154 Dr. W. F. G. Swann on the Magnetic Field produced 



►Section 2. Case ichere K is variable. 



Suppose now that a portion of the space within S (the 

 shaded portion) is filled with material of specific inductive 

 capacity K 2 , the rest of the space being filled with material 

 of S.I.C. equal to K : . By an argument exactly similar to 

 that given above, we arrive at the expression 



Total flux through strip = —,dy\ K(X — vy)d<v. 



c ~ ' J A 



Thus, since K is not the same at all points along the strip, 

 we have 



Total flux through strip 



= K x -j dy (X — vy) do: + K 2 -,- dy I (X — vy) dx 

 c J a c Jc 



v f B 

 l-K^dy] (X — vy)dx 



C J T> 



=Ki I dy f E (X-vy)dx + (K s -K,) £ dyC (X-vy)d.v. 



%J A */ C 



Since I JsLckc is zero, and since, as is very easily verified, 



the quantity vy which itself arises from the motion of the 



system is only of the second order in — compared with X, 



the integrals of the form ^vydx are negligible, and we obtain 



v C D 

 Flux through strip = (K 2 — K x ) -~ dy \ X.d;v 



c J c 



/xr ir \ v 7 fPotential difference"! 

 = (K ? -K 1 ) ? rfy^ between C and D J' 



If we now draw an infinite series of lines perpendicular to 

 the axis of ?/, each line connecting two parts of S, and if V 

 represents the potential difference between the points where 

 these lines enter and leave the intermediate dielectric, we 

 have 



Total Magnetic flux through S = (K 2 -K 1 )^ fwy, 



the integral being taken right across the circuit. 



