Kirchhoff's Loop Rule: Principles & Validity Analysis

What Is Kirchhoff's Loop Rule?

A weightlifter has to do work to move a barbell from the floor (low potential) to arms length overhead (high potential). When he or she lets go of the barbell, gravity does the work moving it back to the ground. The barbell went in a loop. It started on the ground, went to a maximum height above the ground, and then went back to the ground. The net change in gravitational potential energy (GPE) in this loop is zero, because there can be no gain or loss of net energy.

A battery does the same thing, except it moves positive electric charge from the negative terminal of the battery where it wants to be (the ground), to the positive terminal of the battery where it doesn't want to be (high potential). This change in electric potential is known as voltage.

If the battery is connected to a circuit, the positive charge flows through the resistors and capacitors. Kirchhoff's loop rule was developed from the conservation of energy and states that the sum of all voltages in a closed loop has to be zero.

Let's look at an RC (resistor and capacitor) circuit and use voltage measurements to verify the validity of Kirchhoff's loop rule.

RC Circuit

Before we measure voltages across capacitors, we have to make theoretical calculations.

Diagram 1

Initially, both switches are open, which prevents the battery from pushing any current through the circuit. Let's close Switch A and follow the path of current until the capacitors are fully charged, and the current stops flowing. Think of this as the line at a ride at the carnival. Once all of the seats of the ride are filled up, the operator shuts down the line and people have to wait.

Diagram 2 shows the current flowing before the capacitors have accepted all of the charge they can. Notice that the 20 Ω resistor and Switch B are gone. However, they aren't really gone. They have been taken out of the diagram because no current flows through them now that Switch B is disconnected.

Diagram 2. Current still flowing as capacitors charge.

The capacitors in series (3 µC and 2 µC) have different voltages that add up to equal the voltage across the 1.5 µC capacitor because they are in parallel with it. The 1.5 µC capacitor is in parallel with the 15 V battery, so it should have a voltage of 15 V.

Let's determine the charges stored on each capacitor. The charge on a capacitor is Q = CV, where Q is charge in coulombs, C is capacitance in farads, and V is voltage in volts.

The charge on the 3µC capacitor is:

We have to combine the 2µC and the 3µC capacitors into one equivalent capacitor to determine the charge on each. Then we can calculate their voltages. Diagram 3 shows the equivalent capacitor.

Diagram 3

Since they are in series, we use the equation:

Each capacitor in series has the same stored charge, so we use Q = CV again to get the charge on the equivalent capacitor. This gives us:

Now we can calculate the voltage across each. The voltage across the 2µC capacitor is

and the voltage across the 3 µC capacitor is

Adding these voltages we get 15 V, which is what we initially predicted we would get. Now, let's get some voltage measurements to compare to our theoretical calculations.

Steady State

When the capacitors are charged, we can measure the voltage across each capacitor using a voltmeter. Diagram 4 shows the circuit at steady state, which means the capacitors are charged and no current is flowing. The voltmeters are accurate to 0.01 V, and they both read 14.93 ± 0.01 V.

Diagram 4. Circuit at steady state.

Since we have voltage measurements across the capacitors, we can use Kirchhoff's loop rule to prove that the sum of voltages in a closed loop is zero. Diagram 5 shows the loop we'll use to prove this.

Diagram 5

Starting at the top left of the circuit and moving clockwise we get:

0 V across resistor (no current is flowing) - 14.93 ± 0.01 V + 15 V = 0

This equation is invalid because we get 0.07 ± 0.00 V = 0. The reason for this is because electrical energy is lost as heat as it moves through the wires and the resistor. We can calculate the percent error from these values considering the ± 0.01 V uncertainty in our measurement.

For 14.93 V + 0.01 V = 14.94 V, we get:

For 14.93 V - 0.01 V = 14.92 V, we get:

The extremely small percent error range of 0.4 to 0.5% is justification that Kirchhoff's loop rule is valid. Okay, let's continue our analysis by using the loop rule on the right side of the circuit as shown in Diagram 6.

Diagram 6

Let's move counterclockwise from the starting point with:

-14.93 ± 0.01 V + 14.93 ± 0.01 V = 0

which gives us 0% error here shows the validity of Kirchhoff's loop rule using actual measured values.

Lesson Summary

Let's review. Kirchhoff's loop rule is an application of the conservation of energy. If you move through a circuit in a closed loop adding all voltages across the resistors and capacitor, you'll get a net change of zero voltage.

A RC circuit contains resistors and capacitors. When the battery is connected to the circuit, the capacitors begin charging. When the capacitors are charged, the current stops flowing, which is the definition of steady state.

Capacitors in series have the same charge stored on them. Their voltages add up to equal the voltage across a battery or a capacitor in parallel with them.