The electromagnetic force, following from Maxwell's minimal macroscopic elec- tromagnetic stress dyad, which reproduces analytically the measurements made by Coulomb[magnet-magnet force], Biot-Savart[current-magnet force], Ampere-Weber[current- current force] and Faraday-Kelvin[polarization current-magnetization current force], is ¤Corresponding author Email address: akhacha3@calstatela.edu (Armik Khachatourian) 1 presented1. This force leads to a space independent internal torque explaining experi- ments such as the Einstein de Haas e®ect [the rotation of a material in the presence of a uniform external magnetic ¯eld],2or the Barnett e®ect [the creation of the magnetic ¯eld by a rotation of a material]3;4. The cause for the torque should not be, a priori, attributed to the asymmetry in the Cauchy's internal dyad5- a conclusion which is unavoidable since the electromagnetic force is obtained from a symmetric dyad which leads to no internal torque. 2 Introduction The well known elementary magnetic torque per unit volume ~M£~Bis shown to follow from Maxwell's macroscopic stress dyad. This torque could explain the initial cause6of the rotation of the substance, observed more than a century ago2. Hence, in an experimental observation of this mechanism in addition to the intrinsic internal torque (due to the asym- metry of Cauchy's internal stress dyad), the intrinsic external torque (due to the asymmetry of Maxwell's external stress dyad) must be considered. In section 1 the equation of the motion of the macroscopic angular momentum of a substance in the presence of the electromagnetism is brie°y reviewed. In section 2 Maxwell's macroscopic magnetic force and torque are presented. In section 3 Maxwell's macroscopic electromagnetic force and torque are presented, followed with a simple example, discussing the rotation of a soft ferromagnetic sphere in the presence of a uniform static magnetic . 1 The macroscopic angular momentum In a continuum of density ½(~ r; t), the elemental angular momentum of a substance ~`=m~ r£_~ r satis¯es the equation of motion: _~`=m~ r£Ä~ r=dV ~ r£(~r ¢T) (1) where dV(t) is the elemental volume of the substance with the elemental mass m(=½dV) at a position ~ r, velocity _~ rand acceleration Ä~ r[relative to an arbitrary inertial frame]. And T= ^xi^xjTijis the net stress dyad acting within the elemental substance. The net force 3 per unit volume on the elemental substance is given by ~r ¢T=@iTij^xj, i.e. the equation of the motion of mismÄ~ r=dV~r ¢Twith _ m= 0. This force satis¯es the third law since by divergence theorem it can be transformed to an arbitrary surface integral enclosing the elemental volume dV~r ¢T=H @Ad~A¢Tas long as this surface encloses the whole of volume dV but does not enclose any volume of the environment exerting the force. And hence the net force is due to external ¯elds. If the environment is divided into external and internal domains [the elemental mass m belonging to the internal domain], integrating the equation of the motion over the internal space gives MÄ~ rcm=H d~A¢Twhere M=R ½dVis the macroscopic mass of the internal space andÄ~ rcm=1 MR mÄ~ ris the center of mass acceleration. Similarly integrating the Eq. (1) over the whole volume of the internal space gives: _~L=Z dV ~ r£(~r ¢T) (2) The macroscopic torque_~Lis caused by an