230, 231] ELECTROMAGNETISM. 451 



Any self or mutual inductance is the magnetic flux through a 

 circuit due to unit current in its own or another circuit respec- 

 tively. 



We shall first consider a solid of revolution bounded by a 

 surface generated by revolving any closed plane curve about an 

 axis in its plane not cutting it. Such a solid may be called a tore. 

 If the tore be uniformly wound with wire carrying a current, so that 

 every winding lies very nearly in a 

 plane passing through the axis of 

 revolution, Fig. 89, we may very 

 approximately consider the layer 

 of wire as a current sheet, the 

 difference of value of the current 

 function between any two points 

 being ml, where 7 is the current in the wire, and m is the number 

 of turns of wire between the points. By reason of symmetry the 

 lines of magnetic force must be circles whose planes are perpen- 

 dicular to the axis of revolution, and whose centers lie on the axis. 

 Consequently the strength of the field H is a function only of 

 the distance p from the axis, and the line integral of the field- 

 strength around any line of force is equal to the constant value 

 of H on that line times the circumference of the circle. If n be 

 the total number of turns of wire on the tore, any circle lying in 

 the substance of the tore is linked with the current n times in the 

 same direction, so that the value of the above line-integral is 



(i) 4w7rJ = 2>rrpH. 



This gives as the value of the force for internal points 



A circle lying outside the tore, however, is not linked at all with 

 the current, so that the line integral is zero, and therefore the 

 force H must be zero. Such a closed coil or toroidal current-sheet 

 accordingly emits no tubes of force, but all its tubes lie within 

 the doubly-connected space of the tore. The force accordingly 

 has a discontinuity at the sheet equal to 4?r times dW/dn, which 

 is the amount of current crossing unit of length of a circle co- 

 inciding with a line of force, or nl/Zwp. If the tore be filled 

 with a homogeneous medium of magnetic inductivity //, the 

 induction at any point will be pH = ^n^I/p. This whole reasoning 



292 



