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Physics/Kirchoff's Rules, finding current and resistance


QUESTION: I am having difficulty finding the current in the problem. Any help would be greatly appreciated.

Solve the following circuit (in picture)using variables only. Find the expressions for current everywhere.

ANSWER: Hello Nick,

Define current I1 going thru R1 from left to right. And I2 going thru R2 from top to bottom. And I3 going thru R3 from right to left. Using Kirchoff's Current Law at the node between R1 and R2,
I1 + (-I2) + (-I3) = 0
I1 = I2 + I3

To use KIrchoff's Voltage Law, set up a clockwise loop consisting of E1, R1, and R2. The voltage law gives us the equation
E1 - I1*R1 - I2*R2 = 0      #1
Set up another clockwise loop consisting of E2, R3, and R2. The voltage law gives us the equation
E2 - I3*R3 + I2*R2 = 0       #2

Now it's time for some algebra:
Solve the 2 voltage law equations for I2*R2.
I2*R2 = -E1 + I1*R1         #3
I2*R2 = -E2 + I3*R3         #4
So -E1 + I1*R1 = -E2 + I3*R3
E2 - E1 = I3*R3 -I1*R1
Using the current law equation,
E2 - E1 = I3*R3 -(I2 + I3)*R1 = I3*(R3 - R1) - I2*R1      #5

If we solve equation #2 for I2
I2 = (-E2 + I3*R3) / R2
If we substitute that expression equal to I2 into equation #5 we change it from
E2 - E1 = I3*(R3 - R1) - I2*R1
E2 - E1 = I3*(R3 - R1) - ((-E2 + I3*R3) / R2)*R1
E2 - E1 = I3*(R3 - R1) - (R1/R2)*(-E2 + I3*R3)
E2 - E1 = I3*R3 - I3*R1 + E2*R1/R2 + I3*R3*R1/R2
E2 - E1 = I3*(R3 - R1 + R3*R1/R2) + E2*R1/R2

Now you can solve that for I3. Now if you plug that expression in for I3 in equation #4, you can solve for I2. And then you can obtain an expression for I1 using the current law equation.

That's some messy steps of algebra. Check my work and your work carefully.

I hope this helps,

---------- FOLLOW-UP ----------

QUESTION: Thanks for answering but there are still two questions that I have.
The first one is at this line :

"Now it's time for some algebra:
Solve the 2 voltage law equations for I2*R2.
I2*R2 = -E1 + I1*R1  #3" <-------- is it not supposed to be this : I2*R2 = E1 - I1*R1 ?

Furthermore, I do not understand why when we do this part : "If we solve equation #2 for I2
I2 = (-E2 + I3*R3) / R2" we do not use this I2 as the final answer but go about finding it again. Thanks for any help.

Hi Nick,

Good catch on your first question. Look back at my answer to your original question. I have edited it to correct that line and rippled the correction through the following lines.

Regarding your second question: The equation I2 = (-E2 + I3*R3) / R2 is not satisfactory for a value of I2 for your assignment because it is in terms of I3. I do use this equation as a step in the solution of the simultaneous equations. At the end of my corrected answer to your original question, when I point you toward continuing to obtain an expression for I3, your result would be in terms only of resistors and batteries.

I hope this helps,


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Steve Johnson


I would be delighted to help with questions up through the first year of college Physics. Particularly Electricity, Electronics and Newtonian Mechanics (motion, acceleration etc.). I decline questions on relativity and Atomic Physics. I also could discuss the Space Shuttle and space flight in general.


I have a BS in Physics and an MS in Electrical Engineering. I am retired now. My professional career was in Electrical Engineering with considerable time spent working with accelerometers, gyroscopes and flight dynamics (Physics related topics) while working on the Space Shuttle. I gave formal classroom lessons to technical co-workers periodically over a several year period.

BS Physics, North Dakota State University
MS Electrical Engineering, North Dakota State University

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