## Adding resistance in a single loop

Physics Narrative for 14-16

#### More resistors in one loop – take care to identify each

Resistance impedes current, so adding more resistance results in less current. So, perhaps much as you would expect, adding a second resistor to a single loop always increases the resistance of the loop and so decreases the current (assuming the potential difference stays the same – you didn't change the battery).

The resistance in the loop can be varied by adding more and more slices of material to the loop (see Teaching approaches), but you are more likely to alter the resistance by adding extra resistors. We'll always call these *R*_{1} and *R*_{2}, reserving *R* for the total resistance in the circuit.

But what happens to the potential difference? Again, you need to be careful, because here are now three potential differences that you can measure: *V*_{1} across *R*_{1}, *V*_{2} across *R*_{2}, and *V* for the potential difference across everything, and so the potential difference provided by the battery.

There is only one current – it is the same everywhere in a single loop.

#### Analysing a circuit with only series connections

Here is how to analyse such a circuit, idealising it so that neither the connecting wires nor the battery have any resistance. The total resistance in the circuit is just the sum of the individual resistances: *R* = *R*_{1} + *R*_{2}.

Then you can calculate the current in the loop, replacing the two resistors with a single equivalent resistor (either redraw the circuit, or imagine doing so):

*I* = *V**R*

Now step back to the original circuit. You can calculate the potential difference across each resistor by using the relationship between *V*_{1}, *R*_{1} and *I*, or *V*_{2}, *R*_{2} and *I*.

*V*_{1} = *R*_{1} × *I*

*V*_{2} = *R*_{2} × *I*

These two sets of quantities are tied together in particular ways, as shown in this pair of equations. They are mutually constrained. So one might call these kinds of relationship constraint relationships. They are true at all instants, independent of the history of the circuit. The relationship doesn't imply any evolution over time.

#### Colouring can help identify the differences

The potential differences show where energy will be shifted as a result of the current in that part of the circuit. Seeing where the differences are can be helped by colouring all of the wires.

Height differences, as represented on maps, perform a similar function for the shifting of energy from your chemical store to the gravitational store as you climb a hill. Predicting where these differences are is helped by colouring maps by height. Again, it is only the difference in height that has any effect on the energy shifted.

Colouring by height is useful only in so far as it shows where the changes in height occur. So you will have a visual representation of the ways in which the height changes as you go along a path. These changes warn you what lies ahead, so you can predict where energy will be shifted, and how much.

#### Locating the potential differences

Colouring different regions of the circuit diagrams will enable you to see and then calculate the differences. From these potential differences the energy shifted can be calculated. These patterns show what the circuit will do.

These predictions enable you to design circuits to perform particular tasks. That is one of the main reasons why electric circuits are so prevalent in society and in the study of electric behaviours.