50 Prof. R. Clausius on the 



§ 3. Application of the Equation to the Rotating Coil. 



In applying the previous equation to the electromotive 

 force which is induced in the rotating coil, in consequence of 

 its motion, by the currents in the other conductors and by 

 the magnets, we will select for consideration any one of the 

 positions in which the coil is divided, and which shall stand 

 for the above-mentioned conductor a. 



If we follow this portion of the conductor through a whole 

 turn, we have to deal with a series of changes of position, by 

 which that division of the conductor reverts to its original 

 position, so that the final value of AV is equal to the initial value. 

 It would result from this that if the direction in which the 

 electromotive force is to be regarded as positive, were always 

 the same in that portion of the conductor, the electromotive 

 force induced in it during an entire revolution would be partly 

 positive and partly negative, and in the mean would be null. 

 The state of things is, as a matter of fact, different. As stated 

 above, the entire rotating coil is divided by the two brushes 

 in each of its positions into two halves, which form branch 

 circuits between the brushes. The direction in which the 

 electromotive force is to be considered as positive does not 

 traverse the whole coil in the same direction ; but it is in dif- 

 ferent directions in the two halves which we get if we suppose 

 ourselves to start from one brush, and to proceed through 

 both halves to the second brush. It follows from this that, for 

 each separate portion of the conductor, the direction which we 

 regard as positive changes at each half turn ; at the moment, 

 that is to say, at which the portion in passing the brush goes 

 from one half of the coil into the other. When the brushes 

 are properly adjusted, this happens at those places in which 

 the electromotive force changes, so that the sign of this elec- 

 tromotive force is always the same. 



As the inducing action is equal during the two halves of the 

 circuit, then, in order to determine the average electromotive 

 force, we need only consider half a turn — that is, the motion 

 of the division in question from one brush to the other. At 

 the beginning of this half revolution, at the time £', let W 

 have the value W / ; and at the end (that is, at the time t' + Jt, 

 in which t is the time of revolution) let W have the value W x/ ; 

 then, for the determination of the average electromotive force, 

 we get from (2) the following equation: — 



From this expression, which applies to only one portion of the 



