# Kirchhoff's Law

by Jared Rovny

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00:00 Now that we have an understanding of some basic circuit properties and how we actually measure those in practice.

00:05 Let´s discuss Kirchhoff´s law.

00:08 We briefly mentioned something like Kirchhoff´s law when we were introducing resistors and basically Kirchhoff´s law says this, that the total voltage around any path that we could pick through our circuit if we end back where we started has to be zero.

00:23 So let´s actually do this. Let´s pick some starting point in a circuit like this one.

00:27 We´ll start on the bottom right and then let´s walk our way through the circuit, adding up the contributions to the voltage.

00:33 First, we´re going to go through our battery, so we´re highlighting that in red.

00:37 So we move through our circuit, we move through the battery and that is added a voltage.

00:41 It´s put in some more pressure to the circuit.

00:44 So we have a plus V so far and then we keep going.

00:47 When we get to this top part we have to make a decision.

00:49 We have to pick either the top path that goes through the two resistors or the bottom path that goes through the one resistor.

00:55 So picking the top path, we first go through the first resistor and then use Ohm's law which tells us how much voltage we lose by going through that resistor.

01:04 So that would be some delta V, some change in voltage, which of course we could find from the current and the resistance of that resistor and then we continue on and go through the next resistor and lose some more voltage due to that resistor, and after that point we don´t have anything stopping us from going all the way back to our starting point.

01:20 Kirchhoff´s law says that when we´ve added up all of these voltage contributions, a positive value from the battery, adding a voltage, and then the negative values from these resistors taking away voltage, we end up with zero and the reason for this is because if we have voltage that is not zero after we´ve gone all the way around the circuit that means we have a mismatch in our pressure.

01:42 Meaning that we could start at one point and then go back to that point and get a different pressure value, and any time you have a small difference in pressure from one point to the next in a circuit, the circuit will very quickly equilibrate to make sure these pressure differences don´t exist and so with Kirchhoff´s law we know that we don´t have any discontinuities in pressure like this and so we can go through the entire circuit and add up these, what I´m calling pressures or the voltages in the circuit, and see that the total must be zero.

02:08 Similarly, we could have done the exact same thing except instead of going through the top path, we could´ve gone through the bottom path here.

02:14 So, we go through the battery again, go up and go through the bottom path in which case we would have a different delta V, a different change in voltage lost from the other resistor.

02:23 So with Kirchhoff´s law we have two equations and we could use these two equations to find out things about this circuit.

02:31 A few more things that would be important to know if we were going to use Kirchhoff?´s law in a circuit like this is that current is conserved along any wire and as we´ve already mentioned this is because the current doesn´t have anywhere else it can go.

02:43 If the electricity, the electrons are actually flowing through a particular wire, they can´t just leave the wire, jump off, and none can be added either, and so in any contiguous wire, any wire that´s connected, the current can´t go anywhere and so the current will be conserved in that wire.

02:57 So in this example, we have the current flowing out of the battery.

03:02 That current will be the same throughout this entire wire until it splits.

03:06 It has to split at a junction and then some of the current will go down on the lower path and some will go up on the upper path.

03:12 The current that goes up on the upper path will go through both of the upper resistors and that current will be the same in both of those resistors because both of those resistors are on the same wire.

03:22 We could then rewrite Kirchhoff´s law knowing how the current acts throughout our circuit and using Ohm´s law for what the voltage drops would be across each one of these resistors, either the two on the top or the one on the bottom, and again notice in the first expression of Kirchhoff´s law that we have written here, we have I2 times R1 minus I2 times R2.

03:45 In other words, we have the same current in both of these resistors and that´s going to be important as you?re solving problems with this because you don´t want to introduce too many unknowns, you don´t want too many variables that you have to solve for.

03:56 And in this case for a given wire, we only have one unknown, we only have one variable and that would be the current in that wire.

04:02 So when we´re simplifying a circuit, there are some things we should understand about how to break it down and simplify the resistances.

04:09 So for example in a circuit like this, we have two resistances in series of each other, R1 and R2, in parallel with a third resistance. We could simplify this by adding R1 and R2 in the ways we´ve already described so we have an equivalent resistance, R1.2 we could call it.

04:26 And then we could simplify this one more time using our addition laws for resistors that are in parallel.

04:33 The important thing to know here is that right off the bat when you´re trying to add resistances together, we couldn´t for example, have just directly added R1 and R3 and then dealt with R2 because R1 and R3 are actually not in parallel.

04:48 It may seem like they´re in parallel but because we can´t split our path and re-join without going through the second resistance, R2, they´re not really in parallel.

04:58 So what would be more accurate to say is that R1 and R2 are in parallel with R3 but we cannot say that R1 is in parallel with R3, and so we cannot use the parallel addition law for our resistors for just R1 and R3.

05:13 We have to first add the simple system, R1 and R2, because we know that those are in series, there´s no ambiguity there.

05:20 So anytime you find yourself trying to add resistances whether it´s in series or in parallel, make sure you pick the very simplest ones that you can find, the ones that are very obviously in series or the ones that are very obviously in parallel.

05:32 Meaning that you can go through both and then re-join your path and make sure you start by adding those first and then working your way up rather than trying to for example in this case, add R1 and R3 together which would be incorrect.

The lecture Kirchhoff's Law by Jared Rovny is from the course Circuit Elements.

### Included Quiz Questions

1. The sum of the change in voltages around any closed path of a circuit adds to 0
2. The total voltages of all elements in a circuit adds to a total of 0
3. The total voltage of all batteries in a circuit add to 0
4. The sum of the change in voltage across all resistors in a circuit adds to 0
5. The total voltage across all junctions adds to 0
1. There is no addition rule for just these 2 resistors alone
2. They are in series
3. They are in parallel
4. They are partly in series, partly in parallel
5. They are both relatively parallel with B
1. The voltage of the battery will be added to the total voltage since we are going from the negative terminal of the battery which has a lower voltage to the positive terminal which has a higher voltage.
2. The voltage of the battery will be subtracted from the total voltage since there is a reduction in voltage as we go through a battery.
3. Their voltages are skipped and not taken into account.
4. Their voltages are multiplied by the current flowing through the battery.
5. Double the voltage of the battery is added to the total voltage.
1. I = I₁ + I₂
2. I = I₁ = I₂
3. I₁ - I = I₂
4. I₁ + I₂ = 2I
5. I = 2(I₁ + I₂)

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