78 PROPERTIES OF CIRCULAR CURRENTS. 



It follows from this that for a given channel, and therefore for a 

 given weight of metal, if the ratio is constant, the intensity of the 

 field at each point is inversely as the square of the diameter of 

 the wire ; or again (726), is proportional to the square root of the 

 resistance. 



This consequence suggests a curious remark : if R is the resist- 

 ance of the coil, and I the strength of the current, the thermal 

 energy W developed in each unit of time is equal to RI 2 . As the 

 resistance R is proportional to J, or to the square G 2 of the magnetic 

 action for unit current, we see that the thermal energy W is pro- 

 portional to the product G 2 ! 2 that is to say, to the square of the 

 magnetic action of the coil for the current I. It follows that if, 

 varying the diameter of the wire, we modify the intensity so that the 

 magnetic action remains constant, the thermal action will also be 

 constant* 



729. ACTION ON THE Axis. The potential of a current is the 

 same as that of a uniform magnetic shell having the same contour, 

 and the strength of which is equal to the strength of the current. 

 The potential of a circular current of radius r, and strength equal to 

 unity, at a point P on the axis, at a distance x from the centre, is 

 expressed by 



/ x\ 



= 27r( I- - ) , 

 \ / 



V 



putting 



The magnetic force is 



t)V u 2 - X* r*- 



!<=- =27T = 27T . 

 OX U 6 U 6 



Dividing this expression by the length 2irr of the circumference, 

 we get the action of the current for unit length, 



/ \ F r 



which we may call the specific action of the current. 



The action of a coil of length zb and radius r, covered by a layer 

 of wires containing n : turns for unit length, is equal for unit current 



* MARCEL DEPREZ. Comptes rendus de I'Acad., Vol. xciv., p. 431. 1882. 



