Internal Resistance :- 

Let us consider a very simple circuit, i.e. one with a resistance R connected to a source of emf, with emf E . Now most sources of emf have to move internal resistance , e.g. in a cell, the ions have to move against the background of the other ions and of neutral atoms thus giving rise to electrical internal resistance. 

The total work done by the source in taking a unit of charge from the negative to the positive terminal is E. Because of internal resistance, part of this work (Denoted by W') is used up in the source. The remaining namely (E - W') is used to maintain the potential difference  Vexp between the terminals .

Vexp  = E= W'

This is the voltage that would be measured across the terminals PQ of the emf source if current flows through the circuit. Now if Ohm's law is valid for current flow inside the cell B, we can characterize it by an internal resistance r. In a circuit, it can be represented by a resistance r in series with the cell; P and Q are the external terminals of the cell, If a current i flows through the circuit ,

Vexp  = E = W' = E - i r .............................(1)

but Vexp -iR by Ohm's Law . So , we have

                                           E' = Vexp  + i r = i ( R + r ).......................(2)

Resistance in Series | Kirchhoff's Rule | Resistance in Parallel