RESISTANCE AND ELECTROMOTIVE FORCE. 47 



F' the current in the lower branch, R' the resistance of the 

 upper branch, and R" the resistance of the lower branch. The 

 product R 1 1' is the electromotive force between the points A 

 and B, the product R"I" is also equal to the electromotive force 

 between the points A and B, and therefore we have 



R'I'=R"I" (12)* 



The current in the main part of the circuit is equal to the sum 

 of the currents in the various branches into which the circuit 

 divides. Therefore we have the equation 



/=/'+/" (13)* 



It is an easy matter to determine the values of I' and I" 

 [with the help of equations (12) and (13)] in terms of the total 

 current / and the resistances R' and R" of the respective 

 branches. It is important to note that a a definite fractional part 

 of the total current flows through each branch, and equation (12) 

 shows that the currents I 1 and I" are inversely proportional to the ' 

 resistances R' and R" respectively. Thus, if R' is nine times 

 as large as R", then /" is nine times as large as /', so that 

 /" must be equal to nine tenths of /, and /' must be equal to 

 one tenth of /. 



25. Combined resistance of a number of branches of a circuit. 

 (a) The combined resistance of a number of lamps or other units 

 connected in series is equal to the sum of the resistances of the 

 individual lamps. (&) The combined resistance of a number of 



* Equations (12) and (13) express two principles which were first enunciated by 

 Kirchhoff and which are usually called KirchhoflPs laws, as follows : 



(a) Equation (12) may be written R'I f R"I ff ^=o, which means that the 

 sum of the RI drops taken in a chosen direction around the mesh formed by the two 

 branches of the circuit is equal to zero. This relation is true of a mesh of any net- 

 work of conductors. If one side of the mesh contains a voltaic cell of which the 

 electromotive force is , then the sum of the RI drops around the mesh is equal 

 to E. 



(3) Equation (13) may be written / I f I f/ = o, which means that the sum 

 of the currents flowing towards one of the branch points A or B is equal to zero. 

 This relation may be generalized as follows : The sum of the currents flowing towards 

 a branch point in any network of conductors is equal to zero. 



