446 Mr. 0. Heaviside on Electromagnetic Waves, and the 



When e is simply periodic, J, a = makes the external E 

 and H vanish independent of the nature of y. The complete 

 solution is then 



H (in) =4^K^re, E (in) = -4^-K^^^. . (338) 



<J Oa " Oa 



The conduction-current in the tube is Ke per unit area of 

 surface. 



To make the conduction-current vanish by balancing the 

 impressed force against the electric force of the field that it 

 sets up, put an infinitely conducting screen at r=x outside 

 the tube and choose the frequency to make Jj,j,=0, since we 

 now have ?/ = J^/Gr,*. We shall then have the same solution 

 inside and outside, viz. 



H=-i-^i*, E=-^ej . . . (339) 



so that at the tube itself E = — e. This case may be inter- 

 preted as in § 55, the tube being at a nodal surface of E. 



A special case of (338) is when n = 0, or e is steady. Then 

 there is merely the longitudinal H inside the tube, given by 



H = 47rKe. 



61. Cylinder of longitudinal curl of e in a Dielectric, — 

 In a nonconductive dielectric let the impressed electric force 

 be such that its curl is confined to a cylinder of radius a, in 

 which it is uniformly distributed, and is longitudinal. Let/ 

 be the tensor of curl e, and let it be a function of t only. 

 Since E is circular and H longitudinal, we have (314) as 

 operator, in which k is to be zero. This is outside the cylinder. 

 Inside, on the other hand, on account of the existence of curl e, 

 the equation corresponding to (314) is 



H SIM? C P J 0r 



At the boundary r = a both E and H are continuous ; so, by 

 taking r = a in (340) and in the corresponding (314) with 

 k=0, and eliminating E a or H a between them, we obtain the 

 equation of the other. We obtain 



(out) i B =*™y~ ,J i«( J i'-yG., r y, i t ^ ^ 341) 



l-K=i'n-ay- l J la (J 0r —yG iir )(iiv)- 1 if, 5 



in which y, as usual, is to be fixed by an external boundary 

 condition, or, if the medium be boundless, y = i 



We see at once that J, a = 0, with/ simply periodic, makes 

 the external fluxes vanish. We should not now say that it 



