ELECTROSTATICS

The laws of inverse squares and of conservation of charge.-The fundamental law is here, of course, the law of inverse squares; but, the meaning of this law is to some extent a function of the light in which we regard it. We may speak of it in terms of the law of force between certain material bodies, and we may in fact formulate the general law in the following way:

Let there be given an assemblage of bodies. Then it is possible to assign to each element of volume of the bodies a definite number P such that on writing down vectors

for the forces between each pair of corresponding elements, the forces so A obtained when compounded will give the resultant mechanical forces and couples exerted by the bodies on each other.

Moreover, if the bodies be moved to new positions, a similar result holds, and the values of Jpdr for any one of the bodies in the two positions are equal.

Surface distributions may, of course, be included as limiting cases of volume distributions.

This is the law which may be regarded as the experimental basis of electrostatics. In this form the definition of charge density is not made until the law has been recognized, and indeed until then it has no meaning. The superposability of effects, the proportionality of mechanical force to charge, and the conservation of charge are contained in the statement of the law, once the experimental fact has been assumed.

A consideration of the matter will show that all electrostatic experiments designed to test the law of inverse squares are particular cases of a general experiment of this type. An experiment with Coulomb's torsion balance is of course of this type; and while the statement may not be obvious at first sight, a little consideration will show that even experiments having to do with the absence of field inside a hollow conductor are of this form. Thus in its most general form this experiment with the hollow conductor may be taken to show that a charged body placed inside such a shield experiences no mechanical force as a result of the presence of charges outside. Now it is a mathematical fact that whatever distribution p be assigned to the space outside of a surface, it is always possible to assign a function σ varying over the surface in such a way that while fodS=0, the vectorial sum of

is zero for any point which is inside the surface, and whose distances from the various volume elements dr and surface element dS are given by the corresponding values of r. It follows that if the mechanical force upon a body is given by a law of the type above specified, it is certainly

possible to assign a distribution σ which when combined with the distribution p will result in zero mechanical force on a charged body within the surface, and in the limit on the surface. The fact that there are material surfaces (conductors), whose nature is of a kind such that the distribution will automatically come about is something which is outside the scope of assertion of the fundamental law. Once experiment has revealed that there are surfaces inside which a body will experience no force although bodies outside that surface attract and repel each other, we have realized a situation which certainly can be accounted for by a law of the type A preceding, and it is possible to prove with a fairly large degree of generality that a law of this type is the only one possible to account for the phenomenon.

In the above formulation of the law, no mention of a field is made It is, however, a mathematical convenience to define now a quantity E as

4TE=Grad

Pdr.

(1)

This definition involves nothing about the force on a unit charge, and it is automatically contained in the experimental law as above stated that the force on an element of matter charged with density p is pEdт. It may be parenthetically remarked that the possibility of relegating E to a minor role as regards the expression of the law is not one which is confined to the statical case, but can be employed with advantage even in the most general case, and is indeed the only logical way of formulating the laws in a manner capable of experimental proof. Thus, for example, nobody can assign any meaning to the field inside a moving electron when that field is defined as the force on a unit of charge attached to a piece of matter of macroscopic size. In the light of the above statement, one might question the necessity for introducing such a quantity as E at all. It is true that E need play no fundamental role as regards the expression of the fundamental laws; but, there are a number of facts, concerned with the special properties of materials, which are of enormous importance from the practical standpoint although they are not to be regarded in the light of fundamental laws. Such a fact is the existence of a body possessing the properties of what we call a conductor. A conductor is a body in which the charge distribution must adjust itself in such a way that the quantity E as above defined will always be zero within its substance. It is thus in terms of this subsidiary quantity E that this non-fundamental but very valuable property of a conductor becomes expressed.

In spite of what has been written above concerning the advantages to be derived from relegating E to a subsidiary role in the matter of expression of general laws, one must face the fact that it is customary

to speak of a quantity E which is regarded as defined in terms of the usual unit charge, a convenient silence being maintained as regards the realization of this definition in such a case as that of a point inside an electron for example. Without therefore attempting to justify this attitude, or even to remove the logical difficulty in respect of such an imperfect definition, it is of interest to consider what becomes of the law of inverse squares for a case where our starting point is the hypothesis that E is defined or is capable of definition at any point in space. Case where electric field is the quantity fundamentally defined.-The first thing that is necessary is a definition of charge density. Charge per unit volume would only move matters a step farther back, by calling for a definition of charge. Without belaboring this point unduly we may recall that the necessity of making a definition of p which is applicable in all cases, including that of a moving charge, has resulted in its being defined in terms of the field as

With this definition as applied to the statical case, however, a rather remarkable result follows.

It is a result of pure mathematics, and independent of experiment, that any vector E can be expressed at a point P in the form

(Div Edr + Curl Curl Edr

4xB=-Grad [[[Div Ed

r

(3)

where the volume integrals are taken throughout all space, and r is the distance of an element of volume from the point P.

