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the torque on an electric dipole * p * , at rest in a uniform electric field * e * , is @xmath0 the torque on a magnetic dipole * m * , at rest in a uniform magnetic field * b * , is @xmath1 but what if the dipole is moving , at a constant velocity * v * ? it is well known that a moving electric dipole acquires a _ dipole moment @xcite @xmath2 so one might guess that the torque on a moving electric dipole in uniform electric and magnetic fields would be @xcite @xmath3 similarly , a moving magnetic dipole acquires an _ electric _ dipole moment @xcite @xmath4 and one might guess that the torque on a moving magnetic dipole would be @xmath5 but these formulas are incorrect ; in each case there is a third term:@xcite @xmath6 in general , then , @xmath7 our purpose in this note is to derive that result . @xcite    -.1in[.7 ] -.5 in an electric dipole consists of positive and negative charges @xmath8 separated by a displacement @xmath9 ( figure 1 ) . the force on a point charge is @xmath10 so the torque @xcite on a dipole of moment @xmath11 is @xmath12\nonumber\\ & = & q{\bf d}\times\left({\bf e}+{\bf v}\times { \bf b}\right)\nonumber\\ & = & ( { \bf p}\times { \bf e})+{\bf p}\times({\bf v}\times { \bf b}).\end{aligned}\ ] ] using the vector identity @xmath13 , the previous result becomes @xmath14    that is surely the _ easiest _ way to obtain eq . ( 7 ) ( it borrows from an argument by v.  namias@xcite ) . but the magnetic analog is not so straightforward,@xcite so let s repeat the derivation , this time treating the electric dipole as the point limit of a uniformly polarized object.@xcite polarization ( * p * ) and magnetization ( * m * ) together constitute an antisymmetric second - rank tensor : @xmath15 and the transformation rule gives@xcite @xmath16 we use primes for the  proper " ( rest ) system of the dipole ( @xmath17 ) ; no prime means the  lab " frame ( @xmath18 ) , through which the dipole is moving with velocity @xmath19 . notice that if @xmath20 , then @xmath21 , confirming eq . ( 3 ) , and if @xmath22 , then @xmath23 , confirming eq .  ( 5 ) .    now , suppose we have an electric dipole @xmath24 at rest at the origin in @xmath25 : @xmath26 assume first that @xmath24 is perpendicular to * v * , say @xmath27 then @xmath28 and all other components are zero . according to the lorentz law , the force density is @xmath29 where @xcite @xmath30 in our case,@xcite @xmath31 @xmath32,\ ] ] @xmath33 so @xmath34 and @xmath35.\ ] ] the torque is @xmath36,\end{aligned}\ ] ] where @xmath37 . the @xmath38 and @xmath39 components of the integral are zero ; the @xmath40 component is @xcite @xmath41 so @xmath42= ( { \bf p}_0\times { \bf e } ) + { \bf p}_0 \times ( { \bf v } \times { \bf b}),\ ] ] which agrees with eq .  ( 11 ) , and hence confirms eq . ( 7 ) for the case @xmath24 perpendicular to * v*. ( for this orientation the electric dipole moments in @xmath18 and @xmath25 are the same  there is no lorentz contraction . this can be confirmed by integrating eq .  ( 18 ) : @xmath43 . )    if @xmath24 is parallel to * v * , @xmath44 giving @xmath45 with all other components of * p * and * m * being zero . in this case , @xmath46 and @xmath47 so that @xmath48.\ ] ] the torque is @xmath49.\ ] ] because of lorentz contraction , @xmath50as is confirmed by integrating eq . ( 31)and again we recover eq . now , suppose we have a magnetic dipole @xmath51 , at rest at the origin , in @xmath25 : @xmath52 assume first that @xmath51 is perpendicular to * v * , say @xmath53 from the transformation rule , @xmath54 this time @xmath55 @xmath56,\ ] ] @xmath57 so @xmath58,\ ] ] and @xmath59\nonumber\\   & & \quad\quad\quad\times { \bf b}.\end{aligned}\ ] ] the torque is @xmath60 in agreement with eq . ( 8) , for @xmath51 perpendicular to * v*. ( in this orientation , the magnetic dipole moments in @xmath18 and @xmath25 are the same , as we confirm by integrating eq . ( 39 ) : @xmath61 . )    if @xmath51 is parallel to * v * , @xmath62 giving @xmath63 with all other components of * p * and * m * being zero . in this case @xmath64 and @xmath65 vanish , leaving @xmath66,\ ] ] so @xmath67\times{\bf b},\ ] ] and the torque is @xmath68 which is again consistent with eq . ( 8) because in this orientation @xmath69 and , integrating eq . ( 47 ) , @xmath70 . in the presence of an electric field , a magnetic dipole ( a loop of electric current ) carries hidden momentum"@xcite @xmath71 and hence also hidden momentum @xmath72 it is  hidden " in the sense that it is not associated with overt motion of the object as a whole ; it occurs in systems with internally moving parts , such as current loops.