Different emf's in parallel

1. Apr 21, 2012

Scott.P

1. The problem statement, all variables and given/known data

This is a question from my problem set on DC circuits:

30) Determine the magnitudes and directions of the currents through each resistor. The batteries have emfs of E1 = 9.0V and E2 = 12.0V, and the resistors have values of R1 = 15Ω, R2 = 20Ω, and R3 = 30Ω.

2. Relevant equations

Kirchhoff two circuit laws:

Loop Rule - $\Sigma$ Changes in voltage around closed loop = 0
Junction Rule -> For any junction in a circuit, the current entering must equal the current leaving the junction
Ohm's law - V = IR

3. The attempt at a solution

Go around the top loop, use the loop rule. Assume current goes clockwise for both loops. Define I1 to be the current through R1, I2 to be the current through R2, and I3 to be the current through R3:

9.0V - (15Ω)(I1) - (20Ω)(I2) = 0

Go around the bottom loop, with the same assumptions as stated:

12.0V - (30Ω)(I3) - (20Ω)(I2) = 0

And, from the junction rule,

I2 = I1 + I3

So, it appears we have three variables and three equations. Just solve the system of equations right? While it may look simple, I've been struggling to solve the system for the past hour, going through several sheets of paper. I don't know what it is, if I am missing something or if I made an error forming my equations. I am currently taking T3's for a fractured patella, but I don't think they have that much of an effect on my cognitive ability. Any help?

-Scott

2. Apr 21, 2012

tiny-tim

Welcome to PF!

Hi Scott! Welcome to PF!
substitute for I3 in the second equation

you now have two equations with only I1 and I2

3. Apr 21, 2012

ehild

Your equations are correct. Isolate I1 from the first one and I3 from the second one and substitute the expressions for I1 and I3 into the third equation.

4. Apr 21, 2012

Scott.P

Re: Welcome to PF!

Well, no wonder I was having problems seeing the answer, I was trying to make equation one equal equation two and then substitute something in with equation three. Thanks.

5. Apr 21, 2012

Scott.P

I got an answer but it is different from that of the answer key. All of the numbers I gave are as the question says, but I believe the answer key is wrong, because they said that the voltage drop over R3 is I2R3, rather than I3R3as it should be. The mistake is repeated in the next question, which is identical to this except that the two emf's now have an internal resistance of 1Ohm. Am I right or the textbook?

6. Apr 21, 2012

tiny-tim

hmm … looking again, your equations seem wrong …
… that makes no sense, you have to assign a direction for each of the three currents separately

(and mark them on your diagram with arrows, to make sure you don't make a mistake later)

7. Apr 21, 2012

Scott.P

That would be my blunder for not putting the current direction on my diagram. I mixed myself up, I think I meant "going clockwise around each closed loop for the loop rule". On my paper here anyhow, I have chosen that I1 goes right (across R1); I2 goes left (across R2); and that I3 goes up (across R3).

Is that clear now?

8. Apr 21, 2012

tiny-tim

oh, that's ok then
yes, the voltage drop over R3 is definitely R3 times the current through R3