intrinsic volume torque ~ ¿i=¡_~Splus an extrinsic surface torque ~ ¿e, i.e. the right side of Eq. (2) can be transformed to a position independent volume torque [ ~ ¿i=¡_~S] plus a position dependent extrinsic surface torque [ ~ ¿e], hence _~L=R dV ~ r£(~r¢T) =¡R dV T ij^xi£^xj+H ~ r£(d~A¢T) =~ ¿i+~ ¿e=¡_~S+~ ¿e(the intrinsic volume torque ¡_~Sis being caused by an external or internal couple, vanishing if the net stress dyad is symmetric Tij=Tji) leading to _~L+_~S=~ ¿e(3) The intrinsic torque ¡_~Swill then be due to both Cauchy's internal volume torque _~ s= R dV T(int) ij^xi£^xjand Maxwell's volume torque _~ ¾=R dV T(e:m:) ij ^xi£^xj, i.e. T=T(int)+T(e:m:) 4 and_~S=_~ s+_~ ¾. Hence Eq. (3) reads_~L+_~S=_~L+_~ s+_~ ¾=_~J+_~ ¾=~ ¿e=~ ¿e;ext+~ ¿e;intwhere _~Jis the net internal torque (_~J=_~L+_~ s) and where the extrinsic torque ~ ¿eis the sum of the extrinsic torque due to the external environment ~ ¿e;extand the internal environment ~ ¿e;int. If the extrinsic torque ~ ¿e=H ~ r£(d~A¢T) vanishes, the vanishing of the Cauchy's internal volume torque _~ s= 0 does not imply conservation of a net internal angular momentum ~J=~L+~ sby virtue of the presence of Maxwell's volume torque _~ ¾:_~L+_~S=_~L+_~ ¾= 0 . However, turning on the external ¯eld, keeping in mind that the external force can induce permanent changes in the net internal angular momentum, and then turning o® the external ¯eld can lead to the conservation of the net internal angular momentum ~J=~L+~ s: 0 =_~L0field on t= 0+ z}|{! 0 =_~L1+_~ ¾1field on hysteresis z}|{! 0 =_~L2+_~ s2+_~ ¾2field off hysteresis z}|{! 0 =_~L+_~ s=_~J(4) Therefore the external force can turn a substance with a symmetric internal Cauchy stress dyad into an asymmetric one. 2 Maxwell's Macroscopic Magnetic Force Continuity of the ¯elds and their ¯rst derivative is assumed and all ¯elds are relative to an arbitrary inertial frame. Experiments performed by Coulomb, Biot-Savart, Ampere-Weber and Faraday-Kelvin suggest the macroscopic force7[ MKS system of the units are used 5 throughout] ~F(m) macro =Z dV~r ¢~B|{z} 0~H+Z dV(~r £ ~H)£(¹~H) +Z dV[~B¡¹~H]¢~r~H (5) where ~His the force per unit magnetic charge(or per unit magnetic °ux)[Newton V olt second] and ~B is the magnetic °ux per unit area [Tesla] and ¹is the permeability of the medium or the magnetomotive force per unit current [Newton Ampere2]. Hence ~r £ ~His the true current density [The ¯rst term, the Coulomb-Gauss term, although zero is written explicitly to facilitate the transition to the electric sector(see section 3)]. Furthermore, this magnetic force is the divergence of a (minimal) magnetic dyad T(m)given by7: ~F(m) macro =Z dV~r ¢T(m)=Z dV~r ¢[~B~H¡¹ 2~H¢~HI] (6) where T(m)=~B~H¡¹ 2~H¢~HI ;I=^i^i+^j^j+^k^k (7) Hence this force obeys the third law for arbitrary vector ¯elds ~Hand~B. Using Helmholtz' decomposition, the above dyad is written as a combination of a symmetric and asymmetric dyad T(m)="~B~H+~H~B 2¡1 2¹IH2# +~B~H¡~H~B 2(8) and then the symmetric part is diagonalized via: 8 >>< >>:~B=B(^»1cos²¡^»2sin²) ;~H=H(^»1cos²+^»2sin²) I=^»1^»1+^»2^»2+^»3^»3(9) 6 Substituting ~B,~HandIfrom Eq. (9) in the expression for T(m)given by Eq. (8) leads to: T(m)= ( BHcos2²¡1 2¹H2)^»1^»1+BHsin²cos²(^»1^»2¡^»2^»1) + (¡BHsin2²¡1 2¹H2)^»2^»2¡1 2¹H2^»3^»3 = ( ^»1;^»2;^»3)0 BBBBB@BHcos2²¡1 2¹H20 0 0 ¡BHsin2²¡1 