@xcite the _ total _ angular momentum is the sum of  overt " and hidden components : @xmath73 and the torque is its rate of change:@xcite @xmath74 thus , combining eqs . ( 9 ) and ( 53 ) , the  overt " torque on a moving dipole is @xmath75    by contrast , a ( stationary ) _ electric _ dipole has no internally moving parts , and it harbors no hidden momentum.@xcite however , when it is in _ motion _ it picks up a magnetic dipole moment @xmath76 [ eq . ( 3 ) ] that _ does _ carry hidden momentum,@xcite so eq .  ( 54 ) ( with @xmath77 ) applies to this case as well.@xcite co - authored this note in his private capacity ; no official support or endorsement by the centers for disease control and prevention is intended or should be inferred . see , for instance , v.  hnizdo ,  magnetic dipole moment of a moving electric dipole , " am .  j.  phys . * 80 * , 645 - 647 ( 2012 ) . in this paper , except for eqs . ( 1 ) and ( 2 ) , * p * and * m * are the dipole moments of the _ moving _ dipoles_not _ the  proper " moments in the dipoles rest frame . see , for example , k. w. h. panofsky and m. phillips , _ classical electricity and magnetism _ ( dover , new york , 2005 ) , sec . 18 - 4 . in a recent article [ d.  j.  griffiths and v.  hnizdo , mansuripur s paradox , " am .  j.  phys . * 81 * , 570 - 574 ( 2013 ) ] we stated that there is no  third term " in the torque formula . this was misleading : there _ is _ a third term ( in general ) , but it was zero ( in that context ) because @xmath78 at the location of the dipole . if the fields are _ non_uniform , the torque ( 9 ) is supplemented by @xmath79 $ ] , where @xmath80 is the dipole s displacement from the point with respect to which torques are calculated . the term inside the brackets comprises the force on a moving dipole [ g. e. vekstein ,  on the electromagnetic force on a moving dipole , " eur . * 18 * , 113117 ( 1997 ) ] . this additional term vanishes when torques are calculated with respect to the position of the dipole . we shall calculate all torques and angular momenta with respect to the origin . v.  namias ,  electrodynamics of moving dipoles : the case of the missing torque , " am .  j.  phys . * 57 * , 171 - 177 ( 1989 ) . of course , you could model * m * as separated monopoles ( this is what namias did in ref .  7 ) , but it turns out the torque is different for  gilbert " dipoles and  ampre " dipoles ( tiny current loops ) . see ref . this approach is based on an argument by m.  mansuripur , trouble with the lorentz law of force : incompatibility with special relativity and momentum conservation , " phys . lett . * 108 * , 193901 - 1 - 4 ( 2012 ) . a simpler version starts by representing stationary electric and magnetic by their charge and current densities : @xmath81 and lorentz transforming the 4-vector @xmath82 to get @xmath64 and @xmath83 for _ moving _ dipoles [ eqs . ( 22 ) , ( 25 ) , ( 40 ) , and ( 43 ) ] . see , for example , d.  j.  griffiths , _ introduction to electrodynamics _ ( pearson , boston , 2013 ) , eq . ( 12.118 ) . 10 , section 7.3.5 . when attached to a delta function , the prime denotes differentiation with respect to its argument . here we integrate by parts . w.  shockley and r.  p.  james ,  ` try simplest cases ' discovery of ` hidden momentum ' forces on ` magnetic currents ' , " phys . lett . * 18 * , 876 - 879 ( 1967 ) ; w.  h.  furry ,  examples of momentum distributions in the electromagnetic field and in matter , " am .  j.  phys . * 37 * , 621 - 636 ( 1969 ) ; l.  vaidman ,  torque and force on a magnetic dipole , " am .  j.  phys . * 58 * , 978 - 983 ( 1990 ) . hidden momentum is mechanical in nature , even though it arises ( typically ) in electromagnetic contexts . for details see ref .  4 , or ref .  10 ( example 12.13 ) . if the electric field is a function of time , there will be an additional term of the form @xmath84 $ ] , but it vanishes if the torque is calculated with respect to the dipole s location . there is no term involving @xmath85 , since the dipole moments are assumed to be constant . d. j. griffiths,dipoles at rest , " am . * 60 * , 979 - 987 ( 1992 ) . note that a moving electric dipole carries a component of circulating current ; see hnizdo , ref .  1 . a.  l.  kholmetskii , o.  v.  missevitch , and t.  yarman ,  torque on a moving electric / magnetic dipole , " prog . b * 45 * , 83 - 99 ( 2012 ) . these authors claim that the last two terms are new discoveries , but both these terms were obtained already by namias ( ref .  7 , eqs .  ( 1 ) and ( 3 ) ; the 3rd term is implicit in the former equation ) . incidentally , the same result [ eq . ( 54 ) ] holds for @xmath86 and @xmath87 constructed from magnetic monopoles and their currents , though ( curiously ) the roles of the last two terms are reversed : the fourth appears already in the analog to eq . ( 9 ) , while the third is due to hidden momentum .
we calculate the torque on an ideal ( point ) dipole moving with constant velocity through uniform electric and magnetic fields .