation to define a whole class of such objects. In this way one could construct a series of more com- plicated representations.
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These more complicated transformations are called the tensor representations of the group. The particular transformation above is of a tensor of rank 2. By taking products of more than two vectors we get the higher order tensor representations. These are by no means all the possible representations, although they are important ones, as can be seen from the following example. The quanlitie.
In this ease ue have a rather obvious choice to begin with, when we seek for sets of quantities obsiously transforming under a representation of the group, for we can choose i, r as a prototype. The next step is to inquire about analogues of the vector products in ordinary vector analysis. It is, of course, obvious that under the orthogonal group, for example, we could consider, instead of the vector product, the array of nine quantities transforming under the second-order tensor repre- sentation but it is not convenient to do so here because electromagnetic theoiy is not concerned, in general, with such quantities.
They enter, for instance, in rigid mechanics, in the definition of moments and products of inertia, and in certain other branches of applied mathematics, such as elasti- city. Accordingly we shall consider products of jp. By the very way in which these are written down they must be automatically invariant under the orthogonal group in three dimensions but we are con- cerned also in transforming between coordinate systems in uniform relative S TI 1 C only remaining dilTiculty is that the transformation involves, as well as the particular combination of vectors with which wc started, the components of the vector product of the two vectors.
But the transformation of these, A;,Bs-A;B'. Bs-A:iB2, It is interesting to notice that for transformations in which the velocity V is small compared with that of light, eqns. Example I. The remaining results follow directly from the transformations. Then A'. The calculations of the transfomiation of the remaining components are carried out in similar manner. Example 3. Tlicn Example 1 follows, by using the result of Example 2, with 6, A taken as?! We know already that it has the orthogonal group in three dimen- sions as a subgroup.
In order to establish this we have to show that the product of any two elements of the group again belongs to the group. As far as the product of two rotations is concerned this is well known and amounts to the theorem that any member of the rotation group is a rotation in the cle- ntemary sense about a certain axis Euler's theorem.
If we are concerned with the product of a Lorentz transformation and a rotation, it is dear from the way in which weareabic to write our Lorentz transformation in s'cetorial form th. Indeed we tacitly take account of this u hcncM-r rvc choose the. It only remains therefore to consider the product of two Lorentz transformations. The reader may instantly verify this by using the definitions. If the two velocities are not in the same direction, we consider first the case in which they are perpendicular, and we choose these directions as the X- and y-axes. In order to show that the transformation t, x, y to t", x", y" is a transformation of the group we?
Wc now eliminate t between cqns. Pv Pu The chief importance of this result is that by a suitable choice of u, v the angle a, i. This result, combined with that for two transfor- mations in the same direction, suffices to prove that we are dealing with a complete group of transformations.
Tlic first result is that the charge, being simply the number of electrons present, is unchanged by the transformation. Let us adopt two coordinate. We arc concerned witii an element of volume in he new primed coordinate system, which is d. We have already considered the fonnul. Accordingly if one defines the charge and current four-vector as g, J — 2 pb the expression will have as its scalar time component the density m any frame. Accordingly we take the charge-current four-vector as the basis needed to estimate the transformation properties of the whole set of equations.
In discussing the invariance of the equations it is necessary to take up again the method of derivation which was discussed in the last volume. At this stage the main reason for this assumption is that it is logically consistent to assume these three equations, whereas the remaining one cannot hold for non-steady currents. Kow it follows from Example 1 of p. By a similar argument, 5, —E is also a six-vector. But the difference appears, of course, in the constitutive relations. In the.
The second pair, eqns. An immediate consequence of these transformations is the following. Suppose now that a transformation of coordinates of the usual form is made. Another reduction, which is alway. From the. Another interesting application of the transformations is to the e. This dielectric is between the plates of a moving condenser and the plates of this condenser are short-circuited by means of brushes and a wire which passes through a ballistic galvanometer. The condenser moves with a uniform speed in the direction of the x-axis, the x-f plane being chosen parallel to the plates of the condenser.
A magnetic field is then applied in the y-direction see Fig. We have to apply the constitutive relations in the dielec- tric, and accordingly we must transform to a frame of reference in which the dielectric is at rest. The mag- niludc of this charge is in good agreement with the results predicted by the theory. This identification is confirmed by observing that the invariant expression i is the one expected from the usual Lorentz condition [eqn. X and sirrularly for those of O. Obtain expressions for D and B in terms of E, H, e, p when the medium is moving with uniform velocity o.
