Putting together a circuit

Imagine yourself setting up a simple circuit. What do you do?

Head for the store-cupboard, and select some components:

• Choose a cell.
• Select a lamp.
• Pick up some wires to connect in a loop.

Connect the different circuit elements together to form one complete loop, and you get an electric current in the loop: flow of charge.

The lamp will light if enough power is dissipated. Too little power dissipated and the filament will warm up, but not enough to glow.

If it doesn't glow, what do you do? Pick up a lamp with a lower resistance (one with a thicker and/or shorter filament) or a cell that provides a larger potential difference – or even both, if you are careful enough. Both changes will increase the current.

Try building a model

Now think through what you do in terms of representing what is happening in a circuit as a computable model:

• You set the potential difference by choosing the cell.
• You set the resistance by choosing the lamp.

The current is fixed by the potential difference and the resistance. You cannot choose the current directly.

There will be other resistances that impede the flow of charge: it won't only be the filament of the lamp. The wires will present some resistance but very little compared with the filament lamps as they are very much thicker and made of materials that conduct very well. That all of the resistance in the loop is very nearly the same as the resistance of the lamp was argued in detail in the SPT: Electric circuits topic.

The cell also has charge flowing in it – the cell is a part of the loop. The materials that the cell is made of also impede the flow. This resistance is also usually not too large, and in well-designed cells it will be close to zero. However, it is often not zero, and cells with a large flow through them may get hot as a result of this internal resistance. More on this in episode 04.

You could build a number of models to emphasise the importance of these constraints.

Teacher Tip: The potential difference may be a new term to you. It was introduced in the SPT: Electric circuits topic, but only as an extension. We suggest that you use it consistently in teaching older children for the quantity measured in volts, which you might previously have called the voltage. By the end of this episode we hope that you will see why it is a more precise and more evocative term.

Now for the physical model

The same choices are made in modelling an electrical loop with a rope loop. The potential difference is modelled by the pull of the person modelling the cell; the resistance by the sliding frictional force applied by the hands of the person modelling the resistor. The steady movement of the rope comes about as a result of these two forces coming into equilibrium. So the flow of rope (metre second^-1) models the flow of charge (coulomb second^-1 or ampere): the current. There is a short settling down period, after which a steady flow of rope past each point represents the steady flow of charge past each point. Electric circuits also have this short settling down period. Then energy is shifted at a steady rate, here modelled by the battery hands getting tired, and the resistor being warmed by the action of the rope.

Again the flow is fixed by the representatives of the two circuit elements: the cell and the lamp. One person chooses how hard to pull, the other how hard to squeeze the rope. The movement of the rope, representing the current, is a result of these actions.

The current depends on the circuit elements that you place in the circuit

The relationship between the voltage, resistance and current in a complete circuit is one of dependence. This can be shown by writing the relationship in these forms:

I = VR

or write it out

current = potential differenceresistance

or include units,

current/ampere = potential difference / voltresistance / ohm

or abbreviate:

I/A = V / VR / Ω

or

I/ampere = V / voltR /ohm

Choose an alternative to suit your class.

Your actions in building the circuit, the computable model and the physical model all follow this pattern.

(As an aside, remember that this algebraic statement shows the interrelationship of three physical quantities: it is not Ohm's law.)

Algebra is much more flexible, allowing you to represent this relationship in many different equivalent forms. However, overly slick use of algebra by a confident student or teacher can undermine physical understanding. So we suggest keeping it simple and direct, not exploiting this flexibility, and consistently writing:

I = VR

Words follow actions

To make things memorable and to emphasise the physics, just follow what you do.

Talk through your action in building a circuit. The relationship between potential difference, resistance and current for the circuit may then make more sense.

Choose a cell, then choose a lamp. Build them into a loop circuit and find that the current in the lamp, and in the wires, and in the cell, is set by these choices.

So VR tells I what to be.

Written more conventionally:

I = VR

Now perform two quick checks:

As V gets closer to zero, so I gets closer to zero.

As R gets closer to zero, so I becomes very large.

These simple checks show that the relationship correctly represents what you observe in the circuits.

What you can choose:

• A potential difference: select the cell.
• A resistance: choose a resistor or lamp.

What you cannot choose:

• A current – that is the result of your first two choices.

