Self-induction of Wires, 11 



field, where the intensity of force is zero. Let a steady cur- 

 rent exist in the wire, longitudinal of course, and let the 

 return conductor be a close-fitting infinitely conducting sheath. 

 This stops the magnetic field at the boundary of the wire. 

 The sudden discontinuity of the boundary magnetic force is 

 then the measure and representative of the return current. 



The magnetic energy per unit length is -JLC 2 , where C is 

 the current in the wire and L the inductance per unit length. 

 As regards the diminution of the L of a circuit in general, 

 by spreading out the current, as in a strip, instead of concen- 

 trating it in a wire, that, is a matter of elementary reasoning 

 founded on the general structure of L. If we draw apart 

 currents, keeping the currents constant, thus doing work 

 against their mutual attraction, we diminish their energy at 

 the same time by the amount of work done against the 

 attraction. Thus the quantity JLC 2 of a circuit is the amount 

 of work that must be done to take a current to pieces, so to 

 speak ; that is, supposing it divided into infinitely fine fila- 

 mentary closed currents, to separate them against their attrac- 

 tions to an infinite distance from one another. We do not 

 need, therefore, any examination of special formulas to see 

 that the inductance of a flat strip is far less than that of a 

 round wire of the same sectional area ; their difference being 

 proportional to the difference of the amounts of the magnetic 

 energy per unit current in the two cases. The inductance of a 

 circuit can, similarly, be indefinitely increased by fining the 

 wire; that of a mere line being infinitely great. But we can 

 no more have a finite current in an infinitely thin wire than 

 we can have a finite charge of electricity at a point, in which 

 case the electrostatic energy would be also infinitely great, for 

 a similar reason ; although by a useful and almost necessary 

 convention we may regard fine-wire circuits as linear, whilst 

 their inductances are finite. 



Now, as regards our enclosed rod with no external magnetic 

 field, we can in several cases estimate L exactly, as the work 

 is already done, in a different field of Physics. The nature of 

 the problem is most simply stated in terms of vectors. Thus, 

 let h be the vector magnetic force when the boundary of the 

 section perpendicular to the length is circular, and H what it 

 becomes with another form of boundary; then 



H=h + F, and F=— V^. • . ■ (la) 



That is, the field of magnetic force differs from the simple 

 circular type by a polar force F whose potential is fl. This 

 must be so because the curl of H and of h are identical, re- 

 quiring the curl of F to be zero. To find F we have the datum 