Choose 0 in the direction of the ar-axis. Example S. A linearly polarized plane electromagnetic wave is propagated in free space from a transmitter fixed in an inenial frame S. The fields of the transmitted wave observed in. Hence the reflected frequency is as stated. A particle of rest mass and charge e moves from rest in a uniform electric field Ej and a uniform magnetic field Bk, where E cB. Tront the last tsso equations i. Find the possible states of motion of the particle for which the acceleration will be zero.
By considering the four-vector property of Ap, deduce that a, must transform as a four-vector under the Lorentz transformation and that kpXp must be a scalar, so that kp also transforms as a four- vector. S, Describe bricfls a method for transforming tltc components of the electromagnetic field in free space from one Lorentz frame to another. Accordingly the ordinary current in the Maxwell equations is negli- gible compared with the displacement current and we may take the equations oo as so that, since curlJ?
It is dear from the equations, since they arc wave equations, that tlic field will be transmitted with a finite vdocity c. One can. Since the field propagates with a finite speed, the field linc.
In other words, as frequency increases, the field lines tend to separate. Those which are near enough to the dipole will move away from it and back towards it, but there will be some critical surface which separates thc. These more distant ones correspond to the radiation field in which we arc principally interested here. In the ease of the static or slowly changing field the most con- M-menl further restriction i-.
However, it will prove to be more satisfactory to use a different restriction here. It would be very convenient at this point if the vector A also satisfied the wave equation. Let us choose, then.
Now the. Substituting for the poten- tials the field strengths t. This relates the Hertz potential to the distribution of electric dipoles. The first kind of symmetry which springs to mind is that of spherical symmetry. This suggests that. It is therefore necessary to go up to the ne. We shall suppose that the charges present, which produce the field, are located very near to the origin and are fluctuating in some way so as to produce the radiation conditions nccc-ssary. In fact the solution which we shall find is mainly applicable to a dipole whose strength varies with the lime, but we shall also consider the general case.
It is natural to use polar coordinates for the more detailed calculations; because the axis of sym- metry is the z-axis, the angle between this and the radius vector r may be taken as 0. Tlic Hertz vector, JJ, in order to prcsers'c the axial symmetry can only have components radially outwards and parallel to the z-axis. Both of these components can only depend upon the distance from the origin. We shall I'C particularly interested in the.
The distinction between these tw'o parts corresponds to that between the near zone and the distant zone mentioned in our earlier intuitive argument. Taking again the particular ca. The part of the electric field which is important nearest to the origin falls off as the cube of the distance and the part which is important a great way off again falls off inversely as the distance.
Approximately, one can say that the values of the fields near to the oscillating system are M cos 6 iTtEo! These conditions do not depend upon the particular Lorentz frame of reference. The field of a dipole of moment m directed along tlie polar axi? Tfor' 4rreori' so that the electric field corresponds to such a dipole with varying moment m.
Now such a varying moment may be considered as due to a current element. This is in complete agreement with the electrical picture. Thus both the fields imply that the sohnion we have found corresponds to an oscillating dipole at the origin directed up the r-axis. It is to be noticed that from the equations for the field in the near zone the squares of the field strength which enter into the expression for the energy arc inversely proportional to the sixth power of r. Moreover, the electric and magnetic field strength dilTer in phase by T-t. There is therefore an energy-flow outwards.
As a result the energy decreases only as the inverse square of the distance. Thus apart from the two zones dificring by the size of the fields and the manner in which these fields fall off with distance there is also another important distinction in the behaviour of the energy in them. In the near zone the energy flows out and back, so that in so far as the transmission of radiation is concerned the whole behaviour is really a pretence. The region in which the pretence takes place will be larger when the near zone is larger, that is, the lower the frequency or the longer the wavelength.
For a wavelength of about 60 cm the near zone has a radius of about 5 cm, whereas for a wavelength of 6 km the near zone has a radius of about m. The large amount of energy consumed by the long-wave transmitter is mostly used in setting up the pretence of transmission in the near zone which is in fact only a pumping in and out of energy. Certain plausible assumptions nbout the scalar and vector potentials lead to a solution which can then, by looking at the near field, be seen to correspond to an oscillating dipole.
Two different directions suggest themselves for proceeding from this point. In the present section we look at one example of a solution found in this way. Alternatively we miglit seek to find the radiation field from a. In this way the physical significance of what is found is certain, but there is the disadvantage that the calculation of the field in general is not so easy. In illu. Laplace's equation in four dimensions , we can expect a similar elementary solution e. There is, however, a purely formal procedure that enables us to derive from such a solution, with a singular surface, another solution with no singu- larities.
