Kirchhoff's Laws

These two simple laws were drawn up in the Nineteenth Century by Gustav Robert Kirchhoff.  They explain all observations we see in electric circuits.  We can explain everything we have looked at in series and parallel circuits in terms of the two laws.  They can also be used to explain more difficult circuits which cannot be explained in terms of simple series and parallel circuits.

Kirchhoff I

This states that the total current flowing into a point is equal to the current flowing out of that point.  In other words, the charge does not leak out or accumulate at that point.  Charge that flows away must be replaced.  It is conserved.

From this diagram we can easily see that I₃ = I₁ + I₂.

Mathematically we can write this as:

  I₁ + I₂ + -I₃ = 0

Notice that I3 has a minus sign. This means that the current going out is regarded as negative while current coming in is positive.  At no point is there any reference to charge pooling at the junction, for the simple reason that it does not.

In some text books you will see written SI = 0.  The strange looking symbol S is Sigma, a Greek capital letter S, which means "sum of".  So the sum of currents is zero, as we have seen above.

Kirchhoff II

Kirchhoff’s Second Law is not quite so easy to grasp.  It states:

The potential differences around a circuit add up to zero.

  Provided the charge returns to the same place as it started, the gains and losses are equal, no matter what route is taken by the charge. The battery in this circuit has an emf (electromotive force or open terminal voltage) of ℇ.  The curly E is the battery voltage.  We will look at emf later.

  Let us do a journey around the circuit from A to B to C, and back to A.

• From A to B the p.d change is IR₁ volts

• From B to C the p.d. change is IR₂ volts

• From C to A the pd. change is ℇ volts.

  If we add up all the voltages, we can write IR₁ + IR₂ = ℇ or IR1 + IR2 + -ℇ = 0 or Sp.d. = 0

  This is another way of saying that the voltages add up to the battery voltage.  

Although the statement above may seem obvious enough to some, Kirchhoff II does throw a lot of students, so let's have a further look.

The picture shows a simple roller-coaster railway.  The cars leave the station.  They can go down hill the straight way or the wiggly way.  It doesn't matter.  They then have to go back up the potential hill to reach the station they started from.

Some roller-coasters have several hills the cars go up.  As they go up their potential increases.  It's like a circuit with several batteries.  You go up the potential hill as you go pass each battery.  When you go up the potential hill, the sign is negative.

Click HERE to see a more challenging example of what Kirchhoff I and II can solve.  This is NOT on the AQA syllabus and you should only look at this if you are confident with the more simple type of circuit.