# Two-loop circuit using Kirchoff's Laws

1. Feb 21, 2008

### ayeelkaye

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

The circuit in the figure is composed of two batteries (E1 = 4 V and E2 = 10 V) and four resistors (R1 = 110 W, R2 = 40 W, R3 = 40 W, and R4 = 50 W) as shown.

(a) What is the current I1 which flows through R1?

I1 = ___A

(b) What is the current I3 that flows through R3?

I3 = ___A

There are helps given, novel-sized helps:

HELP: Because of the presence of EMFs in more than one branch of the circuit, parts (a) and (b) of this problem can only be solved simultaneously. There is no way around this fact. Equivalent resistance tricks are of no help, except for resistances such as R1 and R4 in this circuit that are in the same "branch" and therefore must carry the same current. Begin by replacing R1 and R4 by an equivalent resistance; call it R14. Next express the current through R2 in terms of I1 and I3 using the Kirchhoff current rule.

HELP: Next write two independent voltage loop equations by going around the left-hand block of the circuit and, separately, the right-hand block. A loop around the entire periphery of the circuit is another possibility, but this does not give independent information because the resulting equation is the sum of the previous two loop equations. Solve the loop equations for I1 and I3. For a review of systematic procedures for solving circuits of this type, consult the essay: Solving Multi-Loop Resistor Circuits.

Substitution of numerical values. When solving multi-loop circuits, the resulting equations for the currents are coupled. That is, several unknowns appear in each equation, except in special circumstances. (Many homework and textbook problems are special to avoid this complication.) Solving N independent loop equations for N unknowns algebraically it is a straightforward task in principle, but it quickly becomes highly tedious in practice. (Here N is the number of independent loops in the circuit, which also equals the number of independent currents after all the Kirchhoff current relationships have been used.) The practice of finding analytical expressions for unknown quantities and only afterward substituting specific numerical values for parameters is extremely useful and strongly favored in science and engineering. For genuinely multi-loop circuits, it is best put temporarily suspend this practice.

Namely, in solving multi-loop circuit problems, you will find it dramatically easier to substitute the numerical values of the EMFs and resistances right after you have written the loop expressions in algebraic form. On a typical test or quiz, you will not have the luxury of time to do otherwise. However, never skip the step of writing the loop equations first in algebraic form. You need equations in algebraic form to check for accuracy and to write computer programs equations to solve such equations.
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2. Relevant equations
I1 = I2 + I3

left-side loop:
E1 - I1R1 - I2R2 - E2 - I1R4 = 0

right-side loop:
-I3R3 + E2 + I2R2 = 0

3. The attempt at a solution

I tried substituting I1-I3 for I2 into both equations, and setting them equal to each other. It just seems to get messy from there, and I don't feel like I'm getting anywhere.. not even sure if that's the most efficient way to do it!

Any help would be great..