Curl in a Vector Field, 531 



flux being solenoidal. We have ~Ddn = dyjr, and since d-ty is 

 independent of s, 



^?=o . (St) 



Eq. (5) becomes 



^t+IM-o (5c) 



The physical meaning of function i|r is as follows : the 

 numerical value of ty is equal to the flux comprised between 

 the line of force under consideration and some other 

 arbitrary line of force selected as a reference ; the flux is 

 understood to be per unit of thickness in the direction 

 perpendicular to the plane of the paper. 



(b) Lines of force are parallel near 0. — In this case 

 H = cc , and 



divD =g (6) 



(c) Lamellar field.— Let <f> be the potential function. 

 Then 



D=g, ...:... (7) 



and eq. (3) becomes 



this is equivalent to Poisson's equation in rectangular co- 

 ordinates. 



(d) Solenoidal lamellar field : 



&♦££- ■ * 



This equation is equivalent to Laplace's equation in 

 rectangular coordinates. 



o 



1 b. Curl in two dimensions. 



By definition, the curl of D at point (fig. 1) is the limit 

 of the ratio of the line integral of D taken (say) along 

 OBB'0'0 to the area enclosed by that path. But the line 

 integrals along BB' and 00' vanish because the field there 

 is perpendicular to the path, so that 



anas on ds on ' 



2N2 



