ELECTRO-MAGNETISM. 



61 



action of a current, P N, extended to 

 an infinite length, upon an elementary 

 portion of a parallel electric current 

 placed at any given distance from it, is 

 in the simple inverse ratio of the shortest 

 distance between them that is, of a 

 line drawn from the one to the other, 

 and perpendicular to both. Thus, in 

 the example just given, where the dis- 

 tance C A is three times the distance 

 B A, the action of the indefinite cur- 

 rent P N is three times greater upon B 

 than upon C. So that if this total 

 action be expressed by F, 

 F _ ab 



~~d" 



(181.) The action, whether attractive 

 or repulsive, of two elementary portions 

 of current, must be conceived as exerted 

 in the direction of the line which joins 

 them, and which, for the sake of distinct- 

 ness, we shall call the medial line. So 

 that in the case of the parallel currents 

 A and B, fig. 98, moving in the same 

 direction, an attraction denoted by the 

 short arrows a and a, takes place in the 



Fig. 98.: i 



Y 



A 



B 



V 

 D 



direction of the medial line AB; and 

 in the case of the currents C and D, 

 likewise parallel, but moving in opposite 

 directions, a repulsion takes place, as 

 shown by the short arrows, in the di- 

 rection of the same line. It should be 

 observed that, in both cases, the action 

 on each current is perpendicular to the 

 direction of that current. 



2. Action of inclined Rectilineal 

 Currents. 



(182.) Let us next suppose that the 

 two currents, still remaining in the same 

 plan?, the direction of one of them, A, 

 fig. 9'J, is changed from parallelism to 

 the position C c, the action will still be 

 perpendicular to that position that is, 

 the current A will be urged to move in 

 the line A a, at right angles to C c; but 

 the force which thus impels it will be 

 diminished in the ratio of radius to the 

 sine of the angle BAG, which the di- 



. 99. 



B 



rection of the current makes xvilh the 

 medial line A B. This will readily ap- 

 pear by resolving the force A b into 

 the two forces, A a, A C, of which the 

 latter, acting counter to the direction of 

 the current, is destroyed, while the only 

 effective force is A a, which is to A b 

 as the sine of B A C to radius. 



(183.) A similar diminution of the 

 force by which the current A reacts 

 upon B, takes place in consequence of 

 its obliquity ; for the portion C n, fig. 

 100, which, when it was parallel to B, 

 acted with its full power, has only the 



'Fig. 100. 



force of the portion m n, when acting in 

 the oblique position I) n, the diminution 

 being proportional to the cosine of the 

 angle C n D, or, what is the same, to 

 the sine of the angle D n B. The mu- 

 tual action of the current, therefore, 

 situated in the same plane, but in ob- 

 lique positions, may, in as far as this 

 obliquity is concerned, be expressed by 

 the following equation, in which and (> 

 denote the angles made by the directions 

 of the currents respectively with the me- 

 dial line. 



/ = sin. . sin. & 



(184.) Let us now inquire into tho 

 modification the formula must receive 

 when the two currents are in different 

 planes, We have seen that in the last; 