Safety through restricted currents

A potential difference of 230 volt is quite safe, but this safety is a trade-off for lower efficiency (we explain why it is more efficient in the next episode).

The hazard posed by any electrical outlet is determined largely by the potential difference. Quite small electrical currents through the human heart can cause it to malfunction, with terminal consequences for the owner. (Death from electric shock is called electrocution.) However, your resistance is quite high (mostly your skin). The largest potential difference that can be applied across you without driving a lethal current in you can be calculated, by modelling you as the resistor in an electrical loop. The outlet itself might also provide some resistance in the circuit, and this internal resistance (internal to the electrical source, or power supply) is the other factor that determines the safety of the outlet.

How much current is lethal depends on the path that the charged particles take as they drift through your body – paths that include the head or heart require lower currents to cause death. 0.1–0.3 ampere is an approximate value for a lethal current: this is thought to vary with the individual as well as with the path. The smallest current that you can feel is about 0.001 ampere (1 mA) – you'd report this as an electric shock (after electrocution you wouldn't be alive to make the report).

Dry skin has a resistance of approximately 100,000 ohm, but wet skin can be 100 times less than this, so only 1000 ohm. The difference made by so little moisture is a major reason for the folk wisdom, captured in the aphorism: electricity and water don't mix. It also explains why normal light switches should not be mounted inside bathrooms. Pure water is not a very good conductor, but you only need a few ions in the water (and the body provides plenty as it perspires) to make it rather a good conductor, as the ions are very mobile, carrying charge as they move. As you are 90 % water, with many ions in your internal transport systems, the electrical resistance of your internal organs is tiny compared with the resistance of your skin. So this simple model, ignoring the resistance of the internal organs, provides a sufficiently accurate prediction of the current.

Sums in the wet and in the dry

Armed with these data and the model, calculations reveal the wisdom of choosing 230 volt.

First calculate for a circuit where dry skin provides all of the resistance.

I = VR

I = 230 volt100 000 ohm

The current is 0.00230 ampere, which is an electric shock.

Now the same calculation, but for wet skin.

I = VR

I = 230 volt1000 ohm

The current is now 0.230 ampere, which is likely to lead to electrocution, if the loop includes the heart.

In other countries, the domestic supply voltage is 120 volt – safer, but less efficient (we will explain more later, in episodes 03 and 04).

For transmission over long distances, the supply voltage is much higher: 11 kilovolt, rising to 110 kilovolt, and even 400 kilovolt for very long distances. This is more efficient, but has to be kept farther away from humans – usually on high pylons. We'll return to the reasons why it is more efficient in episode 02.

These models are simple, and there are significant and complex effects if the current is alternating, rather than direct, as we have assumed. However, at low frequencies, such as are used for the mains electricity supply, these effects do not alter the main conclusions.

One telling indicator of the success of limiting domestic supplies to 230 volt is that more than 90 % of electrocutions each year happen in the workplace, where machinery is routinely supplied with potential differences greater than 230 V.

Safety through restricted currents

An alternating current is driven first one way and then the other by the alternating potential difference: the polarity of the terminals of the supply alternate. Nowadays, nearly all long-distance transmission uses alternating current, but it wasn't always so. In the early days of domestic electrical supply, the proponents of direct and alternating current were locked in a battle for dominance. One device that swung the balance was the transformer. This allowed energy to be shifted by the electrical pathway much more efficiently. The transformer alters the efficiency by altering the character of the pathway. You'll see what this alteration means in episode 02 and how the transformer effects the alteration in episode 03.

As modern houses are so full of electronic devices that require direct current to work, several people have suggested that we might return to having 12 volt sockets in houses. Again, these suggestions have been made to improve efficiency. The arguments one way or the other need to be made carefully.

The differences between alternating and direct current are rather simple. Charge drifts in one direction only in direct current: the charged particles all move round the circuit. In alternating current, the net movement of the charge as a result of the current is exactly zero. So how can the charged particles do anything? Sit on a boat on the surface of the sea and you don't see water moving from left to right, drifting steadily one way, yet you do get lifted up and let down as each wave passes. Influence can be carried without the bulk movement of matter.

The rope model, so useful for accounting for how energy is shifted for direct current circuits, can model this even more closely than the analogy with water waves. We hope you can see how. Think a little for now, then move on to the next step.

