Chapter 4 Electromagnetism 4.1 Introduction This chapter outlines the basic principles of the electromagnetic theory in vacuo. First, the extension of the Lagrangian formalism to functions that depend on more than one variable is tackled: this yields useful tools for the analysis of continuous media. Next, the Maxwell equations are introduced along with the derivation of the electric and magnetic potentials, and the concept of gauge transformation is illustrated. The second part of the chapter is devoted to the Helmholtz and wave equations, both in a finite and infinite domain. The chapter finally introduces the Lorentz force that connects the electromagnetic field with the particles’ dynamics. The complements discuss some invariance properties of the Euler equations, derive the wave equations for the electric and magnetic field, and clarify some issues related to the boundary conditions in the application of the Green method to the boundary-value problem. 4.2 Extension of the Lagrangian Formalism In Sect. 1.2 the derivation of the extremum functions has been carried out with reference to a functional GŒw of the form (1.1). Such a functional contains one unknown function w that, in turn, depends on one independent variable . The result has been extended to the case where the functional depends on several unknown functions w1 , w2 ; : : :, each dependent on one variable only (compare with (1.6)). The extension to more than one independent variable is shown here. To proceed it suffices to consider a single unknown function w that depends on two independent variables ,  and is differentiable at least twice with respect to each. The first and second derivatives of w are indicated with w , w , w , w , and w . Letting  be the domain over which w is defined, and g the generating function, the functional reads © Springer International Publishing AG 2018 M. Rudan, Physics of Semiconductor Devices, DOI 10.1007/978-3-319-63154-7_4 75 76 4 Electromagnetism GŒw D Z g.w; w ; w ; ;  / d : (4.1)  Then, let ıw D ˛ , with .;  / an arbitrary function defined in  and differentiable in its interior, and ˛ a real parameter. Like in the case of one independent variable the choice is restricted to those functions  that vanish at the boundary of , so that w and w C ıw coincide along the boundary for any value of ˛. If w is an extremum function of G, the extremum condition if found by replacing w with w C ˛  and letting .dG=d˛/0 D 0, where suffix 0 indicates that the derivative is calculated at ˛ D 0 (compare with Sect. 1.2). Exchanging the integral with the derivative in (4.1) yields  dG d˛  0 D Z    @g @g @g C  C  @w @w @w d D 0 : (4.2) The second and third terms of the integrand in (4.2) are recast in compact form by defining vector u D .@g=@w ; @g=@w / and using the second identity in (A.16), so that the sum of the two terms reads u
grad  D div.u/  div u. Integrating over  and using the divergence theorem (A.23) yields Z  u
grad  d D Z † Z  u
n d†  div u d ; (4.3)  where † is the boundary of  and n the unit vector normal to d†, oriented in the outward direction with respect to †. The first term at the right-hand side of (4.3) is equal to zero because  vanishes over †. It is important to clarify the symbols that will be used to denote the derivatives. In fact, to calculate div u one needs, first, to differentiate @g=@w with respect to  considering also the implicit -dependence within w, w , and w ; then, one differentiates in a similar manner @g=@w with respect to  . The two derivatives are summed up to form div u. For this type of differentiation the symbols d=d and d=d are used, even if the functions in hand depend on two independent variables instead of one. The symbols @=@ and @=@ are instead reserved to the derivatives with respect to the explicit dependence on  or  only. With this provision, inserting (4.3) into (4.2) yields the extremum condition Z   @g @w d @g d @w d @g d @w   d D 0 : (4.4) As (4.4) holds for any , the term in parentheses must vanish. In conclusion, the extremum condition is d @g d @g @g C D ; d @w d @w @w (4.5) 4.2 Extension of the Lagrangian Formalism 77 namely, a second-order partial-differential equation in the unknown function w, that must be supplemented with suitable boundary conditions. The equation is linear with respect to the second derivatives of w because g does not depend on such derivatives. The result is readily extended to the case where g depends on several functions w1 , w2 ; : : : ; wl and the corresponding derivatives. Defining the vectors w.;  / D .w1 ; : : : ; wl /, w D .@w1 =@; : : : ; @wl =@/, w D .@w1 =@; : : : ; @wl =@ /, the set of the l extremum functions wi of functional Z GŒw D g.w; w ; w ; ;  / d (4.6)  is found by solving the equations d @g d @g @g C D ; d @.@wi =@/ d @.