How to Determine Currents in a Parallel Circuit with Different EMFs?

[Scott.P]

Homework Statement

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.

Homework 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

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

 

[tiny-tim]

Welcome to PF!

Hi Scott! Welcome to PF!

9.0V - (15Ω)(I₁) - (20Ω)(I₂) = 0

12.0V - (30Ω)(I₃) - (20Ω)(I₂) = 0

I₂ = I₁ + I₃

substitute for I₃ in the second equation

you now have two equations with only I₁ and I₂

 

[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.

 

[Scott.P]

Hi Scott! Welcome to PF!


substitute for I₃ in the second equation

you now have two equations with only I₁ and I₂

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.

 

[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?

 

[tiny-tim]

hmm … looking again, your equations seem wrong …

Assume current goes clockwise for both loops.

… 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)

 

[Scott.P]

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)

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?

 

[tiny-tim]

oh, that's ok then

… I believe the answer key is wrong, because they said that the voltage drop over R₃ is I₂R₃, rather than I₃R₃as it should be.

yes, the voltage drop over R₃ is definitely R₃ times the current through R₃