214 Magnetic Field of a Current in a Cylindrical Coil. 



a series of constant values of the function 4> (k, 6) for the 

 plate AB be drawn ; draw OP, and at each point, T, Q, 

 . . . B, of the breadth BT of the coil draw an ordinate, 

 TA, Q^, . , . B/ (fig. 4), equal to the product of r and the 



Fig. 4. 



value of <f>(k, 6) at the corresponding point, t, q, . . . P, of the 

 line OP : these ordinates will form by their extremities a 

 curve, hg . . ./, the area of which multiplied by four times 

 the current-density in the space occupied by the coil is the 

 potential at P due to the upper series of plates, OB, 

 OQ, . . . OT. 



Now if we take the point T 1 such that PP X is equal and 

 parallel to 00', the depth of the coil, the potential at P due 

 to the lower series of plates, OB', . . . OT', is equal to that at 

 Pj due to the upper series. Hence, if A and A : are the areas 

 of the curve hg . . ./ and the corresponding curve for the 

 point P 1? the total potential at P due to the complete coil is 



4C(A-A,) (27) 



The curve //#/ passes, of course, through the point ; and 

 when the line OP coincides with the axis, 00', of the coil, 

 the curve is an hyperbola ; for, in this case 



m 7T 1 — sin# 



andr = OPcot<9. 



We may, if we please, express r in terms of (&, 0), and 

 draw the curve hgf by a different rule. Thus, 



V(A',-# cos <9) 2 + 4# 2 sin 2 <9 

 and we can make the ordinate of the curve equal to 



E — # sin . A, + k'K cos 6 . &' a 

 OP 



V(A'-/:'cos0) 9 + 4£'- 2 sm 2 