@wi =@ / @wi i D 1; : : : ; l ; (4.7) supplemented with the suitable boundary conditions. It follows that (4.7) are the Euler equations of G. Finally, the case where the independent variables are more than two is a direct extension of (4.7). For instance, for m variables 1 ; : : : ; m one finds m X @g @g d D ; dj @.@wi =@j / @wi jD1 i D 1; : : : ; l : (4.8) If g is replaced with g0 D gCdiv h, where h is an arbitrary vector of length m whose entries depend on w and 1 ; : : : ; m , but not on the derivatives of w, then (4.8) is still fulfilled. The replacement, in fact, adds the same term to both sides. For instance, the term added to the left-hand side is ! l m m X X @ d @hr X @hr @ws C ; i D 1; : : : ; l ; (4.9) dj @.@wi =@j / rD1 @r @ws @r sD1 jD1 where the sum over r is the explicit expression of div h. Remembering that h does not depend on the derivatives of wi one recasts (4.9) as m m m l X X @ @hj d X X @hr @.@ws =@r / D ; d @w @.@w =@ / @ j rD1 sD1 s i j j @wi jD1 jD1 i D 1; : : : ; l ; (4.10) where the equality is due to the relation @.@ws =@r /=@.@wi =@j / D ıis ıjr , with ıis.jr/ the Kronecker symbol (A.18). Inverting the order of the derivatives at the right-hand side of (4.10) yields @ div h=@wi , that coincides with the term added to the right-hand 78 4 Electromagnetism side of (4.8). Finally, (4.8) is recast in compact form by defining a vector ui and a scalar si as ui D   @g @g ; ;::: ; @.@wi =@1 / @.@wi =@m / si D @g @wi (4.11) to find div ui D si ; i D 1; : : : ; l : (4.12) If wi depends on one variable only, say , (4.8, 4.12) reduce to (1.7). Using the language of the Lagrangian theory, the comparison between the one-dimensional and multi-dimensional case shows that in both cases the functions wi play the role of generalized coordinates; in turn, the scalar parameter  of (1.7) becomes the vector .1 ; : : : ; m / of (4.8) and, finally, each generalized velocity wP i becomes the set @wi =@1 , : : : ; @wi =@m . 4.3 Lagrangian Function for the Wave Equation P /, i D 1; : : : ; n, It has been shown in Sect. 1.3 that the relations wR i D wR i .w; w; describing the motion of a system of particles with n degrees of freedom, are the Euler equations of a suitable functional. Then, the analysis of Sect. 4.2 has shown that when the unknown functions w1 ; : : : ; wl depend on more than one variable, the Euler equations are the second-order partial-differential equations (4.8). The form (4.8) is typical of the problems involving continuous media (e.g., elasticity field, electromagnetic field). Following the same reasoning as in Sect. 1.3 it is possible to construct the Lagrangian function whence the partial-differential equation derives. This is done here with reference to the important case of the wave equation1 r 2w 1 @2 w D s; u2 @t2 (4.13) where u D const is a velocity and, for the sake of simplicity, s is assumed to depend on x and t, but not on w or its derivatives. It is worth noting that when a differential equation other than Newton’s law is considered, the corresponding Lagrangian function is not necessarily an energy. For this reason it will provisionally be indicated with Le instead of L. To proceed one considers the one-dimensional form of (4.13), @2 w=@x2 .1=u2 / @2 w=@t2 D s and replaces ,  , g with x, t, Le , respectively. Then, one makes the one-dimensional form identical to (4.5) by letting 1 Also called D’Alembert equation in the homogeneous case. 4.3 Lagrangian Function for the Wave Equation 79 @2 Le D @w2t @2 Le D 1; @w2x 1 ; u2 @Le D s; @w (4.14) with wx D @w=@x and wt D @w=@t. The second derivatives of Le .w; wx ; wt ; x; t/ with respect to the combinations of the arguments not appearing in the first two equations of (4.14) are set to zero. The third of (4.14) provides Le D s wCc, with c independent of w. Replacing Le D s w C c into the first two equations in (4.14) and integrating the first one with respect to wx yield @c=@wx D wx C a01 , with a01 independent of wx . Similarly, from the second equation in (4.14), @c=@wt D wt =u2 C a02 , with a02 independent of wt . Also, remembering that c is independent of w, one finds that a01 and a02 do not depend on w either. Considering that all the second derivatives of Le not appearing in (4.14) are equal to zero shows that a01 depends on t at most, while a02 depends on x at most. Integrating @c=@wx D wx Ca01 and @c=@wt D wt =u2 Ca02 one finds cD 1 2 w C a01 .t/ wx C a11 ; 2 x cD 1 2 w C a02 .x/ wt C a12 ; 2u2 t (4.15) where a11 does not depend on w or wx , while a12 does not depend on w or wt . Also, a11 cannot depend on both t and wt due to @2 Le =.@t @wt / D 0; similarly, a12 cannot depend on both x and wx due to @2 Le =.