Kirchoff’s Laws

Pre-Lab: Kirchoff’s Laws

Name: Section:

1. Solve the following set of three equations for the unknowns I1, I2, and I3 in terms of 1= 1.6 V, 2= 1.4 V, R1 = 220 , R2= 230 , and R3= 100 . Note: You will use the result during the lab (although the numbers may be different), so you should make a copy of your answer.

1− I2R2− I1R1 = 0

− 2+ I2R2− I3R3 = 0 I1 = I2+ I3

2. Briefly summarize the procedures you will follow in this lab. Write one or two sentences for each activity.

3. List any part (or parts) of the lab that you think may suffer from non-trivial experimental error, or may otherwise cause you trouble. How might this affect your results?

6

I have a strong resistance to understanding the relationship between voltage and current.

Anonymous

Objectives

• To find a mathematical description of the flow of electric current through different elements in direct current circuits (Kirchhoff’s laws).

• To gain experience with basic electronic equipment and the process of constructing useful circuits while reviewing the application of Kirchhoff’s laws.

Overview

Suppose we wish to calculate the currents in various branches of a circuit that has many components wired together in a complex array. In such circuits, sim- plification using series and parallel combinations is often impossible. Instead we can state and apply Kirchoff’s laws more formally to aid with the solution of such problems. These rules can be summarized as follows:

1. Junction (or Node) Rule (based on charge conservation): The sum of all the currents entering any node or branch point of a circuit (i.e., where two or more wires merge) must equal the sum of all currents leaving the node.

2. Loop Rule (based on energy conservation): Around any closed loop in a circuit, the sum of all emfs and all the potential drops across resistors and other circuit elements must equal zero.

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82 Kirchoff’s Laws

Steps for Applying Rules

1. Assign a current symbol to each branch of the circuit and label the current in each branch I1, I2, I3, etc.; then arbitrarily assign a direction to each cur- rent. (The direction chosen for the circuit for each branch doesn’t matter. If you chose the “wrong” direction the value of the current will simply turn out to be negative.) Remember that the current flowing out of a battery is always the same as the current flowing into a battery.

2. Apply the loop rule to each of the loops by:

a) Letting the potential drop across each resistor be the negative of the product of the resistance and the net current through that resistor. Re- verse the sign to “plus” if you are traversing a resistor in a direction opposite that of the current.

b) Assigning a positive potential difference when the loop traverses from the (–) to the (+) terminal of a battery. If you are going through a battery in the opposite direction assign a negative potential differ- ence to the trip across the battery terminals.

3. Find each of the junctions and apply the junction rule to it. You can place currents leaving the junction on one side of the equation and currents com- ing into the junction on the other side of the equation.

In order to illustrate the application of the rules, let’s consider the circuit in the following figure.

R1

R3

R21 2

I1

I1

I2

I3

I3

Kirchoff’s Laws 83

The directions for the currents through the branches and for I2 are assigned arbitrarily. If we assume that the internal resistances of the batteries are negligi- ble, then by applying the loop and junction rules we find that

1− I2R2− I1R1 = 0 , (6.1)

− 2+ I2R2− I3R3 = 0 , (6.2)

I1 = I2+ I3 . (6.3)

It is not obvious that the loops and their directions can be chosen arbitrarily. Let’s explore this assertion theoretically for a simple situation and then more concretely with some specific calculations. In order to do the following activity you’ll need a couple of resistors and a multimeter as follows:

• 100 resistor

• 220 resistor

• Potentiometer (variable resistor)

• Multimeter

• 2 batteries

• Breadboard

84 Kirchoff’s Laws

Part I

Activity 1.1 Applying the Loop Rule

1. Use the loop and node rule along with the new arbitrary direction for I2 to rewrite the three equations relating values of battery emfs, resistance, and current in the circuit shown below.

R1

R3

R21 2

I1

I1

I�2

I3

I3

2. Show that if I�2 =−I2 then the three equations you just constructed can be rearranged algebraically so they are exactly the same as Equations (6.1), (6.2), and (6.3).

Part I. 85

3. Since you are going to test your theoretical results for Kirchhoff’s law calcula- tions for this circuit experimentally, you should measure the actual values of the two fixed resistors (rated at 100 and 220 ) and the two battery voltages with a multimeter. List the results below.

1 = V 2 = V

R1 = R3 =

4. Carefully rewrite Equations (6.1), (6.2), and (6.3) with the appropriate measured (not rated) values for emf and resistances substituted into them. Use 230 for the value of R2 in your calculation. You will be setting a pot (variable resistor) to that value soon.

5. Solve these three equations for the three unknowns I1, I2, and I3.

6. Show by substitution that your solutions actually satisfy the equations.

86 Kirchoff’s Laws

Comment:

The most accurate

way to measure

current with a digital

multimeter is to

measure the

potential difference

across each of the

resistors and use

Ohm’s law to

calculate I from V

and R.

Activity 1.2 Verifying Kirchhoff’s Laws Experimentally

1. Use the ohmmeter feature of the digital multimeter to measure the total resistance across a pot that is labeled 500. Then measure the resistance between the center tap on the pot and one of the other taps. What happens to the ohmmeter reading as you change the setting on the pot?

2. Set the pot so that there is 230 between the center tap and one of the other taps.

3. Wire up the circuit pictured in the previous figures. Use the pot (set at 230 ) as R2. Measure the current in each branch of the circuit and compare the measured and calculated values of the current by computing the % error in each case.

4. Fill in the follwing table with your measurements.

Measured R Measured V Calculated I Measured I % Error [ ] [V] [mA] [mA]

R1 R2 230 R3

5. What do you predict will happen to each of the currents as the resistance on the pot is decreased? That is, will the currents I1, I2, and I3 increase or decrease? Explain your predictions.

6. What actually happens to each of the currents as you decrease R2? How good were your predictions?