First suppose that the original singular event occurs, not at the origin, but at the time to and the position ro. Now U, V can only be singular if - 0. Now that the complex numbers have done their work they can be dis- regarded,' the linearity of the wave equation is what makes this method possible. Of course, by a real change of origin it is possible to make y. There arc no singularitic. The curve is obsaousiy of the form showm in Fig. Thus a disturbance builds up, changes somewhat, and disappears again. It seems to lack some additional restriction that will serve to ensure that fields originate only from sources like charges and magnets.
But we do not know how to modify the theory so as to rectify this defect. A common method in some theories c. In point of fact, there are known to exist c-xactly eleven such coordinate systems, of which cartc. Verj' little is known about the separable solutions of this equation. Accordingly we must have recourse to other methods. Consider then the scalar wave equation above. To find such a particular integral let us diWdc the space up into very small portions and find the separate fields produced by the elementary electromagnetic theory, VOL.
The resultant of all these fields will then be a particular integral. For any given volume element dr containing a charge o dr we have then to solve the homogeneous wave equation every- where, except in the immediate neighbourhood of the volume element where the charge corresponds to a point charge q dr which is a function of time. But it by no means follows from this rough appeal to causality that the same choice is always required.
We have to choose it so as to make the potential correspond to the correct value of the charge, that is, to make the potential have the correct value very near the origin, as r — 0. To this field we should in general add any solution of the homogeneous wave equation which is then chosen so as to satisfy the conditions of the problem. For our purposes, however, it will be sufficient merely to consider in detail the particular integral found called the retarded solution. By vvriting the vector wave equation in cartesians, a similar argument on each compo- nent leads to the result But their chief importance is that they express the potentials of the field of a moving charge in an explicit form which, as we can see from the expression, depends on its velocity but not on its iicccleration.
Suppose that we have a single charge, moving in any manner. However, a little care is needed in the limiting procedure. We cannot simply replace [p]dr by e since we have initi. It is not true that it makes no difference whether we carry out the limiting operation before or after taking retarded values. There is only one frame of reference in which this is true — that in which the charge is iiv tantancously at rest. In order to see the value in any other frame of reference it is necessary to express everything in four-vector form. The velocity four-vector, for the charge, is defined by 9,?
TliC'C arc the Lidnard-Wicchert potentials. Some care i. The second,, however, depends on the acceleration, and so is non-zero even for a charge which is instantaneously at rest. This fact was to be expected of the radiation parts of the fields but it is. We began by solving the wave equation for the potentials,, found a solution that depended only on velocity and not on acceleration,, and then differentiated this solution in order to derive a field that depended both on velocity and on acceleration.
This suggests that, if we had begun with the Hertz potential being independent of the acceleration, then the- ordinary potentials would have depended on acceleration, and the fields on. Does our argument forbid tliis? Some discussion has taken place on this subject, and the matter is still not clear. It is our opinion, however, that the argument we have given contains a hidden assumption. When we solved the scalar wave equation L c- dfi we sought a particular integral that was spherically symmetric, about an origin chosen at the position of charge involved.
Instead we would need axially symmetric solutions, of which,, of course, there are many, involving Legendre polynomials. Conversely, when we make the assumption of spherical symmetry, we are assuming that the accelerations are unimportant here. The solution that we have RADIATION I m derived must then rely, for its complete justification, on its experimental confirmation, and fortunately there is ample experimental verification that it is, in fact, true.
I Using formula I3. In fact. Thus the direction in which there is no flux of radiation is the direction in which E vanishes. For the circular motion there is one obvious way of satisfying this, by making parallel too. Since b is along the radius to the circle this shows at once, from Fig. Each has dipole moment and the position sector of P- relative to P, is ak. Show that the ratio of the total energy radiated to tliat radiated by two non-interfering oscillators is j L-. With reference to Fig. Tlicn P.
Since, from the test. A wire of negligible resistance lies along the z-axis and carries a varying charge g z,? Tlic scalar potential is most easily ound by noting the Lorenu condition. The Lorentz condition fne-s. Since If r. The equ. What is the relation be- tween these solutions? Obtain the corresponding formulae for the components of the electric intensity; and prove that the lines of electric force arc the meridian curves of the surfaces sshcre QdSISn — constant. Show that these equations arc satisfietl if E, If. A current j flows in a homogeneous region V of dielectric constant K and permeability ft.
Introduction Tfic motion of electrically charged particles in electromagnetic and craviiaiional fields is of interest in various studies of astronomical and engineering problems. During the past five decades research into the motion of charged particles in the terrestrial magnetic and gravitational fields has greatly improved our understanding of the structure and extent of the terrestrial magnetic field. In electronics, devices such as the magnetron were dc'. However, much care must be taken when attempts arc made to generalize the results for a single particle to the ease of an ionized gas, which consists of a multitude of charged panicles, in a magnetic field, since the interactions betsseen charged particles can modify or completely change the calculated and observed results concerning a single particle.