@x @wx / D 0. On the other hand, as both (4.15) hold, a11 must coincide (apart from an additive constant) with the first two terms at the right-hand side of the second equation in (4.15), and a12 must coincide with the first two terms at the right-hand side of the first equation. In conclusion, cD 1 2 w 2 x 1 2 w C a01 .t/ wx C a02 .x/ wt ; 2u2 t (4.16) with a01 .t/, a02 .x/ arbitrary functions. The last two terms in (4.16) are equal to d.a01 w/=dx C d.a02 w/=dt, namely, they form the divergence of a vector. As shown in Sect. 4.2 such a vector is arbitrary, so it can be eliminated by letting a01 D 0, a02 D 0. The relation Le D s w C c then yields Le D 1 2 w 2 x 1 2 w C sw : 2u2 t (4.17) The generalization to the three-dimensional case (4.13) is immediate, 1 Le D j grad wj2 2 with j grad wj2 D w2x C w2y C w2z . 1 2u2  @w @t 2 C sw: (4.18) 80 4 Electromagnetism 4.4 Maxwell Equations The Maxwell equations, that describe the electromagnetic field, lend themselves to an interesting application of the results of Sect. 4.3. The first group of Maxwell equations reads div D D % ; rot H @D D J; @t (4.19) where D is the electric displacement and H the magnetic field.2 The sources of the electromagnetic field are the charge density % and the current density J. When point-like charges are considered, they read %c D X j  ej ı r  sj .t/ ; Jc D X j  ej ı r  sj .t/ uj .t/ ; (4.20) where index c is used to distinguish the case of point-like charges from that of a continuous charge distribution. In (4.20), ej is the value of the jth charge, sj and uj its position and velocity at time t, respectively, and r the independent positional variable. If the spatial scale of the problem is such that one can replace the pointlike charges with a continuous distribution, one applies the same procedure as in TheP number of charges belonging to a cell of volume V centered at r is RSect. 23.2. 0 3 0 % d s D j ej , where the prime indicates that the sum is limited to the charges V c P that belong to  at time t. Then one defines %.r; t/ D 0j ej =V. The continuous distribution of the current density is obtained in a similar manner, 1 JD V Z 1 X0 Jc d s D ej uj D % v ; V j V 3 0 P0 j ej uj v D P0 ; j ej (4.21) with v.r; t/ the average velocity of the charges. If all charges are equal, e1 D e2 D : : : D e, then % D e N, with N.r; t/ the concentration, and J D e N v D e F, with F.r; t/ the flux density (compare with the definitions of Sect. 23.2). P If the charges are different from each other it is convenient to distribute the sum j over the groups made of equal charges. In this case the charge density and current density read % D %1 C % 2 C : : : ; J D %1 v1 C %2 v2 C : : : ; (4.22) The units in (4.19, 4.23, 4.24) are: ŒD D C m 2 , Œ% D C m 3 , ŒH D A m 1 , ŒJ D C s 1 m 2 D A m 2 , ŒB D V s m 2 D Wb m 2 D T, ŒE D V m 1 , where “C,” “A,” “V,” “Wb,” and “T” stand for Coulomb, Ampère, Volt, Weber, and Tesla, respectively. The coefficients in (4.19, 4.23, 4.24) differ from those of [10] because of the different units adopted there. In turn, the units in (4.25) are Œ"0  D C V 1 m 1 D F m 1 , Œ0  D s2 F 1 m 1 D H m 1 , where “F” and “H” stand for Farad and Henry, respectively, and those in (4.26) are Œ' D V, ŒA D V s m 1 D Wb m 1 . 2 4.5 Potentials and Gauge Transformations 81 where %1 , v1 are the charge density and average velocity of the charges of the first group, and so on. Taking the divergence of the second equation in (4.19) and using the third identity in (A.35) yield the continuity equation @% C div J D 0 : @t (4.23) Apart from the different units of the functions involved, the form of (4.23) is the same as that of (23.3). The meaning of (4.23) is that of conservation of the electric charge. The second group of Maxwell equations is div B D 0 ; rot E C @B D 0; @t (4.24) where B and E are the magnetic induction and the electric field, respectively. Here the Maxwell equations are considered in vacuo, so that the following hold D D "0 E ; B D 0 H ; p 1 D c; "0 0 (4.25) with "0 ' 8:854  10 12 F m 1 and 0 ' 1:256  10 6 H m 1 the vacuum permittivity and permeability, respectively, and c ' 2:998  108 m s 1 the speed of light in vacuo. 4.5 Potentials and Gauge Transformations Thanks to (4.25), the electromagnetic field in vacuo is determined by two suitably chosen vectors—typically, E and B—out of the four ones appearing in (4.25). This amounts to using six scalar functions of position and time. However, the number of scalar functions is reduced by observing that while (4.19) provides relations between the electromagnetic field and its sources, (4.24) provides relations among the field vectors themselves; as a consequence, (4.24) reduces the number of independent vectors. In fact, using the properties illustrated in Sect. A.9, one finds that from div B D 0 one derives B D rot A, where A is called vector potential or magnetic potential. In turn, the vector potential transforms the second of (4.24) into rot.E C @A=@t/ D 0; using again the results of Sect. A.9 shows that the term in parentheses is the gradient of a scalar function, that is customarily indicated with '. Such a function3 is called scalar potential or electric potential. In summary, 3 The minus sign in the definition of ' is used for consistency with the definition of the gravitational potential, where the force is opposite to the direction along which the potential grows. 