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AP Physics 2: Exam Prep26 chapters | 138 lessons

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Lesson Transcript

Instructor:
*Sarah Friedl*

Sarah has two Master's, one in Zoology and one in GIS, a Bachelor's in Biology, and has taught college level Physical Science and Biology.

Ohm's Law can be useful for analyzing simple circuits, but when circuits are more complex, as they often are, we can instead analyze the circuits using Kirchhoff's rules. Learn about Kirchhoff's rules in this lesson.

Up to this point in our lessons on series and parallel circuits, we've covered basic direct current circuits in a pretty good amount of detail. But, the types of circuits we've discussed so far have been fairly simple. They consisted of one loop, one power source, and only a handful of resistors.

But, we can't always make our circuit be just one single, easy to analyze loop. Sometimes our circuits are more complicated, which may lead us to the work of a man named Gustav Kirchhoff. What? You've never heard of him? That's unfortunate because he was a very smart man and a talented physicist. For example, he calculated that an electrical signal travels along a resistance-less wire at the speed of light! He also worked with Robert Bunsen, the inventor of, that's right, the Bunsen burner!

Kirchhoff developed two rules, or laws, regarding circuits. The first of **Kirchhoff's rules** states that the sum of the currents entering a junction must equal the sum of the currents leaving the junction. Because this law deals with the current at junctions in the circuit, it's also sometimes called Kirchhoff's current law or Kirchhoff's junction rule. All this law is saying is that we have a conservation of charge - the current may split and go in different directions at the junction, but if you added up the total amount of current in each branch they would equal the amount of current that originally came into that junction.

The second of **Kirchhoff's rules** states that the algebraic sum of the voltage differences across all elements of a closed loop must equal zero. This rule is based on the law of conservation of energy, and reminds us that as current encounters a resistor there's a 'voltage drop' and that the sum of all voltage drops is equal to the total supplied voltage. Since this rule deals with voltage across loops in the circuit, it's also sometimes called Kirchhoff's voltage law or Kirchhoff's loop rule.

So, now that you know what Kirchhoff's rules are, let's look at how you might use them to help you analyze a more complicated circuit. Say, for example, that you have a complex circuit like this:

Here, we use the letter *I* to represent current, the letter *R* to represent resistance, and the letter *E* to represent the power supply. You can tell by the arrows which way the current is flowing and, in some cases, how much current is flowing in that direction.

But, let's say that some part of the current flow is unknown. We can use Kirchhoff's first rule, the current law, to help us. In this circuit, we can see that 15A enters junction *A*, and that 7A goes through the right branch toward Resistor 2. How much current flows toward Resistor 3? We simply find the difference between the left branch and the total incoming current. Therefore, *I*1 - *I*2 = *I*3, or 15A - 7A = 8A.

It works the same no matter how many incoming sources of power or outgoing branches you have. If, for example, you have the type of junction where there are two sources of incoming current and three exit paths for the current to flow through, your equation would look like this: *I*1 + *I*2 = *I*3 + *I*4 + *I*5.

And, if you know how much current is leaving the junction in each branch, but only know how much is coming in from one of the branches, say *I*2, simply rearrange your equation so that you get that unknown value by itself. Then, solve for *I*3 + *I*4 + *I*5 - *I*2 = *I*1, and you'll get your answer!

The voltage rule is a little bit more complicated because we have to pay close attention to which direction we're heading through the circuit as well as through the power source. Current flows from the negative terminal of our power supply, through the circuit, and back to the positive terminal of the power supply as it completes the loop. As current passes through a resistor along the circuit there's a 'voltage drop,' which means there is a loss in voltage.

The total loss of voltage around a circuit loop will equal the total voltage of the power supply. So, if you have a 12V battery powering the circuit and you have three resistors along that circuit, you'll have a total loss of 12V across the resistors. In contrast, as current flows through the power supply, it passes from the positive terminal to the negative terminal, and there's a 'voltage rise,' or increase in voltage. Again, this will be the same voltage as the power supply.

Let's apply this idea to a simple circuit to see how we would use Kirchhoff's second rule for circuit voltage analysis. In this circuit, we have a power supply of 100V and four resistors, three of which have known values of voltage loss: 25V, 30V, and 15V.

Since these values are losses, they're all negative values, but the power supply is a positive value.

We know that the algebraic sum of the voltage differences across all elements of a closed loop must equal zero, so in equation form we can write our problem like this: *E* = -*R*1 - *R*2 - *R*3 - *R*4. But, since we know *E* and are trying to solve for *R*4, let's rearrange this a bit. Now, we have *E* - *R*1 - *R*2 - *R*3 = *R*4, and plugging in our values we find that 100V - 25V - 30V - 15V = 30V.

You can use this same idea to solve for the power supply if that is the unknown. Simply plug in your values and solve for *E*. And, while this example was for a single loop, you can apply this rule to more complex circuits that contain multiple loops, multiple resistors, and even multiple power sources.

Circuits can be quite complex, but luckily analyzing them doesn't have to be. Thanks to **Kirchhoff's rules** we can analyze current and voltage in circuits that have multiple elements. The first rule states that the sum of the currents entering a junction must equal the sum of the currents leaving the junction. This shows us that the charge is conserved - no matter how many ways it's divided, the total amount of current stays the same.

The second rule states that the algebraic sum of the voltage differences across all elements of a closed loop must equal zero. This reminds us that the total 'drop' in voltage across all resistors will be the same as the total voltage supplied by the power source. To use this rule, you'll just need to keep in mind that voltage drops as it passes from the negative end to the positive end and rises as it passes from positive to negative.

Finish the lesson and increase your capacity to do the following:

- Discuss Gustav Kirchhoff and some of his accomplishments
- State the two rules of Kirchhoff
- Apply both rules when solving problems with complex circuits

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AP Physics 2: Exam Prep26 chapters | 138 lessons

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