242 THE MAGNETIC CIRCUIT [ART. 70 



(Fig. 45). The energy contained in this tube is dW =\Mpdd> p \ but 

 M p = J Hdl, and d<D=BdA. Since d0 is the same through all cross- 

 sections of the tul)o, </</>* can be introduced under the integral sign, and 

 we have dW = \ C HdlBdA =J C HBdV, the integration being extended 



over the volume of the tube. The total energy of the circuit is found 

 liv intending the integration over the volume of all the tubes of the 

 field. The other equations are proved in a similar manner. 



70. The Longitudinal Tension and the Lateral Compression 

 in a Magnetic Field. The existence of mechanical forces in a 

 magnetic field is well known to the student. He needs only 

 to be reminded of the supporting force of an electromagnet, of 

 the attraction and repulsion between parallel conductors carrying 

 electric currents, of the torque of an electric motor, etc. These 

 mechanical forces must necessarily exist, if the magnetic field 

 is the seat of stored energy. This is because, if we deform the 

 circuit, we must in general change the stored energy and hence 

 do mechanical work. The lines of force tend to shorten them- 

 selves and to spread laterally, so as to make the permeance of 

 the field a maximum, with the complete linkages. Where there 

 are partial linkages, it is the total stored energy that tends toward 

 a maximum (Art. 57). This fact is entirely consistent with 

 the hypothesis of whirling tubes of "force, because the centrifugal 

 force of rotation produces exactly the same effect, that is, a lateral 

 spreading and a tension along the axis of rotation. A good 

 analogy is afforded by a short piece of rubber tube filled with 

 water and rotated about its longitudinal axis. 



(a) The Longitudinal Tension. Consider again the simple 

 magnetic circuit (Fig. 1), and let it be allowed to shrink, due 

 to the longitudinal tension of the lines of force, so as to reduce 

 its average length by Al, without changing the cross-section A. 

 Let at the same time the current be slightly decreased so as to 

 keep the same total flux as before. Let F{ be the mechanical 

 tension along the lines of force, per square cm. of cross-section 

 A] then the mechanical work done against the external forces 

 which hold the winding stretched is (F t f .A] Al. The density 

 of energy W f remains the same because B is the same, but the 

 total stored energy is decreased by W(AAT), because the volume 

 of the field is decreased by AM. Since the change was made 