82 4 Electromagnetism B D rot A ; ED grad ' @A ; @t (4.26) showing that for determining the electromagnetic field in vacuo it suffices to know four scalar functions, namely, ' and the three components of A. To proceed, one replaces (4.26) into (4.19) and uses the third relation in (4.25), to find r 2' C @ div A D @t % ; "0 rot rot A 1 @2 A D c2 @t2 0 J C 1 @' : grad 2 c @t (4.27) Thanks to the first identity in (A.36) the second equation in (4.27) becomes r 2A 1 @2 A D c2 @t2 0 J C grad  ;  D div A C 1 @' c2 @t (4.28) while, using the definition (4.28) of  , one transforms the first equation in (4.27) into r 2' 1 @2 ' D c2 @t2 % "0 @ : @t (4.29) In conclusion, (4.29) and the first equation in (4.28) are a set of four scalar differential equations whose unknowns are ' and the components of A. Such equations are coupled because  contains all unknowns; however, they become decoupled after suitable transformations, shown below. To proceed, one observes that only the derivatives of the potentials, not the potential themselves, appear in (4.26); as a consequence, while the fields E, B are uniquely defined by the potentials, the opposite is not true. For instance, replacing A with A0 D A C grad f , where f .r; t/ is any differentiable scalar function, and using the second identity in (A.35), yields B0 D rot A0 D B, namely, B is invariant with respect to such a replacement. If, at the same time, one replaces ' with a yet undetermined function ' 0 , (4.26) yields E0 D grad.' 0 C @f =@t/ @A=@t. It follows that by choosing ' 0 D ' @f =@t one obtains E0 D E. The transformation .' ; A/ ! .' 0 ; A0 / defined by '0 D ' @f ; @t A0 D A C grad f : (4.30) is called gauge transformation. As shown above, E and B are invariant with respect to such a transformation. One also finds that (4.29) and the first equation in (4.28) are invariant with respect to the transformation: all terms involving f cancel each other, so that the equations in the primed unknowns are identical to the original ones. However, the solutions ' 0 , A0 are different from ', A because, due to (4.30), their initial and boundary conditions are not necessarily the same. The difference 4.6 Lagrangian Density for the Maxwell Equations 83 between the primed and unprimed solutions is unimportant because the fields, as shown above, are invariant under the transformation. Using (4.30) in the second equation of (4.28) shows that  transforms as  0 D div A0 C 1 @' 0 D  C r 2f c2 @t 1 @2 f : c2 @t2 (4.31) The arbitrariness of f may be exploited to give  0 a convenient form. For instance one may choose f such that  0 D .1=c2 / @'=@t, which is equivalent to letting div A0 D 0 ; (4.32) called Coulomb gauge. The latter yields r 2'0 D % ; "0 1 @2 A0 D c2 @t2 r 2 A0 0 J C 1 @ grad ' 0 ; c2 @t (4.33) the first of which (the Poisson equation) is decoupled from the second one. After solving the Poisson equation, the last term at the right-hand side of the second equation is not an unknown any more, thus showing that the equations resulting from the Coulomb gauge are indeed decoupled. Another possibility is choosing f such that  0 D 0, which is equivalent to letting div A0 D 1 @' 0 ; c2 @t (4.34) called Lorentz gauge. This transformation yields r 2'0 1 @2 ' 0 D c2 @t2 % ; "0 r 2 A0 1 @2 A0 D c2 @t2 0 J : (4.35) that are decoupled and have the form of the wave equation (4.13). Another interesting application of the gauge transformation is shown in Sect. 5.11.4. 4.6 Lagrangian Density for the Maxwell Equations To apply the Lagrangian formalism to the Maxwell equations it is useful to use the expressions (4.26) of the fields in terms of the potentials. It follows that the functions playing the role of generalized coordinates and generalized velocities are ', Ai , and, respectively, @'=@xk , @Ai =@xk , @Ai =@t, with i; k D 1; 2; 3, k ¤ i. The Lagrangian density, whose units are J m 3 , then reads Le D "0 2 E 2 1 B2 2 0 %' C J
A; (4.36) 84 4 Electromagnetism with E2 D  @A1 @' C @x1 @t 2 C  @' @A2 C @x2 @t 2  @' @A3 C @x3 @t  @A1 @x2 C 2 (4.37) and 2 B D  @A3 @x2 @A2 @x3 2 C  @A1 @x3 @A3 @x1 2 C @A2 @x1 2 ; (4.38) To show that (4.36) is in fact the Lagrangian function of the Maxwell equations one starts with the generalized coordinate ', to find @Le =@' D %. Then, considering the kth component, @Le ="0 @E2 =2 @E2 =2 @' @Ak D D D D C @.@'=@xk / @.@'=@xk / @.@Ak =@t/ @xk @t Ek D Dk : "0 (4.39) Using (4.8) after replacing g with Le , j with xj , and wi with ' yields div D D %, namely, the first equation in (4.19). Turning now to another generalized coordinate, say, A1 , one finds @Le =@A1 D J1 . As Le depends on the spatial derivatives of A1 only through B2 , (4.38) and the first of (4.26) yield @B2 =2 @A1 D @.@A1 =@x3 / @x3 @A3 D B2 ; @x1 @B2 =2 @A1 D @.