If we make the assumption that the vector E has no curl, and we may take this as our definition of what we mean by limiting ourselves to an electrostatic system, we have

If p is defined as Div E, this is nothing more nor less than the expression of the law of inverse squares for the entity whose density is given by p. If, for example, we have a point 0, and we assign a field which varies inversely as the cube of the distance from 0, it is obvious that the field in question will give rise to definite values of Div E throughout space, and the above mathematical theorem shows that the amount of this is just such that the field may be regarded as one of the inverse square type, produced however by a distribution of charge which is not confined to the point 0.1 It is useless to say that there is no charge save

1 The infinite charge which will be necessitated at O may be avoided by specifying the inverse cube law as applying only outside of a sphere of finite radius.

at 0, for in terms of Div E as the definition of p, the very act of assigning such a field as one varying according to the inverse cube will automatically carry with it the assignment of the charge distribution necessary to represent this field as consistent with an inverse square law. These remarks are particularly pertinent to such cases as those where it is customary to speak of the forces between electrical charges in atoms as obeying laws of force other than the inverse square, even in the statical case. Such laws need be regarded as in no sense inconsistent with the classical theory, and indeed cannot be.

What then remains of the experimental proof of the law of inverse squares for the type of formulation which defines p as Div E? We must ask what it is that we wish to prove. The most reasonable answer is that we wish to prove that there are certain conditions in which the field in a given region can be expressed entirely in terms of fields due to charges outside of the region, which charges act according to the inverse square law. From the standpoint we have adopted, however, this result will follow as a consequence of pure mathematics provided that we have Div E zero throughout the region. The whole experimental law can thus be formulated in the statement that there are regions in which Div E is zero.

Our search for a region in which Div E is zero must of course be made in the light of a precise definition of E. As regards phenomena on a sufficiently large scale the usual definition in terms of unit charge will suffice. We now observe that if we can find a case in which E is zero throughout a region, (although not of course zero everywhere outside, for this would give a trivial result), we shall have found a region in which Div E is zero. Now experiment shows that there are regions, hollow conductors, in which we can have E zero while it is not zero outside; and from the standpoint which we are now considering, it would appear that all that this experiment of the hollow conductor may be taken to establish is that there are regions in which Div E is zero while it is not zero outside. We have apparently not made much use of the fact of the boundary of the region's being a conductor. Indeed the only essential thing is the existence of a region in which E and consequently Div E is zero. It is however possible to go on and show that if such a region exists, its boundary must be an equipotential surface, although this is not a fact of which the proof makes direct use.

Although the definition p= Div E would carry with it the law of inverse squares for the field in the foregoing sense, an extra hypothesis to the effect that the mechanical force on an element of matter is proportional to the charge on it is necessary; for, the definition of E as the force on a particle of matter containing unit charge does not carry with it as a necessary consequence, the proportionality of the force on the matter

and the charge when the latter is defined in terms of a density given by Div E. The assumption of the said proportionality would therefore be one subject to the necessity of experimental investigation. This experimental fact, combined with the assumption, which must be regarded as based upon experiment, that certain types of charge distribution exist, that in fact charge is only to be found in the places where we choose to specify it in any given problem, these are the only facts relying upon experiment for a field whose curl is everywhere zero, and in which the charge density p is defined as Div E. The law of inverse squares is for this case a result of pure mathematics.1

Dielectrics. As regards the role played by dielectric phenomena in electrostatics little is to be said. The fundamental assumption is that the dielectric may be represented as regards its external action by replacing it by a distribution of polarization P. This turns out to be the mathematical equivalent of a fictitious volume distribution of charge with density Div P, and a fictitious surface distribution which at each point of the surface has a density equal to Pn, where the subscript n is to be taken as denoting the component in the direction of the outward normal. Thus, in order to include dielectrics, no further extension is needed in the experimental law as formulated above with the electric field at an external point occupying the position of a subordinate quantity introduced after the truth of the law has been recognized. As regards the field at a point inside the dielectric, this is defined as the quantity obtained by drawing a small cavity about the point, measuring the field therein, and subtracting the portion of the field due to the fictitious charge on the wall of the cavity, this latter being the only part of the field within the cavity which does not attain a definite limit as the size of the cavity is made infinitesimal. We may parenthetically remark that, although it is not customary to do so, we may avoid completely the mention of a cavity by defining the field at a point within the dielectric as the actual field which would be measured at that point if we were to remove the dielectric, and replace it by its fictitious. volume and surface charges. The smoothing out process implied in the specification of the fictitious charges carries with it the requirements necessary for insuring a definite limiting value for the field at the point, without resort to the use of a cavity. This method suffers in no respect in comparison with that invoking the use of a cavity in that it requires specification of the fictitious charges before the field can be obtained;

As regards conservation of charge, perfect generality of view-point would hardly call for a more restrictive hypothesis than the assumption of a constant and zero value for the integral of Div E for the universe as a whole. Denial of the possibility of coalescence of the ultimate positive and negative units would permit of a more restrictive hypothesis in the form that the volume integrals of Div E taken separately for positive and negative values of Div E, and in a true differential as distinct from a macroscopic sense, are each individually constant for all time, and are equal, though opposite in sign.