- #1

- 4

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- Homework Statement
- Can I solve this using Kirchoff's Law?

- Relevant Equations
- Kirchoff's Law

We were given a circuit with 7 resistors and 3 voltage sources

For emf sources, ##E_1=120V##, ##E_2=60V##, and ##E_3=30V## while for resistors in ohms,##R_1=10##, ##R_2=5##, ##R_3=20##, ##R_4=8##, ##R_5=12##, ##R_6=6##, ##R_7=8##. Nodes are indicated in small letters (a-j). Loops ##abefa##, ##abcdefa## and ##aghcdijfa## are assumed clockwise while the loop ##cbedc## is counterclockwise.

I applied Kirchoff's Current Rule in nodes ##b## and ##e##:

$$

I_1+I_3 = I_2\\

I_4+I_5 = I_2

$$

And then I apply the Voltage Rule:

Loop ##abefa##:

$$120-I_1R_1-I_2R_2-30-I_4R_4=0$$

Loop ##cbedc##:

$$60-I_3R_3-I_2R_2-30-I_5R_5=0$$

Loop ##abcdefa:##

$$120-I_1R_1+I_3R_3-60+I_5R_5-I_4R_4=0$$

Since I have 5 equations already with 5 unknows, hopefully I should solve these values.

From 1st and 2nd equations, ##I_4 = I_2-I_5 = I_1+I_3-I_5##. Setting ##I_4##, and ##I_2## in terms of ##I_1##, ##I_3## and ##I_5## only and substituting them to equations 3, 4 and 5, I get:

$$

23I_1+13I_3-8I_5=90\\

18I_1-12I_3-20I_5=60\\

5I_1+25I_3+12I_5=30

$$

Luckily my calculator supports systems of equations for 3 uknowns, but then I get a math error, which could indicate infinitely many solutions? Maybe I incorrectly applied Kirchoff's rule but I don't know where. Is it valid to apply Kirchoff's rule here?