@A1 =@x2 / @x2 @A2 D @x1 B3 : (4.40) In contrast, Le depends on the time derivative of A1 only through E2 , as shown by (4.39). To use (4.8) one replaces g with Le and wi with A1 , then takes the derivative with respect to x3 in the first relation in (4.40), the derivative with respect to x2 in the second relation, and the derivative with respect to t of the last term in (4.39). In summary this yields 1 0  @B3 @x2 @B2 @x3  @D1 D J1 ; @t (4.41) namely, the first component of the second equation in (4.19). 4.7 Helmholtz Equation Consider the wave equations (4.35) and assume that the charge density % and current density J are given as functions of position and time. In the following, the apex in ' and A will be dropped for the sake of conciseness. The four scalar equations (4.35) are linear with respect to the unknowns and have the same structure; also, their coefficients and unknowns are all real. The solution of (4.35) will be tackled in this 4.7 Helmholtz Equation 85 section and in the following ones, basing upon the Fourier transform whose general properties are depicted in Sect. C.2. This solution procedure involves the use of complex functions. The starting assumption is that the condition for the existence of the Fourier transform with respect to time holds (such a condition is found by replacing x with t in (C.19)). Then one obtains r 2 Ft ' C !2 Ft ' D c2 1 Ft % ; "0 r 2 Ft A C !2 Ft A D c2 0 F t J : (4.42) Indicating with f the transform of ' or Ai , and with b the transform of %="0 or 0 Ji , i D 1; 2; 3, and letting k2 D ! 2 =c2 , each scalar equation in (4.42) has the form of the Helmholtz equation r 2 f C k2 f D b : (4.43) The solution of (4.43) is sought within a finite domain V (Fig. 4.1), for a given set of boundary conditions defined over the boundary S of V, and for a given righthand side b defined within V and over S. Let r D .x; y; z/ be a point external to V, q D .; ; / a point internal to V, and gDr q;  g D .x /2 C .y /2 C .z /2 1=2 (4.44) where, by construction, it is g > 0. In the following calculation, r is kept fixed while q varies. As a consequence, the derivatives of g are calculated with respect to , , and . It is easily shown that in a three-dimensional space the auxiliary function4 Fig. 4.1 The domain V used for the solution of the Helmholtz equation (4.43). The three possible positions of point r are shown: external to V, internal to V, or on the boundary S r q r V S n r ε n 4 Function G is also called Green function. If k D 0, this function fulfills the Laplace equation r 2 G D 0. As shown in Sect. 4.12.4, the form of the Green function that solves, e.g., the Laplace equation depends on the number of spatial dimensions considered. 86 4 Electromagnetism G.g/ D 1 exp. i k g/ ; g k real, (4.45) fulfills the homogeneous Helmholtz equation r 2 G C k2 G D 0. Using the procedure that leads to the second Green theorem (Sect. A.5, Eq. (A.25)) yields the integral relation  Z Z  @G @f f dS D G b dV ; (4.46) G @n @n V S where the unit vector n over S is oriented in the outward direction and, by construction, point r is external to V. 4.8 Helmholtz Equation in a Finite Domain The relation (4.46) would not be applicable if r were internal to V, because G diverges for g ! 0 and, as a consequence, is not differentiable in q D r. On the other hand, in many applications r happens to be internal to V. In such cases one must exclude from the integral a suitable portion of volume V; this is achieved by considering a spherical domain of radius  centered on r and internal to V (Fig. 4.1). Letting V , S be, respectively, the volume and surface of such a sphere, and considering the new volume V 0 D V V , having S0 D S [ S as boundary, makes (4.46) applicable to V 0 , to yield Z  @f G @n S @G f @n  dS C Z S .: : :/ dS D Z V G b dV Z G b dV ; (4.47) V where the dots indicate that the integrand is the same as in the first integral at the left-hand side. Over S it is G D .1=/ exp. i k /, with the unit vector n pointing from the surface towards the center of the sphere, namely, opposite to the direction along which  increases. It follows @G=@n D @G=@ D .i k C 1=/ G. Letting Œf  and Œ@f =@n be the average values of f and, respectively, @f =@n over S , and observing that G and @G=@ are constant there, yields Z  @f G @n S f @G @n  dS D 4  exp. i k / R    @f  @n  .1 C i k / Œf  : (4.48) As for the integral I D V G b dV it is useful to adopt the spherical coordinates (B.1) after shifting the origin to the center of the sphere. In the new reference it is r D 0, so that the radial coordinate coincides with g. It follows 4.8 Helmholtz Equation in a Finite Domain ID Z 0  Z 0  Z 87 2 g sin # exp. i k g/ b.g; #; '/ dg d# d' : (4.49) 0 Taking the absolute value of I and observing that g and sin # are positive yields jIj  2   2 supV jbj. To proceed, one assumes that f and b are sufficiently smooth as to fulfill the conditions   @f D 0; lim  2 sup jbj D 0 : (4.50) lim  Œf  D 0 ; lim  !0 !0 !0 @n V Thanks to (4.50) one restores the original volume V by taking the limit  ! 