404 Dr. Oliver Lodge on Opacity. 



" At the sheet we have 



Ei + E 3 = E 2 

 H 1 +H 3 =H 2 +47r^E 2 , 

 k being the conductivity of the sheet of thickness z. Therefore 



Eg __ H 2 Ej -f E 3 __ 1 



E x H! - E7~ s= l+2«/i*8w' ' 



(15) 



wherefore — ( t«H equals the curl of E. (The statement of this second 

 circuital law is entirely due to Mr. Heaviside ; it is now largely adopted 

 and greatly simplifies Maxwell's treatment, abolishing the * need for 

 vector potential.) 



The first of the above two fundamental equations, on the other hand, 

 depends on the fact that a current round a contour excites lines of mag- 

 netic force through the area bounded by it, and states the law that the 

 total magnetomotive force, or line-integral of the magnetic intensity 

 round the boundary, is equal to 4n- times the total current through it j 

 the total current being the " ampere-turns " of the practical Engineer. 



Expressing this law in terms of current density c, we write 



MMF = f Hds = 4ttC = ff 4*cd$ 



J cycle J-.' 



so always current-density represents the curl of the magnetic field due 

 to it, or curl H=47rc. 



Now in a conductor c = kE, but in an insulator c = D, the rate of change 

 of displacement or Maxwell's " displacement-current " ; and the dis- 

 placement itself is proportional to the intensity of the electric field, 



D = — E ; hence the value of current density in general is 



47T 



c=*E+*E, 



tt7T 



whence in general 



curl H = 47r&E+KE = (4ttA'+K/?)E, 



and in an insulator the conductivity k is nothing. 



The connexion between " curl " so defined and Vv is explained as 

 follows. The operator v applied to a vector R whose components are 

 X Y Z gives 



. d , .. d , 7 . d 

 ~dy 



which, worked out, yields two parts 



Q /dX,dY.dZ\ 



also called convergence, and 



T7 ./ dZ dY\ , ■ /dX dZ \ , , laY dX\ 



or say «| -\-jt] + Jc£, where £ rj £ are the components of a spin-like 

 vector <». Now a theorem of Sir George Stokes shows that the normal 

 component of co integrated over any area is equal to the tangential com- 

 ponent of R integrated all round its boundary ; hence Vv and curl are 

 the same thing. 



(•a +>s +*£)(*+.*+»>. 



