Kirchoff's Law Question; I'm very close

[eriadoc]

Homework Statement

Two batteries and three resistors are connected as shown in the figure. How much current flows through the 6.0V battery when the switch is closed? How much current flows through the 9.0V battery when the switch is closed? How much current flows through the 6.0V battery when the switch is open? How much current flows through the 9.0V battery when the switch is open?

Homework Equations

V=IR; I1+I2+I3=0

The Attempt at a Solution

I labeled the loop on the left as LOOP1, the right side loop LOOP2, and the total circuit LOOP3. I've labeled the current leaving the 6V potential as I_1, and it splits into I_2 (going down the middle leg) and I_3, which continues around the outer leg. I've gone round and round with this problem, so I'm going to start the question at LOOP1. Without using Kirchoff's Law at all, I can easily surmise that the total resistance in the loop is 6ohms. The two resistors (4ohm and 2ohm) are in series. I know the answers to the questions "How much current flows through the 6.0V battery when the switch is open? How much current flows through the 9.0V battery when the switch is open?" are -0.5A, which fits perfectly with what I know from using V=IR: (6-9)=I(6).

But when I set it up following the book example and that given by the professor in class, the resistance is not added, but subtracted. So I do the following:

ε - I_2R + I_1R=0 => -3 - 4I_2 + 2I_1=0

Obviously I know this is wrong, and it should be 4+2, but I don't know why.

For LOOP2 (closed switch):

ε - I_3R + I_2R=0 => (6+9) - 5I_3 + 4I_2 =0

From here, I do the loops out, rearrange the equations algebraically, and substitute to solve for each I. I'm coming up with consistent numbers, but nothing close to the answers. In shooter's parlance, I'm grouping really well, but waaaay over there, LOL.

I think if I can figure out how to set up LOOP1, I'll be able to work through the rest. Thanks for any advice and help.

 

[gneill]

The first thing to keep in mind is that loop currents don’t split. Loop currents are continuous, and pass entirely around their loop unchanged. Second, to analyze a complete circuit you need just enough loops so that every component that has current flowing through it is included in at least one of them.

Your circuit requires only one or two loops to cover all components, depending upon whether or not the switch is closed.

So, if you begin with the case where the switch is open , you have just one loop comprising the two batteries and the 2Ω and 4Ω resistors. What's an equation for this loop?