0. Observing that lim!0 Œf  D f .r/, one finds 4  f .r/ D Z  @f G @n S f @G @n  dS Z G b dV ; (4.51) V that renders f .r/ as a function of b, of the boundary values of f and @f =@n, and of the auxiliary function G. It is easily found that if r were on the boundary S instead of being internal to V, the left-hand side of (4.51) would be replaced by 2  f .p/. Similarly, if r were external to V, the left-hand side would be zero. In conclusion one generalizes (4.51) to Z  @f !r f .r/ D G @n S @G f @n  dS Z G b dV ; (4.52) V where !r is the solid angle under which the surface S is seen from r considering the orientation of the unit vector n. Namely, !r D 0, !r D 2 , or !r D 4  when r is external to V, on the boundary of V, or internal to V, respectively. Letting k D 0 in (4.45), namely, taking G D 1=g, makes the results of this section applicable to the Poisson equation r 2 f D b. It must be noted that (4.52) should be considered as an integral relation for f , not as the solution of the differential equation whence it derives. In fact, for actually calculating (4.52) it is necessary to prescribe both f and @f =@n over the boundary. This is an overspecification of the problem: in fact, the theory of boundary-value problems shows that the solution of an equation of the form (4.43) is found by specifying over the boundary either the unknown function only (Dirichlet boundary condition), or its normal derivative only (Neumann boundary condition). To find a solution starting from (4.52) it is necessary to carry out more steps, by which either f or @f =@n is eliminated from the integral at the right-hand side [69, Sect. 1.8–1.10]. In contrast, when the solution is sought in a domain whose boundary extends to infinity, and the contribution of the boundary conditions vanish as shown in Sect. 4.9, the limiting case of (4.52) provides a solution proper. More comments about this issue are made in Sect. 4.12.3. 88 4 Electromagnetism 4.9 Solution of the Helmholtz Equation in an Infinite Domain The procedure shown in Sect. 4.8 is readily extended to the case V ! 1. Here one may replace S with a spherical surface of radius R ! 1, centered on r; this makes the calculation of the integral over S similar to that over the sphere of radius " outlined in Sect. 4.8, the only difference being that the unit vector n now points in the direction where R increases. Shifting the origin to r and observing that !r D 4  yield Z  @f G @n S f @G @n  dS D 4  exp. i k R/     @f R C .1 C i k R/ Œf  ; @n (4.53) where the averages are calculated over S. To proceed one assumes that the following relations hold,    @f lim Œf  D 0 ; C i k Œf  D 0 ; (4.54) lim R R!1 R!1 @n that are called Sommerfeld asymptotic conditions. Due to (4.54) the surface integral (4.53) vanishes. Shifting the origin back from r to the initial position, the solution of the Helmholtz equation (4.43) over an infinite domain finally reads f .r/ D 1 4 Z b.q/ 1 exp. i k jr qj/ 3 d q; jr qj (4.55) R where 1 indicates the integral over the whole three-dimensional q space. The k D 0 case yields the solution of the Poisson equation r 2 f D b in an infinite domain, f .r/ D 1 4 Z 1 b.q/ 1 jr qj d3 q : (4.56) 4.10 Solution of the Wave Equation in an Infinite Domain The solutions of the Helmholtz equation found in Sects. 4.8, 4.9 allow one to calculate that of the wave equation. In fact, it is worth reminding that the Helmholtz equation (4.43) was deduced in Sect. 4.7 by Fourier transforming the wave equations (4.35) and i) letting f indicates the transform of the scalar potential ' or of any component Ai of the vector potential, ii) letting b indicates the transform of %="0 or 0 Ji , i D 1; 2; 3. As a consequence, f and b depend on the angular frequency ! besides the spatial coordinates. From the definition k2 D ! 2 =c2 one may also assume that both k and ! have the same sign, so that k D !=c. Considering for simplicity the case V ! 1, applying (C.17) to antitransform (4.56), and interchanging the order of integrals yields 4.11 Lorentz Force F 1 f D 89 1 4 Z 1 1 g  1 p 2 Z C1 b.q; !