Modelling alternating current

The heating effects of an alternating current can still be modelled with the rope loop, just as for direct current. Consider the modelling of a battery: instead of applying a steady driving force, to counter the resistive force offered by the resistor, apply an alternating force.

Pull first one way and then the other. The hand modelling the resistor is passive as before: we don't need to change a resistor to put it in this circuit. The rope will no longer move steadily but will, instead, move to and fro in response to the pair of forces (provided by the battery and the resistor).

Results: the rope moves to and fro, and the resistor hands get warm – all as in the direct current model.

Predictions from the model: in spite of the average potential difference being zero, and the average current being zero, energy can be shifted by the action of an alternating current.

New measurements for existing quantities

As a consequence of the similarities between the alternating and direct current cases, we choose to find new ways of measuring alternating current and alternating potential difference so that the values give the same energy shifted as if the measurements were done in a direct current circuit. 1 volt predicts 1 joule per coulomb, for alternating or direct current. It is just that you mustn't take the arithmetic average of the potential difference over time as this would be zero (the rope spends as much time being pulled one way as the other: so equal lengths of time moving one way or the other). A more complex average is used that accurately predicts the power in the electrical pathway.

A catch-all term for complex interactions

The resistance of a circuit element is a term that summarises many complicated microscopic interactions between the flowing charge carriers and the material, the result of which is to impede the flow of charge. In this way it is just like sliding friction, which gives a tidy macroscopic description of many messy dissipative interactions between two moving surfaces that impede movement.

However, resistance isn't force-like. There is no active opposition to the action of the voltage. The current is reduced by increased resistance, but it cannot be reduced to zero.

The dissipative effects increase with increased flow precisely because there is resistance.

That it is a dissipative effect suggests a connection with shifting energy, which in turn suggests thinking about potential difference. As the flow increases in a resistor, so a larger potential difference appears across the resistor.

The potential difference across the resistance appears when there is a flow of charge, and, where there are only two circuit elements, it is always balanced by the potential difference provided by the cell. This pair of differences produce a steady flow: a constant current. A steady current in the circuit is the result.

As complicated as drag

A more complete understanding of resistance will allow a connection to be made between the structure of the material through which the charged particles pass to the value of the resistance. The best basis we have is quantum theory, which predicts the forces on the electrons by calculating the interactions between them and the material using very advanced mathematics. (This is not a simple story about charged particles bouncing off atoms.) However, it does predict the values of the resistance (even if not very well), which the colliding particles model (perhaps appropriate as a story for 11–14 year olds) doesn't.

Perhaps we should no more try to give a detailed account of resistance than of drag forces. Both are a macroscopic summary of many complex interactions, and all the more useful for that. We can make carefully engineered components that impede the flow more or less independently of the current. These are resistors. Putting one of these in a constant flow sets up a potential difference across the resistor as a result of this impeding action.

A constrained triplet: I, V and R

Potential difference, current and resistance are constrained. All three are interlinked, in that changing any one is likely to change at least one of the others – but which one, and how?

You need to do some careful physical reasoning about this question to get sensible answers. Take particular care to identify the resistance, voltage and current under consideration.

Let's start with a complete circuit, as this is what we have studied so far. We have already discussed this: just consider what you do in setting up the circuit. Clearly the current responds to changes in the other two. The resistance and the potential difference jointly set the current. There is only one current, one resistance and one potential difference. Changing either the potential difference or the resistance, or both, (very nearly) instantaneously changes the current – certainly the change is faster than you can measure with school meters. The change is not a straightforward jump from one state to another – there is a bit of a fluctuation, as the shock wave caused by the change ripples around the circuit (at nearly the speed of light; the teaching approach using the big circuit in the SPT: Electric circuits topic emphasises this speed of change). So, I = VR is a connection that is true all at one time. It is not a case of first one, then the other in this relationship. The three quantities are constrained, so you can only change the current through changes to either of the other two quantities.

Calculating for a resistor that is a part of a loop

But what about doing calculations for a resistor that is just a part of a circuit loop? First use the relationship I = VR for the complete loop to find the current (we'll always use simple R for the resistance of the complete loop and I for the current in that loop). We'll leave the details of how to calculate the value of R from many resistors until later. For now it's important to get the idea that one resistor can replace many, without changing the current in the loop.

