ELECTRO-MAGNETISM. 



(42.) When the conducting wire W, 

 23, is situated in any part of the line 



v s 



\s 





N 



WC, at right angles to the axis of the 

 needle, the tangential force acting on the 

 pole S, in the direction represented by 

 the arrow, at right angles to WS, may 

 be supposed to be transferred to the 

 centre, C, of the needle, and to be re- 

 presented by the line C s. The force 

 acting upon N being in like manner re- 

 presented by C n ; the resultant of these 

 two forces will be a force represented 

 by the diagonal C a of the parallelogram, 

 having Cs and Cn for its two sides, 

 and tliese sides being equal, and equally 

 inclined to the line WC, this diagonal 

 will coincide with that line : hence the 

 force will be such as to move the centre 

 of the needle directly towards W, that is, 

 the needle will appear to be attracted by 

 it. If either the current had followed 

 an opposite course, or the poles of the 

 needle had been reversed, the forces 

 would have acted in the opposite direc- 

 tion, and would have been represented 

 by the lines G s', C n', forming a paral- 

 lelogram, of which the diagonal is C r, 

 indicating a motion of the centre of the 

 needle from the wire, and resembling 

 repulsion. This effect also takes place 

 under the original circumstances of the 

 experiment when the wire is on the other 

 side of the needle, that is, in any part of 

 the line CW; so that the needle will 

 always appear to be attracted by the 

 wire on one side and repelled on the 

 other. 



(43.) The intensity of the force which 

 thus impels the needle, either towards 

 or from the wire, diminishes as their dis- 

 tance is increased. Two causes con- 

 spire to produce this diminution ; the 

 one is that the component forces them- 

 selves are inversely proportional to the 

 distances of the points on which they 



act from the wire ; and the other is that 

 the angles they form with one another 

 become more obtuse as that distance 

 increases. Mathematically speaking, 

 the tangential force applied to each pole, 

 when referred to the direction of the 

 line joining the wire and the centre of 

 the needle, is directly as the cosine of 

 the angle formed between the axis of 

 the needle and the line connecting the 

 pole and the wire; and it is also in- 

 versely as this line ; so that calling the 

 force referred to that direction a, the dis- 

 tance from the wire to the centre of the 

 needle d, the distances of the wire from 

 the respective poles S and N, * and n y 

 and the length of the needle m ; and and 

 /3 being the angles between the axis of 

 the needle and the lines connecting the 

 respective poles with the wire, we have 

 the following equation : 



_cos. u. cos. /3 

 a ~ s + n 



But as we have taken the case of W 

 being placed on the line drawn from the 

 centre of the needle at right angles to 

 its axis, the two angles above mentioned 

 are equal, and every part of the line is 

 equidistant from S and N, that is, 

 a = p, and s = n; 



hence the equation becomes a= '. 



Now 



CS m 



cos . *=_=*_; 



which value of cos. being substituted 

 in the former equation, the formula be- 

 comes 



that is, the force of apparent attraction 

 is directly as the length of the needle, 

 and inversely as the square of the dis- 

 tance of the wire from each pole. 



(44.) In order to estimate the attraction 

 with relation to the distance of the wire 

 from the centre of the needle, or d, we 

 must substitute for s 2 its equal d z +^ m* ; 

 so that the formula becomes 



But when the distance of the wire is 

 very great compared with the length of 

 the needle, the quantity m* may safely 



be neglected ; and 



- may be taken 

 a 2 



without any sensible error as the ex- 

 pression of the attractive force. 



(45.) This may be experimentally illus- 

 trated by suspending a magnetic needle, 

 SN,/g-. 24, from its" centre by a thread, 



