Kirchhoff's Loop Rule: Principles & Validity Analysis

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  • 0:04 What Is Kirchhoff's Loop Rule?
  • 1:15 RC Circuit
  • 4:28 Steady State
  • 6:46 Lesson Summary
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Lesson Transcript
Instructor: Matthew Bergstresser
There are two Kirchhoff's rules: the junction rule and the loop rule. In this lesson, we'll focus on the loop rule and how it's based on the law of the conservation of energy. We'll use data obtained from an RC circuit to prove the loop rule is valid.

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
D1


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.
S1


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:


q1.5


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
Ceq


Since they are in series, we use the equation:


C_eq


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:


Q_eq


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


v2


and the voltage across the 3 µC capacitor is


v3


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