REPRODUCTION OF MAGNETICALLY RECORDED SIGNALS 1167 



tapes of the sort to which equation (23) applies. One of these tapes is at 

 z = 8/2 and the other at z = —8/2. 



The problem then is to rewrite equation (23) for a very thin tape and 

 in terms of surface density of charge. As 5 approaches zero, equation (23) 

 reduces to 



% = - ^ 4^/.(2,r6/X).-^''^^^ cos (co/) (28) 



at M + 1 



From equation (7), the volume density of charge in this tape is 



p = - {lirlm/X) cos (Ittx/X) 



But as 5 approaches zero, the longitudinally magnetized tape to which 

 equation (28) applies becomes equivalent to a surface distribution of mag- 

 netic charge of surface density equal to 8p. This amounts, for the thin 

 longitudinally magnetized tape, to a surface charge density of 



(71 = -(27r5A) cos (Ittx/X) (29) 



But the charge density on the top side of the perpendicularly magnetized 

 tape is given by equation (26). Comparing these two values shows that the 

 surface charge density in the thin longitudinally magnetized tape is just 

 27r5/X times as great as the surface charge density on top of the perpendicu- 

 larly magnetized medium. This means that d4>x/dt due to the top side of the 

 perpendicularly magnetized tape can be obtained by dividing the right hand 

 side of equation (28) by 2t8/\. This gives 



dt M + 1 



^irvln.e-'''"' cos (a,/) (30) 



due to the top side of the tape. 



The contribution from the bottom side is obtained from equation (30) 

 by replacing d by d -\- 8 (since the bottom side is spaced d -\- 8 from the 

 reproducing head) and changing the sign. Adding these two contributions 

 gives for the total 



'^= --JL- i^Ui - e-""\-'"" cos (0,0 (31) 



dt M + 1 



This is the same as equation (23) and so the desired result has been 

 established. 



Note from equations (6) and (24) that in order to get the same result for 

 the perpendicular and longitudinal cases it was necessary to assume a 90- 

 degree phase difference between 7^ and 7, . The usual type of recording 

 head lays down a pattern of magnetization which is neither purely per- 



