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The Electromagnetic Field E9uations CHAPTER 4 ' THE ELECTROMAGNETIC FIELD E9UATIONS § 26. The first pair of Maxwell's equations From the expressions H = curl A, E = - 1oA ~ai -grad cp it is easy to obtain equations containing only E and H. To do this we find curl E: 1 0 curl E = - -c -;-vt curl A- curlgradcp. But the curl of any gradient is zero. Consequently, loH curlE = - --. (26.1) c ot Taking the divergence of both sides of the equation curl A = H, and recalling that div curl = 0, we find ~H=Q ~~ The equations (26.1) and (26.2) are called the first pair of Maxwell's equations. t We note that these two equations still do not completely determine the properties of the fields. This is clear from the fact that they determine the change of the magnetic field with time (the derivative oHlot), but do not determine the derivative oE/ot. Equations (26.1) and (26.2) can be written in integral form. According to Gauss' theorem Idiv HdV = fH. dr, where the integral on the right goes over the entire closed surface surrounding the volume over which the integral on the left is extended. On the basis of (26.2), we have fH'df=O. (26.3) The integral of a vector over a surface is called the flux of the vector through the surface. Thus the flux of the magnetic field through every closed surface is zero. According to Stokes' theorem, IcurlE. df= fE. dl, where the integral on the right is taken over the closed contour bounding the surface over t Maxwell's equations (the fundamental equations of electrodynamics) were first formulated by him in the 1860's. 66 § 27 THE ACTION FUNCTION OF THE ELECTROMAGNETIC FIELD 67 which the left side is integrated. From (26.1) we find, integrating both sides for any surface, ~ E. dl = - ~8 H .df. (26.4) j cot f The integral of a vector over a closed contour is called the circulation of the vector around the contour. The circulation of the electric field is also called the electromotive force in the given contour. Thus the electromotive force in any contour is equal to minus the time derivative of the magnetic flux through a surface bounded by this contour. The Maxwell equations (26.1) and (26.2) can be expressed in four-dimensional notation. Using the definition of the electromagnetic field tensor Fik = oAk/ox/-oAdox'<, it is easy to verify that of ik oFkl oFu - 0 ' (26.5) OXI + OXI + OXk - . The expression on the left is a tensor of third rank, which is antisymmetric in all three indices. , The only components which are not identically zero are those with i =f:k =f:I. Thus there are altogether four different equations which we can easily show [by substituting from (23.5)] coincide with equations (26.1) and (26.2). We can construct the four-vector which is dual to this antisymmetric four-tensor of rank three by multiplying the tensor by e.iklmand contracting on three pairs of indices (see § 6). Thus (26.5) can be written in the form iklm oFlm = 0, (26.6) e OXk which shows explicitly that there are only three independent equations. § 27. Tbe action function of the electromagnetic field The action function S for the whole system, consisting of an electromagnetic field as well as the particles located in it, must consist of three parts: S = Sf+Sm+8mf' (27.1) where Smis that part of the action which depends only on the properties of the particles, that is, just the action for free particles. For a single free particle, it is given by (8.1). If there are several particles, then their total action is the sum of the actions for each of the individual particles. Thus, 8m= - L: mc f ds. (27.2) The quantity Smf is that part of the action which depends on the interaction between the particles and the field. According to § 16, we have for a system of particles: (27.3) 8"jf = - L: ~ f Akdxk: 68 THE ELECTROMAGNETIC FIELD EQUATIONS § 27 In each term of this sum, Ak is the potential of the field at that point of spacetime at which the corresponding particle is located. The sum Sm+ Sml is already familiar to us as the action (16.1) for charges in a field. Finally S1 is that part of the action which depends only on the properties of the field itself, that is, Sf is the action for a field in the absence of ch~rges. Up to now, because we were interested only in the motion of charges in a given electromagnetic field, the quantity SI' . which does not depend on the particles, did not concern us, since this term cannot affect the motion of the particles. Nevertheless this term is necessary when we want to find equations determining the field itself. This corresponds to the fact- that from the parts Sm+Sml of the action we found only two equations for the field, (26.1) and (26.2), which are not yet sufficient for complete determination of the field. To establish the form of the action S1 for the field, we start from the following very important property of electromagnetic fields. As experiment shows, the electromagnetic field satisfies the so-called principle of superposition. This principle consists in the statement that the field produced by a system of charges is the result of a simple composition of the fields produced by each of the particles individually. This means that the resultant field intensity at each point is equal to the vector sum of the individual field intensities at that point. Every solution of the field equations gives a field that can exist in nature. According to the principle of superposition, the sum of any such fields must be a field that can exist in nature, that is, must satisfy the field equations. As is well known, linear differential equations have just this property, that the sum of any solutions is also a solution. Consequently the field equations must be linear differential equations. From the discussion, it follows that under the integral sign for the action S1 there must stand an expression quadratic in the field. Only in this case will the field equations be linear; the field equations are obtained by varying the action, and in the variation the degree of the expression under the integral sign decreases by unity. The potentials cannot enter into the expression for the action SI' since they are not uniquely determined (in Sml this lack of uniqueness was not important). Therefore S1 must be tlle integral of some function of the electromagnetic field tensor Fik' But the action must be a scalar and must therefore be the integral of some scalar. The only such quantity is the product Fik Fik. t Thus SI must have the form: S 1 = a IIFikFikdV dt, dV = dx dy dz, where the integral extends over all of space and the time between two given moments; a is some constant. Under the integral stands Fik Fik = 2(H2 - £2). The field E contains the derivative oAjat; but it is easy to see that (oAjot)2 must appear in the action with the positivesign (and therefore E2 must have a positivesign). For if (oAjt)2 appeared in SI t The function in the integrand of Sf must not include derivatives of F.k, since the Lagrangian can contain, aside from the coordinates, only their first time derivatives. The role of "coordinates" (i.e., parameters to be varied in the principle of least action) is in this case played by the field potential Ak; this is analogous to the situation in mechanics where the Lagrangian of a mechanical system contains only the coordinates of the particles and their first time derivatives. As for the quantity e1klmF.kF.m(§ 25), as pointed out in the footnote on p. 63, it is a complete four- divergence,'so that adding it to the integrand in Sf would have no effecton the -f'equationsof motion". It is interesting that this quantity is already excludedfrom the action for a reason independent of the fact that it is a pseudoscalar and not a true scalar. § 2S THE FOUR-DIMENSIONAL CURRENT VECTOR 69 with a minus sign, then sufficiently rapid change of the potential with time (in the time interval under consideration) could always make Sf a negative quantity with arbitrarily large absolute value. Consequently Sf could not have a minimum, as is required by the principle of least action. Thus, a must be negative. The numerical value of a depends on the choice of units for measurement of the field. We note that after the choice of a definite value for a and for the units of measurement of field, the units for measurement of all other electromagnetic quantities are determined. From now on we shall use the Gaussian system of units; in this system a is a dimension- less quantity, equal to -(1/16n).t Thus the action for the field has the form (27.4) Sf = - I6~C f F/kFikdo., dO.= c dt dx dy dz. In three-dimensional form: (27.5) Sf = sInf (E2_H2) dVdt. In other words, the Lagrangian for the field is (27.6) Lf = SInf (E2_H2)dV. The action for field plus particles has the form (27.7) S = - Lf mcds- Lf ~ Akdxk- I6~C f FikFikdo.. We emphasize that now the charges are not assumed to be small, as in the derivation of the equation of motion of a charge in a given field.Therefore Ak and Fikrefer to the actual field, that is, the external field plus the field produced by the particles themselves; Ak and F/know depend on the positions and velocities of the charges.
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