ELECTROMAGNETIC ACTION ON A CURRENT ELEMENT. 437 



But Qo is the flow of force cut by the arc BDA, Q x the flow cut 

 by the arc ACB in the displacement ; hence we may say that the 

 work of electromagnetic forces is equal to the excess of the flow cut 

 by one of the portions of the circuit over the flow cut by the other. 

 If the forces traverse the plane of the figure from front to back, the 

 values of the flow are positive for the direction of the current 

 indicated by the arrow. 



For the elements of the curve ACB, the motion is to the right 

 of an observer who is placed in the current, and who looks in the 

 direction of the force, and the flow of force cut enters into the 

 expression of the work with the - sign. For the curve BDA the 

 motion is towards the left, and the flow of force cut is taken with 

 the + sign. 



If we agree to give the sign + to the flow of force cut by the 



circuit when the motion is towards the left of the observer, and the 



- sign when it is to the right, we may say that the total work is 



equal to the algebraical sum of the flow of force cut by the current. 



458. ELECTROMAGNETIC ACTION ON A CURRENT ELEMENT. 

 We are thus led to consider the action exerted on a current as 

 resulting from the actions which would be exerted on each of the 

 elements into which we may suppose it to be decomposed ; it is the 

 same problem as for a magnetic shell (344). To apply the result 

 obtained to currents, we must replace the magnetic power of the 

 shell by the strength of the current, and the force exerted on each 

 element is expressed by 



(8) IF<&sina = L/A, 



dA being the area of the parallelogram constructed on Fds. Thus : 

 The action exerted on a current element placed in a magnetic field, 

 is equal to the product of the intensity of the current into the area of 

 the parallelogram, drawn on a right line which represents the intensity 

 of the field, and on the current element. This force is perpendicular to 

 the parallelogram, and is directed to the left of the observer placed in 

 the current, and who is looking in the direction of the force. 



If the field is due to a single pole of mass m placed at P 



(Fig. 103), at a distance r from the element, we have F = ; it 



follows that the reciprocal action of a current element and of a pole 

 is expressed by 



(9) d(f> = 



