Em induction, sliding rod with resistance? 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.  
 A: Your formulas are (almost) fine: They relate the induced voltage (or electromotive force aka emf, if you must) to the current and total resistance of your circuit. Where you went wrong is that this will help you determine the induced voltage. You need to calculate it first to then, optionally, use your formula, corrected to use the total resistance, i.e. $U = (R_1+R_2) I$, to determine the current it creates.
What determines your induced voltage is the magnetic field, the velocity of your sliding rod, and its length. If the field is constant along its length and perpenticular to both the motion and the orientation of the rod, the formula will be $U = l \  v \  B$ where $l$ is the length of the rod between the contact points. If the conditions are not met, you will need to integrate the vector cross product $\vec{v} \times \vec{B}$ along the (relevant) length of the rod, i.e. again only between the contact points.
A: 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.
