Consider the digram below: 
This consists of a rod sliding with velocity $v$. I want to find the voltage developed between $A$ & $B$. To do this I drew this 'equivalence circuit': 
From this (assuming that we know $\epsilon$) we can work out the voltage between $A$ and $B$ as follows: $$V_{AB}=R_{2}I=\epsilon \frac{R_{2}}{R_{1}+R_{2}}$$ The question I am working through says, however, that the answer is: $$V_{AB}=R_{1}I=\epsilon \frac{R_{1}}{R_{1}+R_{2}}$$ I cannot see where my logic is wrong. The only emf is induced in the (blue) rod so we must put the equivalent voltage source between $A$ and $B$. Since this is the only place where the electrons feel any force. (I assume I have got the answer wrong) So please can you explain why we do not include the induced emf in between the terminals $A$ and $B$.
Here is a little more of my reasoning: if $R_2=0$ then a current will flow around the red part of the circuit but no voltage will be dropped. My answer will indicate this, giving $V_{AB}=0$ by the answer stated in the question will not, it will give $V_{AB}=\epsilon$ which to me seems wrong.
Answer
Are you sure that the numbering of $R_1$ and $R_2$ in your diagram is the same as in the answer key? It seems to me that it is more conventional to label the first resistor (the one on the left) $R_1$.
The equations you wrote appear to me to be correct: the current is indeed given by the e.m.f. divided by the series resistance, and the voltage developed across each element must be equal to the current times the resistance.
Having said that - these problems are often tricky, and I have run into problems with them myself. There is a famous Walter Levin lecture on a similar topic - see https://www.youtube.com/watch?feature=player_detailpage&v=eqjl-qRy71w#t=234s. Mind blowing.
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