Then focus on a resistor that is one of many in a loop. You have chosen the resistor and the current is known. So now use the all-at-one-time connection linking R₁, V₁ and I arranged as V₁ = R₁ × I to find the potential difference across this one resistor. We'll always use V₁, V₂, etc and R₁, R₂, etc for potential differences and resistances that contribute to a whole loop. We suggest that you are also consistent about this, to help keep track of what is being discussed.

So you need to take care to deal either with the whole loop, using I = VR, or with just a component in the loop, in which V₁ = R₁ × I is often useful.

See support sheet for sample calculation laid out (see below).

Designing a circuit to shift energy

Electric circuits are very useful because we can design them to shift energy in a controlled way. You can measure or predict both current and potential difference. Combine these two measures to calculate which circuit components will shift this energy.

One part of the loop where we do not want energy shifted is the wires. As we have no choice but to have current in those wires, we need to fix it so that the potential difference across the wires is as small as possible. This is true for all circuits that are designed to shift energy – from those on the bench top to those serving major cities as a part of the National Grid. The current in the loop is the same everywhere, so make the resistance of the energy-shifting components large compared with the resistance of the wires. Ideally the resistance is so much larger that the potential difference across the wires is tiny. You can do this by making the wires as thick as you can afford, and from a good conductor.

The importance of difference in potential difference

Energy is shifted wherever there is a potential difference, so identifying these differences is useful. Measure the potential difference from a chosen zero (usually the negative terminal of the battery or cell in simple circuits).

This is very much like mapping out the terrain – you have to choose a zero (for the UK Ordnance Survey, this is at Newlyn in Cornwall). Then you can choose to report spot heights, draw contours or shade within bands. All are used in map making. Traversing a circuit (walking, riding or driving) results in a climb and a fall to leave you back at the same height.

For these maps, the energy changes are calculated for a particular rider and bicycle. You could equally well present a map that would work for any bicycle and rider – just quote the energy change for each kilogram hauled up or down the hill. Multiply by the mass of the particular bike and rider and you'll have the energy shifted. The energy change per kilogram is the gravitational potential difference, and the concept is similar to the electrical potential difference. This measure was introduced in the SPT: Electric circuits topic as the number of joules/coulomb or the voltage.

Up and down hill

Think of cycling round such a loop. As you climb, so you are shifting energy; as you freewheel downhill, so more energy is shifted. It is the act of changing height that alters the energy in each store.

Hillwalkers interpret topographical maps to generate similar expectations: their chemical stores will be depleted as they gain height.

A potential difference across a circuit element shows that energy will be shifted by that component. This is why the hills analogy, introduced in the SPT: Electric circuits topic, was so useful.

Being at a particular height has no implications for shifting energy; nor does being at a particular potential. Engineers assess the viability of micro-hydro schemes by looking for changes in height, not just high places.

Differences in height and potential differences set where the energy is shifted

Freewheeling downhill or toiling uphill shift energy: it's not the being uphill but the getting there that determines the energy shifted. The same applies to electrical circuits – the differences in potential are the key to calculating the energy shifted.

As you step across the circuit elements in the same direction as the current in the element, the potential difference can be positive or negative. A positive potential difference shows that an energy store is being emptied by the electrical working; a negative potential difference shows that an energy store is being filled by the electrical working. As there is no energy accumulation in the circuit, the increases in potential difference must be equal to the decreases in potential difference. So in any one loop, you end up back at the potential that you started with. The sum of the potential differences around any one loop is zero. (In the study of physics post-16, this is called Kirchhoff's first law.)

Crossing countour lines and potential difference

Go round an electrical loop, starting anywhere, and you end up back where you started – the change in energy in the gravity store is zero, so you also end up with a potential difference of zero. This should not be a surprise: going for a circular walk doesn't result in a change in height, even though there may be plenty of uphill and downhill along the way. Crossing contour lines warn you of the working mechanically: the potential difference performs the same function for a journey along part of an electrical loop. It's always potential differences, and if someone does not mention where the difference is measured from, then there is a good chance that they forgot to mention that they took the negative terminal of the battery as a zero, just like the Ordnance Survey use Newlyn.

One big difference with the contour lines is that gravity is always there. The these electrical potentialal differences, perhaps highlighted by colouring in parts of the circuit, only exist while there is something flowing round the circuit: disconnect the cell and you have no electrical hills. Another is that electrical working depends on complex interactions between many charged particles and is not at all like having lifted a mass up a hill and parked it there.

