156 The Energy in the Electromagnetic Field. 



and _ l'[ p ] 



*=yfd»- 



(14) is the usual expression for the electric intensity in 

 terms of the vector and electric potentials A, yjr, whilst the 

 three equations (15) give 



(P, Q, R)= ? [VH], .... (16) 



where H=curlA, .... (17) 



(16) being the well-known value for the force on a charge 

 due to its motion in a magnetic field given in terms of the 

 vector potential by relation (17). 



Differentiation of equations (14) combined with the known 

 values of F, G, H, and the relation (17), gives the remain- 

 ing equations 



ldX + 4^ = d7_^ &c 

 c 'dt c ~dy d^' 



The above is not, of course, an independent derivation of 

 Maxwell's equations, since it has been necessary to introduce 

 the assumption of propagation with finite velocity c. It is, 

 however, significant that, when this assumption is made, 

 Maxwell's equations are derived by applying Lagrange's 

 equations to a Lagrangian function obtained by assuming 



T(H 2 + G 2 ) 



the kinetic energy in the field to be I - — 5 dv, and the 



J on 

 potential energy to be equal to the electrostatic energy of 

 the system. 



In the case of slow uniform motion, neglecting the finite 

 velocity of propagation, we had a, j3, y given by 



and 



{ a U> (fo _! 1 \pJ dv ( &c, 



Gby -2RY 



_ \ % L(> dv + | 1 (> dv +* i (e?L dv \, 



I ^wcj r dye J r dzcj r J 



