Magnetic and Electric Energies as a Pressure. _ 6G3- 



magnetic and electric energies are contrasted forms, the 

 contrast is not exactly that between potential and kinetic 

 energy in dynamics, since the electric energy is not dependent 

 on position alone when there is motion — i. e. the analogy with 

 dynamics is imperfect. 



It is remarkable that the theorem has escaped attention in 

 its application to the simple problem of reflexion of a plane 

 wave at a fixed plane surface. Accordingly I take this case 

 first, then deal with a moving plane surface, and finally take 

 a general electromagnetic field near the surface of any perfect 

 conductor in motion. 



§ 1. If z constant is the surface on which a plane wave is 

 incident, the conditions at the surface are X=Q r Y = : and 



,, . -, n 1 "be dX BY -_ 



these involve c = U, since fr ^— = ^r ^— ;■ Hence the 



\ ot og ov 



normal pressure 



Z,= l(a 2 + /r-r-)+l(X 2 + Y 2 -Z 2 )=i(« 2 + /--Z 2 j,] 



and (1) 



L=i(a 2 + & 2 + c 2 )-i(X 2 + Y 2 + Z 2 )=i(a 2 -f& 2 -Z 2 ),! 



have the same value at thesAirface: moreover X s = -ZX-ac 

 vanishes, as also Y«, so that the pressure is purely normal. 



Consider the composition of these functions with reference 

 to the separate waves. If ('X, Y 1 Z x ) refer to the incident 

 wave, then 



l l X 1 + m l Y 1 + n 1 Z l = 0, a x — m^— %Y l5 .... (2) 



For the reflected wave take the argument to be l 2 x + m 2 y 

 —n 2 z—Yt, so that (l 2 , m 2 , n 2 ) = (/ 1? m u m). At the surface 

 then we have the relations 



(X 2 ,Y SJ Z 2 ) = (-X 1 , -Y^Zj, (a ft ?„c,) = (a 1 ,ft 1 ,-c 1 3. (3) 



The value of L at the surface is that of product terms, viz. r 



L = a x a 2 + b x h. 2 + c L c 2 — (Xx-Xa 4- Y : Y 2 + ZxZ 2 J 



= ai' + b i *-<: 1 *+X 1 ' + Y 1 '-Z 1 * (4a> 



On the other hand product terms in Z z vanish at the 

 surface, since 



XiXo -f Yj Y 2 — ZjZs + a { a 2 + hj> 2 — c Y c 2 



= -(X 1 2 + Y l 2 + Z 1 2 )+a 1 2 +> 1 2 + c 1 2 . 

 The pressure is shown in the Z~ formula by the sum of 



