POTENTIAL OF AN UNLIMITED RECTILINEAR CURRENT. 423 



equal to that of an unlimited rectilinear current. Hence we arrive 

 at the following law of Biot and Savart : 



The action of an unlimited rectilinear current on a pole is perpen- 

 dicular to the plane passing through the current and the pole, is directed 

 towards the left of the current, and is inversely as the distance of the 

 current from the pole. 



A simpler experiment, at any rate in theory, leads to the same 

 result. Suppose that a portion of the circuit is vertical, and a 

 magnet placed in any given way, upon an apparatus movable 

 about an axis which coincides with that of the current. It will be 

 seen that the movable system is at rest for all positions of the 

 magnet, whatever be the direction and strength of the current. 

 It follows hence that the sum of the moments, in reference to 

 the axis, of the actions exerted on the different masses of the 

 magnet, is null. 



If m is the magnetic mass at a distance a from the axis, we 

 shall have 



If we suppose the magnet reduced to two masses m equal and 

 of contrary signs, at the distances a and a' from the current, the 

 equation reduces to 



m(<f>a (f> f a') = Q, or <$>a = const., 



that is to say, to Biot and Savart's law. 



The experiment carries with it its own verification, for if we cease 

 to make the axis of rotation coincide with the axis of the current, 

 the system is displaced, and tends to turn in one or the other 

 direction to obtain its position of equilibrium. 



445. POTENTIAL OF AN UNLIMITED RECTILINEAR CURRENT. 

 We shall proceed to show that the magnetic field of a current is 

 defined by a potential that is to say, by a function whose partial 

 differentials, in reference to the axis of the co-ordinates, represent 

 the respective components of the force taken with contrary signs. 



In the case of a rectilinear current, the equipotential surfaces are 

 planes passing through the current. Let us take the current for the 

 2-axis, and a plane perpendicular to the current passing through the 

 point P (Fig. 96) for the plane xy. If we suppose that the current 

 goes behind the figure, the force <f> at a point P of the plane, from 

 Ampere's rule, is perpendicular to PO, and would tend to turn this 