In fact the essential feuturc of the motion of a charged particle in a magnetic field is the tendency of the particle to. This is regarded as the counicrpari of the magnctohydrodynamic or ionized-gas phenomenon in which a highly electrically conducting material can flow freely along the lines of magnetic force but motion of the material perpendicular to the nugnctic field cirrics the magnetic field lines with the material.
In this chapter, w first give an account of the non-rclativistic motion of a charged particle in uniform cro. ThLs is folloued by a brief account of the flow of charged particles when space ch. However, in this chapter we give only an elementary account of the motion in crossed electric and magnetic fields. For more detailed investigations in cases where the magnetic field is non-uniform or gravita- tional fields are present the reader should consult more advanced works devoted entirely to this subject.
In fact, taking the scalar product of eqn. Further, if the magnetic field B is constant, the path of the particle is a circular helix described at constant speed. R- arc constants of integration. These arc the parametric equations of a circular heli,x. In fact the particle moves so that the resolute v,.
Tlic radius of the circular cylinder on which the spiral lies is called the Lamior radius of the p. In the following examples we discuss a number of phenomena which bear on problems of physical interest. Example 1. The particle initially moves in a plane perpendicular to Oxy. Further the vector fj. A beam of electrons passes through a small aperture into a uniform magne- tic field of strength B. If the electrons have a speed v at the aperture and are moving in directions nearly parallel to the magnetic field, prove that the beam will focus at a distance TjtmvJleB from the aperture, where —e and m denote the charge and mass of the electron.
Neglect interactions between the electrons. The beim will focus where A'i-r. An electron of negligible mass carries a charge e and moves with unifonn velocity 0 in a medium which offers a resistance kv to its motion. A conducting medium may be regarded as composed of ji large number ii of electrons per unit volume, whose motion is resisted according to the above law, with k given by s fixed framework of positively Charged particles, die total chargC'dcnsity of the medium being zcm.
Then using eqn. With the given expression forB, B we find. Therefore 5 ]4. The particle is subjected to a resistance mPa? Shotv that the sufasequent motion is given fay pm mp'. Tlic initial conditions arc. Equation I -i eqn.
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As t or, vx,-,-. A potential difference is mvaintained bewcen the conduc- ton. Assuming that the electric potenti. Then the energy equation From eqns. Examplr 6. Tiie equation of motion of a charged particle moving in uniform constant eketrie and magnetic fields E and U is? The p. Integrating we find i. TaUng the scalar product of both sides of cqn. B ri where A is a constant scalar. Example 7. A particle, of mass m carrying a charge e, moves in vacua in an electric field E and a magnetic field with magnetic flux density vector B. Show that the velocity of the particle parallel to Oz remains constant.
Therefore v. Tlicreforc cqn. The use of the Lagrangian Show that the equation of motion of a charged particle may be written where A. V, 1 luivc the rectangular rcsolutcs of the force acting on the particle as the right-hand sides. These equations are the same as those given on p. An electron of mass m and charge -e is emitted with negligible velocity from a thin straight wire, and thereafter moves under the action of a magnetic field of intensity B parallel to the wire and an electric field of potential Er, where t is the distance from the srirc and E and B are constants.
Show that the electron describes a cardioid with angular velocity cBIliit about the wire, 3, A bead of mass rtu and bc,iring a charge c, is free to slide on a smooth, circular, non- conducting loop of radius a, fixed horizontally. A magnetic field, parallel to the r-axis, whose magnitude depends only on distance from tltc r-,axis, ho! Show that the srxfcd of the particle cm be increased by the action of the electric field induced by varying the magnetic field, without any change in the radius of the orbit, provided tlu!
An ion of m. Assuming a termin. IJcctrons of mass m and clinrgc -r arc released with negligible velocity from the inner cylinder. Neglecting variation of mass witli velocity, show that, in cylindrical polir Civrdin. An infinite earthed wire of negligible radius lies along the axis of an infinite circular cylinder of radius a whose potential is F; a uniform magnetic field of induction B is applied parallel to the wire. Show that it cannot reach the cylinder if 9. The unit vectors i,j, k denote the directions of the coordinate axes. A particle of mass m and charge e moves in the magnetic field of a current J flowing in an infinite straight wire.
A complete discussion of the dynamics of a fully or partially ionized gas or plasma is beyond the scope of this book ; as indicated on p. However, we derive below two results of physical importance from the elementar ' equations. Suppose now that in an element dr of the volume contained within a closed surface T there arc N charges of magnitude -f e.
It follows that the rate of working of the electromagnetic held on the charges within dr is j-E dr and.