/ expŒi ! .t  g=c/ d! d3 q ; 1 (4.57) with g D jr qj. Now, denote with a the antitransform of b, a.q; t/ D F 1 b. It follows that the function between brackets in (4.57) coincides with a.q; t g=c/. As remarked above, when f represents ', then a stands for %="0 ; similarly, when f represents a component of A, then a stands for the corresponding component of 0 J. In conclusion, 1 '.r; t/ D 4  "0 0 A.r; t/ D 4 Z Z %.q; t 1 jr J.q; t 1 jr jr jr qj=c/ qj qj=c/ qj d3 q ; d3 q ; (4.58) (4.59) that express the potentials in terms of the field sources % and J, when the asymptotic behavior of the potentials fulfills the Sommerfeld conditions (4.54). The functions rendered by the antitransforms are real, as should be. Note that jr qj=c > 0 is the time necessary for a signal propagating with velocity c to cross the distance jr qj. As t jr qj=c < t, the above expressions of ' and A are called retarded potentials.5 4.11 Lorentz Force It has been assumed so far that the sources of the electromagnetic field, namely charge density and current density, are prescribed functions of position and time. This is not necessarily so, because the charges are in turn acted upon by the electromagnetic field, so that their dynamics is influenced by it. Consider a test charge of value e immersed in an electromagnetic field described by the vectors E, B generated by other charges. The force acting upon the test charge is the Lorentz force [10, Vol. I, Sect. 44] F D e .E C u ^ B/ ; (4.60) where u is the velocity of the test charge and E, B are independent of u. The expression of the Lorentz force does not derive from assumptions separate from Maxwell’s equations; in fact, it follows from Maxwell’s equations and Special 5 Expressions of ' and A obtained from (4.58,4.59) after replacing t jr qj=c with t C jr qj=c are also solutions of the wave equations (4.35). This is due to the fact that the Helmholtz equation (4.43) can also be solved by using G instead of G, which in turn reflects the time reversibility of the wave equation. However, the form with t jr qj=c better represents the idea that an electromagnetic perturbation, that is present in r at the time t, is produced by a source acting in q at a time prior to t. 90 4 Electromagnetism Relativity [52, 135]. The extension of (4.60) to the case of a number of point-like charges follows the same line as in Sect. 4.4: considering the charges belonging to a cell of volume V centered at r, one writes (4.60) for the jth charge and takes the sum over j, to find fD P0 j Fj V D % .E C v ^ B/ ; (4.61) where %, v are defined in (4.21) and f is the force density (Œf D N m 3 ). The fields in (4.61) are calculated in r and t. Consider a small time interval ıt during which the charge contained within V is displaced by ır D v ıt. The work per unit volume exchanged between the charge and the electromagnetic field due to such a displacement is ıw D f
ır D % .E C v ^ B/
v ıt D E
J ıt ; Œw D J m 3 ; (4.62) where (A.32) and (4.21) have been used. When the scalar product is positive, the charge acquires kinetic energy from the field, and vice versa. Letting ıt ! 0 yields @w D E
J; @t (4.63) where the symbol of partial derivative is used because (4.63) is calculated with r fixed. 4.12 Complements 4.12.1 Invariance of the Euler Equations It has been shown in Sect. 4.2 that the Euler equations (4.8) are still fulfilled if the generating function g is replaced with g0 D g C div h, where h is an arbitrary vector of length m whose entries depend on w and 1 ; : : : ; m , but not on the derivatives of w. This property is a generalization of that illustrated in Sect. 1.2 with reference to a system of particles, where it was shown that the solutions wi ./ are invariant under addition to g of the total derivative of an arbitrary function that depends on w and  only. 4.12.2 Wave Equations for the E and B Fields The Maxwell equations can be rearranged in the form of wave equations for the electric and magnetic fields. To this purpose, one takes the rotational of both sides 4.12 Complements 91 of the second equation in (4.24). Using the first identity in (A.36) and the relation D D "0 E provides @ rot B=@t D rot rot E D grad div.D="0 / r 2 E. Replacing div D and rot H D rot B=0 from (4.19) and using "0 0 D 1=c2 then yield r 2E @J 1 1 @2 E D grad % C 0 : c2 @t2 "0 @t (4.64) Similarly, one takes the rotational of both sides of the second equation in (4.19). Using the relation B D 0 H provides "0 @ rot E=@t C rot J D rot rot H D grad div.B=0 / r 2 H. Replacing div B and rot E from (4.24) yields r 2H 1 @2 H D c2 @t2 rot J : (4.65) 4.12.3 Comments on the Boundary-Value Problem Considering relation (4.52) derived in Sect. 4.8, one notes that the right-hand side is made of the difference between two terms; the first one depends on the boundary values of f , @f =@n, but not on b, while the second one depends only on the values of b within V and over the boundary. In these considerations it does not matter whether point r is external to V, on the boundary of V, or internal to it. In latter case the two terms at the right-hand side of (4.52) balance each other. Q and thereby the value of the second If b is replaced with a different function b, integral changes, it is possible to modify the boundary values in such a way as to balance the variation of the second integral with that of the first one; as a consequence, f .r/ is left unchanged. A possible choice for the modified b is bQ D 0; by this choice one eliminates the data of the differential equation and suitably modifies the boundary values, leaving the solution unaffected. An observer placed at r would be unable to detect that the data have disappeared. The same process can also be carried out in reverse, namely, by eliminating the boundary values and suitably changing the data. An example is given in Prob. 4.4 with reference to a one-dimensional Poisson equation where the original charge density differs from zero in a finite interval Œa; b. The charge density is removed and the boundary values at a are modified so that the electric potential ' is unaffected for x  b. Obviously ' changes for a < x < b because both the charge density and boundary conditions are different, and also for x  a because the boundary conditions are different. 