EXAMPLES OF SOLENOIDAL FIELDS. 



37 



122. Solenoidal Field in Space with Neutral Point. It will be useful to show 

 the simplest case of a solenoidal field in space having a neutral point. Correspond- 

 ing to the two-dimensional field of section 120, we shall then consider a field with 

 the rectangular components 



(a) A x = ax A y = by A t = cz 



The integral of the normal component of the vector is easily formed for a surface of 

 parallelepipedic form having sides parallel to the coordinate planes. The solenoidal 

 condition is seen to be fulfilled if 



a+b+c = o 

 In order to simplify we shall further set b = a, which gives c = 2a. We then have 

 the field 



(b) A x = ax A y = ay A t =2az 



Composing the components A x and A y we get a resultant contained in a plane 

 passing through the axis of z. Calling r the distance of any point in this plane from 

 the axis of z and R the resultant of A x and A y , we get instead of the two first equa- 

 tions R = ar. The field will then be completely given by the two components 



(c) R = ar A z = 2az 



y 



% 



Fig. 41. Addition of translation-field and defor- 

 mation-field. 



Fig. 42. Lines of flow and curves of equal inten- 

 sity, 1, 2, 3, of a symmetrical deformation-field 

 in space. 



The field is thus symmetrical around the axis of z, and the vector is contained 



in the meridian planes passing through this line. Substituting the values of R and z 



in the differential equation 



dr _ dz 



R~A, 

 and integrating, we get the equation of the vector-lines 

 (d) r"z = const. 