2¹H20 0 0 ¡1 2¹H21 CCCCCA0 BBBBB@^»1 ^»2 ^»31 CCCCCA+ +(^»1;^»2;^»3)0 BBBBB@0 BHsin²cos²0 ¡BHsin²cos² 0 0 0 0 01 CCCCCA0 BBBBB@^»1 ^»2 ^»31 CCCCCA(10) The symmetric part in the above equation gives the principle stresses, while the asym- metric part gives the couple1 2~B£~H. Hence7" (1) a pressure equal in all directions1 2¹H2, (2) a tension along the line bisecting the angle between the magnetic force [ ~H] and the magnetic induction [ ~B]BHcos2², (3) a pressure along the line bisecting the exterior angle between these directions BHsin2² , and (4) a couple tending to turn every element of the substance in the plane of two directions from the direction of the magnetic induction to the direction of the magnetic force1 2BHsin 2² .". The couple leads to the intrinsic torque, i.e. the net magnetic external torque, relative to some origin, follows as: ~ ¿m:;ext macro =Z dV ~ r£(~r ¢T(m)) =Z dV ~ r£· ~r ¢½ ~B~H¡1 2¹H2I¾¸ =I ~ r£½ d~S¢· ~B~H¡1 2¹H2I¸¾ ¡Z dV³ ~B£~H´ (11) 7 the ¯rst term [in the second line above] is a surface term which involves the external ¯elds, the second term is Maxwell's intrinsic torque which acts within the body. Accordingly, if the magnetic ¯eld ~Band the magnetic ¯eld intensity ~Hare not propor- tional to one another, they give rise to an elemental internal magnetic torque of the form dV~B£~H, even in the absence of the true current density ~r £ ~H= 0 [See also the example at the end of the section 3.] . Maxwell's torque, dV~B£~H, can explain the Einstein-de Haas e®ect2, i.e. the rotation of the ferromagnetic material in the presence of an external magnetic ¯eld. In a similar manner it follows that a forced rotation of a substance could lead to the creation of a magnetic ¯eld, which can be observed by noting Maxwell's equation for the conductors in motion, which is given by Eq. (16). The induced ~Band ~Hwithin the substance, when exposed to the external ¯eld, could then, when the external ¯eld is removed _~ ¾!0, lead to a permanent asymmetry in Cauchy's original symmetric stress dyad. In this light then when the external ¯eld is turned o® _~ ¾= 0 the internal angular momentum of the substance ~Jwill remain constant, according to Eq. (4) 3 Maxwell's Macroscopic Electromagnetic Force Maxwell's Magnetic force can be transformed into an electric force by virtue of the known Coulomb forceR dV(~r ¢~D)~E. Therefore, in equations (5) and (6), the vectors ~Hand~Bare replaced with the vectorsq ² ¹~Eandp¹ ²~Drespectively, resulting in the generalized electric 8 force: ~F(e) macro =Z dV(~r ¢~D)~E+Z dV(~r £ ~E)£(²~E) +Z dV[~D¡²~E]¢~r~E =Z dV~r ¢· ~D~E¡1 2²E2I¸ (12) [page 155 Vol.1 and page 258 Vol.2 (third edition) ]7, where ²is the permittivity of the medium [Farads/meter], ~Eis the electric ¯eld [Newton/Coulomb] and ~Dis the displacement vector or the true electric °ux per unit area [Coulomb/meter2]. Hence the general macro- scopic electromagnetic force motivated by Maxwell's Magnetic force and Coulomb's electric force follows as ( ~Fe:m: macro =~F(m) macro +~F(e) macro [from Eq. (6) and (12)] )7: ~Fe:m: macro =Z dV~r ¢T(e:m:)=Z dV~r ¢· ~D~E+~B~H¡1 2(²E2+¹H2)I¸ =I d~S¢· ~D~E+~B~H¡1 2(²E2+¹H2)I¸ (13) where the permittivity ²and the permeability ¹are that of the medium. The net electromagnetic external torque, relative to some origin, follow as: ~ ¿e:m:;ext macro =Z dV ~ r£½ ~r ¢· ~D~E+~B~H¡1 2(²E2+¹H2)I¸¾ = =I ~ r£½ d~S¢· ~D~E+~B~H¡1 2(²E2+¹H2)I¸¾ ¡Z dV³ ~D£~E+~B£~H´ (14) the second term [in the second line above] is Maxwell's torque which acts within the body. This torque vanishes if the dyad is symmetric, i.e. when ~Band ~Eare along ~Hand ~D respectively. Maxwell's torque can be written in the familiar form [ ~H=~B ¹¡~M ; ~D=²~E+~P]: ~ ¿Maxwell;ext macro =Z dV³ ~D£~E+~B£~H´ =Z dV³ ~P£~E+~M£~B´ (15) 9 This is a remarkable ¯nding[by Maxwell]: a classical macroscopic torque leading to a macro- scopic spin. It states that the ¯eld outside the substance [which itself gets modi¯ed] can induce an electric displacement as well as a magnetic induction within the substance, which can then lead to a rotation of the substance. An elementary example (involving two parameters ¯and°) is given in this conclusion: Maxwell's macroscopic equations for a substance moving with variable velocity ~ vis given by [all vector ¯elds are relative to an inertial frame]: ~r £ ~E= ¡@~B @t+ [¡(~r ¢~B|{z} 0)~ v+~r £(~ v£~B)] ~r £ ~H=~Jf+@~D @t+ [+( ~r ¢~D)~ v¡~r £(~ v£~D)](16) Consider the case where ~E=~D=~Jf= 0. The above equations simplify to: 0 =¡@~B @t+~r £(~ v£~B) ; ~r £ ~H= 0 Let the above equations hold within and without a solid soft ferromagnetic sphere of radius R, initially in the presence of a space and time independent magnetic ¯eld B0(=¹H0) along the z axis. The following ¯elds satisfy the above equations: ~ v=!rsinµ^Á;~Bin= 3B0^z+B0 ¯rsinµ^Á;~Hin=H0 (¡°)r^r; 0·r·R ~ v= 0 ; ~Bout=¹~Hout=B0h ^z+R3 r3(2 cos µ^r+ sin µ^µ)i ;R·r· 1(17) where the angular velocity !(t) is independent of space but may depend on time, explicitly. The parameters ¯and°have units of length. ¯is a constant independent of time and space and °(t) is independent of space but may depend on time explicitly. The Cartesian and spherical base vectors are (^ x;^y;^z), (^r;^µ;^Á) respectively. Here ~ r=r^ris relative to the 10 geometric center of the sphere and ~ v=_~ ris the velocity of a point within (without) the sphere. The above solution satis¯es the following boundary conditions for the magnetization ~Mjr=R: ¡^r¢~Min=RH 0 (¡°)¡3B0cosµ ¹ ¡^r£~Min=3B0sinµ ¹µ ^Á+R 3¯^µ¶ The force follows from equation (13) which vanishes: ~F(m) em =I d~S¢[~Bout~Hout¡1 2¹I~Hout¢~Hout] =Z¼ 0Z2¼ 0R2sinµdµdÁ [^r¢~Bout~Hout¡1 2¹^r~Hout¢~Hout]jr=R= 0 The torque relative to the geometric center of the sphere follows from equation (14): ~ ¿ext=I ~ r£½ d~S¢· ~B~H¡1 2¹H2I¸¾ ¡Z dV³ ~B£~H´ =I ~R£½ d~S¢· ~Bout~Hout¡1 2¹H2 outI¸¾ | {z } ~ ¿e;ext=0¡Z dV³ ~Bin£~Hin´ | {z } _~ ¾=¡8B0H0 15¯(¡°)¼R5^z = (B0H04¼R3 3)(2 5R21 ¯(¡°))^z (18) where the factor2 5R2is the moment of inertia of a uniform solid sphere per unit mass. From Euler's equation of motion [_~L+_~S=_~L+_~ s|{z} 0+_~ ¾=~ ¿e=~ ¿e;ext|{z} 0+~ ¿e;int|{z} 0= 0 where Cauchy's stress dyad is assumed to be symmetric _~ s=0 and ~ ¿e;int= 0]: @Icm~ ! @t+~ !£~L|{z} 0+_~ s|{z} 0+_~ ¾= 0 (19) Hence, equations (18) and (19) give @ @tIcm!(t) = (B0H04¼R3 3)(2 5R2 1 ¯(¡°(t))) (20) 11 Introducing the new time dependent magnetic ¯eld b(t) ¡gqe 2me@ @tIcmb(t) = (B0H04¼R3 3)(2 5R2 1 ¯(¡°(t))) (21) and comparing Eq. (21) with Eq. (20) gives !=¡gqe 2meb (22) which is a well known macroscopic result connecting the angular velocity !of a "°uid" to the magnetic ¯eld binvolving Thomson's parameterqe me. 12