# Two-loop circuit using Kirchoff's Laws problem

• ayeelkaye
In summary, the circuit consists of two batteries and four resistors. The current through R1 is represented by I1 and the current through R3 is represented by I3. The problem can only be solved by using simultaneous equations. The process involves replacing R1 and R4 with an equivalent resistance, writing two independent voltage loop equations, and solving for I1 and I3. It is important to write the loop equations in algebraic form first before substituting numerical values.
ayeelkaye

## Homework Statement

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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## Homework Equations

I1 = I2 + I3

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

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

## 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..

Keep in mind that for the left loop the current through R2 = (I1 - I3) and for the right loop, the current through the resistor is (I3 - I1). Otherwise your equations look good, but what do you mean by "setting them equal to each other"? Please show these steps.

Hello ,

Thank you for reaching out for help with this problem. It is understandable that you are feeling stuck and unsure of the most efficient approach to solving this circuit using Kirchoff's Laws.

First, let's review the steps that you have already taken. You correctly identified that the currents I1 and I3 must be solved simultaneously, as they are dependent on each other due to the presence of multiple EMFs in the circuit. You also correctly used the Kirchoff current rule to express the current through R2 in terms of I1 and I3.

Next, let's take a look at the loop equations that you have written. Your equations for the left-side and right-side loops are correct, but there are a couple of small mistakes in the coefficients. In the left-side loop, the coefficient of I2R2 should be -I2 instead of -I1. Similarly, in the right-side loop, the coefficient of I2R2 should be I2 instead of I3.

Now, let's move on to solving the equations. As you correctly identified, substituting I1-I3 for I2 into both equations is the most efficient approach. This is because it eliminates the need to solve for I2 as an intermediate step. After substituting, you should get the following equations:

E1 - I1R1 - (I1-I3)R2 - E2 - I1R4 = 0

-I3R3 + E2 + (I1-I3)R2 = 0

From here, you can solve for I1 and I3 using any algebraic method of your choice (substitution, elimination, etc.). Once you have obtained their values, you can use the Kirchoff current rule to solve for I2.

I hope this helps guide you towards finding the solution to this problem. Remember to always double-check your equations and coefficients for accuracy, and don't hesitate to reach out for further assistance if needed. Good luck!

Best,

Scientist

## 1. What are Kirchoff's Laws and how do they apply to a two-loop circuit?

Kirchoff's Laws are fundamental principles in circuit analysis that govern the behavior of currents and voltages in a closed loop circuit. The first law, also known as Kirchoff's Current Law, states that the sum of all currents entering a node must equal the sum of all currents leaving that node. The second law, also known as Kirchoff's Voltage Law, states that the sum of all voltage drops around a closed loop must equal the sum of all voltage sources in that loop.

## 2. How do I set up a two-loop circuit problem using Kirchoff's Laws?

To set up a two-loop circuit problem using Kirchoff's Laws, first draw a circuit diagram with all the components and label all the nodes and loops. Then, apply Kirchoff's Current Law at each node and Kirchoff's Voltage Law around each loop. This will result in a system of equations that can be solved to determine the values of currents and voltages in the circuit.

## 3. What are the steps to solve a two-loop circuit problem using Kirchoff's Laws?

The steps to solve a two-loop circuit problem using Kirchoff's Laws are as follows:

1. Draw a circuit diagram and label all the nodes and loops.
2. Apply Kirchoff's Current Law at each node to write equations for the currents.
3. Apply Kirchoff's Voltage Law around each loop to write equations for the voltages.
4. Combine the equations and solve for the unknown variables using algebraic methods.
5. Check the solution by substituting the values back into the equations to ensure they satisfy Kirchoff's Laws.

## 4. Can Kirchoff's Laws be applied to any circuit, regardless of its complexity?

Yes, Kirchoff's Laws can be applied to any circuit, regardless of its complexity. These laws are based on the fundamental principles of conservation of charge and energy, and are applicable to all types of circuits, including two-loop circuits.

## 5. Are there any limitations to using Kirchoff's Laws to solve two-loop circuit problems?

The only limitation to using Kirchoff's Laws to solve two-loop circuit problems is that the circuit must be a closed loop. This means that there must be a complete path for current to flow from the source back to the source. If there are any open branches in the circuit, Kirchoff's Laws cannot be used to solve the problem.

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