Thinking carefully about a steadily glowing lamp

A lamp glowing steadily is shifting energy at a steady rate. In a simple circuit, where the wires connect one lamp to one cell, energy is being shifted to or from stores at two places: the lamp and the cell. The same change is calculated at both: energy is shifted from the chemical store associated with the cell, and to the thermal stores around the lamp. That the change is the same seems like too much of a coincidence for there not to be a strong connection, and, of course, there is. The wires make the physical connection (there is a similar connection in the energy description). The same electrical working empties the chemical store and fills the thermal stores. This is what electric circuits are good at: connecting stores so that we can arrange the rates of depletion and accumulation.

Electrical working is the pathway that empties the chemical store at a calculable number of joules/second or watts. Electrical working is also the pathway that fills the thermal stores at the same rate. This power in this pathway is a constant number of watts. It is the link between the stores.

The energy shifted each second is the power in the pathway

The energy shifted every second in this circuit depends on both the current and the potential difference. In this simple circuit there is only one value for the current and one value for the potential difference. In more complex circuits you need to be more careful when identifying and describing the currents and potential differences. This was covered in episode 01. In this episode you will learn how to combine these quantities to find out how much energy is shifted each second by different circuit elements.

The calculation done for each pathway shows the power: the rate at which stores of energy are emptied or filled. Finding the power in the electrical pathway is very simple – you multiply the potential difference by the current.

(Think about fundamentals again, from the SPT: Electric circuits topic. The current is the charge each second, and the potential difference is the energy for each charge. In this episode we approach the same end point by thinking about power.)

Arranging such a circuit to shift energy at a particular rate is then simply a question of choosing the potential difference of the cell and the resistance of the lamp or resistor. These jointly determine the current, as shown in episode 01. The current and the potential difference then together set the power.

Energy is shifted to and from stores by the action of the electrical pathway for as long as the circuit is connected. The rate at which this happens is the power, measured in watts. This power is set by the potential difference and the current.

Compensation – one factor diminishing to compensate for growth in another

The power in a pathway depends on both the current and the potential difference. This allows a choice: there can be the same power in different characters of pathway. You can choose cells and lamps or resistors to get a large current but a small potential difference; or a small current but a large potential difference. This is a good example of compensation in action. Compensate for changes in one quantity by making inverse changes in the other. For example, making the current larger allows you to reduce the potential difference, but still have the same power in the pathway.

Back to why circuits are so pervasive

Later we'll see how choosing the character of the pathway allows particular tasks to be performed more effectively. Electric circuits are so common because we can control how and where the power is switched, so any chance to control that power better should be taken. Altering the character of the pathway, by selecting a large or small potential difference, with a resulting small or large current, is one such clear chance.

To shift energy over long distances, say for the National Grid, choosing the character of the pathway determines how much power stays in the electrical pathway and how much is switched to other pathways. We don't want any power in the heating by particles or heating by radiation pathways en route – only when the loop passes the consumer. At this point in the loop, consumer appliances switch to these pathways.

Be consistent and careful in your choice of prepositions

Here are some phrases where we think consistency pays off: consistent use of the following will help children to head along the right lines.

Teacher Tip: … potential difference across X…

Teacher Tip: … current in Y…

Teacher Tip: … power dissipated by X…

Teacher Tip: … energy shifted by Y…

As the job of the prepositions (across, by, in) is to contextualise the named objects, you should not use them interchangeably. In electricity and energy, this is perhaps particularly important because many of the things that you want to locate are intangible, so anything that can help the imagination should be exploited.

When discussing physical circuits, as above, aim always to have a circuit (physical or diagram) so that you can support these phrases by pointing to objects or their representations, to reduce ambiguity.

Always try to communicate with more than words

When discussing situations using the ideas in energy, you may not have any physical object in mind. In particular, pathways and stores may not be associated with any one physical object. So we suggest that it's even more important and helpful to be precise about language:

Teacher Tip: … energy shifted to a store…

(Perhaps, in being slightly more relaxed, one might say, energy in a store, really meaning the energy that has been added to or subtracted from the store.)

Teacher Tip: … power in a pathway…

Support this with clear diagrams or models, where appropriate. We will give examples of this.

Using units carefully can help you reason fruitfully

Calculate the power in an electrical pathway in watts. Remind yourself that one watt is just one joule of energy shifted each second (discussed in detail in the SPT: Energy topic).

Remember also, covered in detail in the SPT: Electric circuits topic, that: one ampere = one coulombone second and one volt = one jouleone coulomb.

Now perhaps you can see why the electrical power has to be calculated by multiplying potential difference and current: just check on the units.

Checking for consistency in units is always a good guide to talking sense about a physical situation: the measures coded by the units are not arbitrary, nor obvious, and took a good deal of hard work to settle on in the early days of investigation of electric circuits.

Because you can only equate or add similar things, the units must be identical, or be able to be shown to be so.

How to find the power dissipated in circuits with only series connections

In episode 01 you saw how to find the current in this loop (with only series connections like this, there is only one value for the current), using the relationship I = VR and then the potential differences across the lamps, using the constraining relationships V₁ = R₁ × I and V₂ = R₂ × I.

Now you need only add to this analysis to find out the power dissipated in each lamp. This value is the power in the electrical working pathway, calculated by P = I × V.

But, and it can be a big but, you need to take great care to use the correct values of V and I. Careful and explicit labelling will help you and those you teach to keep a clear head.

We emphasise again that V, I and R, without modifiers, are best reserved for entire circuits.

Keeping the calculations in order

You have to repeat a sum with variations two or three times, depending on how much you remember about the conservation of energy:

For the top lamp, P₁ = I × V₁.

