231, 232] ELECTROMAGNETISM. 455 



tion from uniformity of the field at any part of the solenoid 

 calculated. In a long solenoid the field is very nearly uniform 

 for a considerable distance from the middle of its length. By 

 differentiating the expressions for V l and F 2 with respect to r, 

 the distance from the center of either, multiplying by the element 

 of the area of a sphere of radius r, and integrating, we may find 

 the flux due to either end through a circle perpendicular to and 

 with center in the axis, and hence the correction due to the end 

 to be made in the mutual inductance of the coil with another 

 circuit of a single turn, and thence by another integration with 

 respect to any concentric coil. 



232. Pair of Rectangular Circuits. In the case of two 

 linear circuits, we may use Neumann's formula for the mutual 

 inductance 



,, 



/7 



in those cases which are simple enough for us to effect the 

 integration. If the two circuits are 

 equal rectangles ABGD and A'B'C'D', 

 Fig. 90, of length ^ and breadth 1 2 

 with corresponding sides parallel, and 

 the lines joining corresponding cor- 

 ners perpendicular to their planes and 

 of length a, then for pairs of sides 

 which are perpendicular the integral 

 vanishes, while for pairs of parallel 

 sides the cosine is either plus or minus unity, according as we 

 consider corresponding or opposite sides in the two rectangles. 

 For the sides AB, A'B' we have 



The integration of the logarithms in the second integrand may 

 be performed by taking as a new variable the quantity whose 

 logarithm is to be integrated, and then integrating by parts, the 

 result being 



= 2 Jo - 



1, . log *' + 



