494 Electromagnetic Theory of Moving Charges. 



8. The mutual energy of a moving charge and external 

 magnetic field has been given by Mr. Heaviside for the case 

 of motion which is very slow compared with the 4 velocity of 

 radiation. It is eu A . cos (uA), S where A is the circuital 

 vector potential of the external field. Mr. Larmor, in the 

 second part of his " Dynamical Theory " (Phil. Trans. 1895, 

 p. 717), concludes that the same expression holds good for 

 motion at any speed. He seems, however, to overlook the 

 fact that in the general case the displacement-currents in the 

 medium — being no longer derivable from a potential function 

 — will make their appearance in the result as well as the 

 convection-current eu. 



If (F G H) is the vector potential, the part of the energy 

 corresponding to the displacement currents will be 



which in the case we have been considering becomes 



J\ dz dz dz J 



dxdz dy dz dz 1 ) 



But by a well-known transformation, when we take the 

 integral through all space, we have 



K 



dx dz dy dz dz* ) 



' J dz \dx dy dz ) 



= since (F GrH) is circuital. 

 .*. The expression for this part of the energy reduces to 



-«(l-^)jHgf^ = -^JHg^. 



Therefore if the velocity u ceases to be negligible in com- 

 parison with V, we have a correction of the second order in 



u 

 the ratio ^ in addition to the expression involving the 



convection- current simply. It also appears from the above 

 that the force on the moving charge cannot, unless this term 

 be neglected, be expressed in terms of the magnetic intensity 

 at the charge, but will depend on the entire field* 