For the bottom lamp, P₁ = I × V₁.

For the cell, P = I × V.

Now start with E = E₁ + E₂, which is just the conservation of energy, and remember that the energy shifted in any one second is just the power.

You'll get P = P₁ + P₂, and you could use this relationship in place of one of the three multiplications.

You might like to review the process by completing the model provided in the support sheet (see below).

Potential difference, power and safety

As smaller potential differences seem safer, why not use them? Why not have mains wiring at 12 volt throughout the house, rather than 230 volt? And why shift energy around the country using 11 kilovolt, 110 kilovolt and even 400 kV transmission lines? After all, damp skin and 230 V are best kept apart, as explained in episode 01.

Here we'll build a simple model of a circuit, using what you have learnt about working in series circuits to explain why large-voltage, small-current pathways are better suited to shifting large quantities of energy than small-voltage, large-current pathways. The character of the different pathways determines their possible uses, even though the power in both pathways may be identical.

For domestic distribution, we already know that 230 volt is an upper limit, chosen so that lethal currents are not, in normal circumstances, driven through people. The key question here is why a smaller potential difference is not used. For long–range distribution, from power station to town, why use anything greater than 230 volt, if that is the final required potential difference? The answers are linked, and the simple model used illuminates some of the factors to be considered in engineering a system to shift energy from one store to another store, often very remote from the first, using electrical pathways of different characters.

An illuminating calculation

Here you can do calculations using the simple model. These support an account, using the understanding of series circuits developed in episode 01, to explain how some electrical pathways work better than others for shifting energy where you want it.

How to find the power dissipated in circuits with only parallel connections

As with series circuits, start with the analysis in episode 01. This provides values for the currents in the loops and the potential differences across each circuit element. Again, taking care to identify the values, multiply the correct values of potential difference and current to find the power:

For the left hand lamp, P₁ = I × V₁.

For the right hand resistor, P₂ = I × V₂.

For the cell, P = I × V.

Again, you need only do two of the three multiplications to be able to work out the third, by simple addition or subtraction. Once more, the conservation of energy comes to your assistance: the pathway shifting energy from the chemical store and the sum of the pathways shifting energy to the thermal stores are of equal power.

Refer to the support sheet where the calculations are laid out (see below).

Something for nothing revisited

Now you are in a position to revisit the puzzle that adding loops in parallel seemed to give something for nothing (SPT: Electricity and magnetism topic, episode 03). You do get something: the thermal stores are being filled more rapidly, here by the extra glowing provided by the additional lamps. But not for nothing: the chemical store is being emptied at a much greater rate.

The necessary link between the depleted and augmented stores of energy is the electrical pathway, where the power sets the energy shifted to stores as well as from stores over the selected duration. The same power, so the same energy shifted in and out. Energy is conserved: the pathway is the mechanism that links the two stores.

A kitset for thinking about circuits

First you'll have to identify the lkind of circuit you are dealing with.

To analyse a circuit usually means that you calculate all of the potential differences and currents from the cell potential difference and the resistances in the circuit. Sometimes there will be other puzzles, involving designing for a particular current, and so choosing a resistor, but the principles remain the same. From this analysis you will be able to find what the power is in each pathway – simply multiply the current by the potential difference for that pathway.

And here are the simple steps to follow:

1. Identify each loop in the circuit.
2. If a resistor appears in more than one loop, take special care (see episode 01).
3. If more than one resistor appears in a loop, add together the resistances that do appear to give a single equivalent resistance. Then work with one resistor in the loop, of this calculated value.
4. Calculate the current in each loop (loop_n, where n indicates one of the loops), using I_n = VR_n.
5. Add together the currents in each loop to find the current in the battery.
6. You can find out the single equivalent resistance by dividing the battery potential difference by this current.

Keeping the algebra under control

There are two relationships for any given lamp or resistor: V₁ = R₁ × I, the constraint relationship that links the potential difference, voltage and current; and P₁ = I × V₁, which relates the power in the electrical pathway to the potential difference and current.

You might be tempted to combine these, in all kinds of ways. And doing so could be correct, as it is the same potential difference, and the same current.

Such algebraic manipulations could give you:

P = I² × R and P = V²R.

Keeping the physical relationships to the fore

However, although you might derive pleasure from such manipulations, we don't recommend using such complicated short cuts with all children. Such short cuts are for the confident who are already familiar with the countryside, but may often turn into long cuts for those who are not so certain in their navigational abilities. Such manipulations would earn their place in teaching at this level if they helped to focus attention on the physical situation and the fundamental relationships, but we think their purpose is rather to avoid such a focus, jumping to a solution instead.

So these may be best avoided for now, being left for later study, when the fundamentals are much more secure. We suggest that you exploit the opportunity to work and talk through the fundamental relationships in making sense of a circuit. Once your students are comfortable with these, they may then benefit from suitable exposure to the elegance afforded by algebraic manipulations.

Teacher Tip: Teach the physics first.

It is all about power in the electrical working pathway

Think about the effect that your choices in assembling the circuit have on the loops. That's the place to start analysing circuits.

Understanding electrical working is the key to mastering this episode

Carefully set up a circuit to perform a particular quantity of working, and see the energy shifted in a controlled fashion, with the correct quantity accumulating just where you want it. To set it up correctly, you will have to choose the resistances in the different parts of the circuit, and the potential difference of the supply. As you have seen, choosing to place different resistances in different parts of the circuit determines where the electrical working happens, by fixing the currents and potential differences.

Don't forget that the sum of the powers in all of the dissipative pathways must be equal to the energy shifted from the supply in each second. No amount of cunning engineering can violate the conservation of energy.