(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Given the following circuit:

Calculate the currents I_{1}, I_{2}, and I_{3}

The given resistances are:

r: 1.05 ohm

R1: 11.7 Ohm

R2: 10.5 Ohm

R3: 7.20 Ohm

R4: 14.3 Ohm

R5: 18.8 Ohm

The given emf's are:

E_{1}= E_{2}: 9.0 V

E_{3}: 3.0 V

2. Relevant equations

We know Kirchhoff's laws for current and voltage. Namely, I_{in}= I_{out}for any junction, and voltage around any loop must = 0.

3. The attempt at a solution

I attempted to use Kirchhoff's voltage laws around the big loop (all the way around the outside), the upper (top) loop, and the lower (bottom) loop. Then, I put these into a matrix, and tried to use the reduced row echelon form (rref) to solve for the currents.

Big loop:

E_{3}- I_{3}r - I_{3}R5 - I_{1}R1 + E_{1}- I_{1}r - I_{1}R3 - I_{3}R4 = 0

This is equivalent to: I_{1}(-R1 - r - R3) + I_{2}(0) + I_{3}(-r - R5 - R4) = -E_{3}- E_{1}

Top loop:

E_{2}- I_{2}r - I_{2}R2 - I_{1}R1 + E_{1}- I_{1}r - I_{1}R3 = 0

Equivalent to: I_{1}(-R1 - r - R3) + I_{2}(-r - R2) + I_{3}(0) = E_{2}- E_{1}

Bottom loop:

E_{3}- I_{3}r - I_{3}R5 + I_{2}R2 + I_{2}r - E_{2}- I_{3}R4 = 0

Equivalent to: I_{1}(0) + I_{2}(R2 + r) + I_{3}(-r - R5 - R4) = -E_{3}+ E_{2}

Resultant matrix

However, the rref of this matrix ends up with a row of all 0's (implying infinite solutions). Therefore, I'm assuming there is a mistake in how I went around the loops and setup the equations. Can anyone help?Code:[(-R1 - r - R3) 0 (-r - R5 - R4) ] [ I1 ] [ -E3 - E1 ] [(-R1 - r - R3) (-R2 - r) 0 ] [ I2 ] = [ E2 - E1 ] [ 0 (R2 + r) (-r - R5 - R4) ] [ I3 ] [ -E3 + E2 ]

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# Current in a loop (Kirchhoff's Laws)

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