92 4 Electromagnetism 4.12.4 Green Function for the Upper Half Plane The solution of the Laplace equation in two dimensions is considered in this section, using the method of the Green function introduced in Sect. 4.7. Using the Cartesian coordinates, let P D .x; y/ be a point of the upper half plane y > 0, Q D .; 0/ a point of the x axis, and r D jP Qj D Œ. x/2 C y2 1=2 > 0 the distance between them. Treating  as a parameter, the function G.x; yI / D log.r2 / (4.66) is harmonic in the upper half plane. Indeed, using a suffix to indicate a partial derivative, it is Gx D 2 .x /=r2 and Gy D 2 y=r2 , whence Gxx D 2 Œy2 .x /2 =r4 and Gyy D 2 Œ.x /2 y2 =r4 . Adding the last two relations yields r 2 G D 0: From the above it follows that also log r D G=2 is harmonic in the upper half plane. On the contrary, G is not everywhere harmonic along the x axis because G.x; 0/ diverges as j xj ! 0. Let E.x/ be a piecewise continuous function defined along the x axis. It is also assumed that E is bounded, namely, that some number M > 0 exists such that jEj  M for all x. The integral '.x; y/ D '0 1 2 Z C1 E./ G.x; yI / d ; (4.67) 1 with '0 an arbitrary constant, is harmonic in the upper half plane due to the properties of G. In addition, its derivative with respect to y along the x axis is E. More precisely, if E is continuous at x, then limy!0C @'=@y D E.x/, whereas if E has a first-order discontinuity at x, E.x/ in the above must be replaced with ŒE.xC / C E.x /=2. In fact, from the definition of G it follows @' D lim C y!0 @y lim y!0C Z C1 1 . E./ y= d ; x/2 C y2 (4.68) where x is kept fixed. Consider an interval of length s centered at x. The contribution to the integral of this interval is Z xCs=2 x s=2 E./ y= d D hEis . x/2 C y2 Z xCs=2 x s=2 . y= d ; x/2 C y2 (4.69) with hEis the average value of E over the interval. Letting  D . x/=y, the integral at the right-hand side of (4.69) becomes Z Cs=.2 y/ s=.2 y/   1= 2 s ; d D arctan 2 1C  2y (4.70) Problems 93 whose limit for y ! 0C is 1. The contribution to the integral of the interval from 1 to x s=2 is zero because ˇZ ˇ ˇ ˇ ˇ x s=2 1 ˇ Z ˇ E./ y= ˇ  M d ˇ . x/2 C y2 ˇ x s=2 1 . y= d : x/2 C y2 (4.71) The integral at the right-hand side of (4.71) is equal to .M=/ Œarctan . s=.2 y// arctan. 1/, whose limit for y ! 0C is 0. Similarly, the integral over the interval from x C s=2 to C1 does not contribute. In conclusion, lim y!0C @' D @y hEis : (4.72) The proof is completed by observing that s can be taken as small as we please. In conclusion, the integral (4.67) defining ' provides the solution of the Laplace equation r 2 ' D 0 in the upper half plane, supplemented with the prescription of the normal derivative along the x axis (Neumann boundary conditions). The results found in this section have a useful application to a method for measuring the conductivity of a simply connected, flat sample of material of arbitrary shape. The method is described in Sect. 25.7. Problems 4.1 Solve the one-dimensional Poisson equation d2 '=dx2 D %.x/="0 , with % given, using the integration by parts to avoid a double integral. The solution is prescribed at x D a while the first derivative is prescribed at x D c. 4.2 Let c D a in the solution of Prob. 4.1 and assume that the charge density % differs from zero only in a finite interval a  x  b. Find the expression of ' for x > b when both the solution and the first derivative are prescribed at x D a. 4.3 In Prob. 4.2 replace the charge density % with a different one, say, %. Q Discuss the conditions that leave the solution unchanged. 4.4 In Prob. 4.2 remove the charge density % and modify the boundary conditions at a so that the solution for x > b is left unchanged. 4.5 Using the results of Probs. 4.2 and 4.3, and assuming that both M0 and M1 are different from zero, replace the ratio %="0 with  ı.x h/ and find the parameters , h that leave M0 , M1 unchanged. Noting that h does not necessarily belong to the interval Œa; b, discuss the outcome for different positions of h with respect to a. 4.6 A particle of mass m and charge q enters at t D 0 a region where a constant magnetic induction B D B k is present, with B > 0. The velocity u0 at t D 0 is normal to B. Find the particle’s trajectory for t > 0.