This might be a very good reason not to employ the hills analogy in your teaching, and even to consider whether colouring the circuits to reveal potential difference earns its keep.

Teacher Tip: Potential difference is one of the hardest ideas taught in pre-16 physics and you need to think carefully about the models or analogies you use. There is further discussion in the teaching and learning issues strand.

Aim for a coherent and consistent account of energy

One reason for controlling the flow of charge round an electrical circuit is so that energy can be shifted to the particular stores that you want. In the intricacies of modelling simple electrical loops, it's probably important to re-emphasise this reason, asking questions about what happens over a duration for which there is a flow. As the modelling and calculations are all for constant currents, ask questions that explicitly take a slice of duration from within this steady state. As emphasised in the SPT: Energy topic, you will need to pose the questions carefully, specifying clear start and end points, and describing the physical situation carefully, before asking for the energy description. Ask about the energy shifted as a ball bounces and you ought to expect a wide variety of answers. Ask again, but this time specifying exactly where the ball starts and where it ends up, and you can much more reasonably expect a very limited range of answers.

Please avoid electrical energy and light energy, as these are not stores. The reasoning is given in the SPT: Energy topic. (In episode 02 there is more on the flow of energy through electrical working.) The circuit doesn't contains energy in an electrical store that it shifts to the thermal store: the rope loop doesn't contain a supply of energy in a kinetic store that is depleted to warm the hands of the individual modelling the resistor. Both the electrical loop and the mechanical loop provide a physical mechanism that connects changes in one store with changes in another – both are examples of pathways.

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 R₁ and R₂, 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: V₁ across R₁, V₂ across R₂, 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 = R₁ + R₂.

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 = VR

Now step back to the original circuit. You can calculate the potential difference across each resistor by using the relationship between V₁, R₁ and I, or V₂, R₂ and I.

V₁ = R₁ × I
V₂ = R₂ × 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.

Making any parallel connection increases the current

Adding more routes for the charge to take will result in an increased flow, so more current, as there will be less resistance. The same potential difference is available for each route, so, in addition to the current driven down the original route, there will be additional current for each route added. More charge is set in motion in each extra loop. The end result is that more loops lead to more charge in motion – a larger current in the battery. As the potential difference hasn't changed, this can only mean that the resistance of the entire circuit has dropped. As you add more routes, the resistance of the whole network falls further.

One step in analysing a circuit is to reduce it to a single loop: the whole network of resistors is replaced by a single resistor that results in the same current in the battery as the network it replaces. If the network is made with parallel connections, so providing additional routes, then the single resistor that replaces the network will always be of a lower value than any resistor in the network.

Separate loops are the key to unlocking parallel circuits

Circuits with only one parallel connection are best analysed by thinking of the two separate loops that make up the circuit. The introduction to parallel connections in circuits was made in this way in the SPT: Electric circuits topic. The loops have nothing in common, other than the battery, so you can treat them as two separate loops.

Then simply use the relationship

I = VR

for each loop to find the current in each. The potential difference and the resistance in the loop together set the current in each loop. The current in the battery is the sum of the current in each of the loops, as the battery is in both loops. The interactive keeps things very simple because the two arms of the parallel circuit are identical.

You can model this process rather effectively using a modelling package, to show on a single screen all of the calculations that you perform and the relationships between the different currents, resistances and the potential difference. Here you can easily vary the resistance of the two loops independently.

Domestic circuits

Domestic circuits are more complex than those studied in the school laboratory. As with alternating current, however, simple models of these circuits illuminate the principles. Here we show how inserting fuses into circuits and earthing those circuits both contribute to making the circuits safer for us to use. Understanding more modern devices, such as miniature circuit breakers (MCB) – often now used to replace fuses – and earth leakage circuit breakers (ELCB), takes us beyond simple models of circuits, so these are not explained here.

A wire warms a little when there is an electric current in the wire. A larger current warms the wire more. And very large currents in a wire makes the wire very hot. It is also the case that large currents often indicate that the appliance is not working properly, suggesting that a fault has developed. Later, in episode 03, we find out how to calculate the normal working current for an appliance. If the current goes above that value, there may be a fault, so that we want to break the circuit, and isolate the appliance from the supply. That is the job of the fuse: it is a current-sensitive switch, and a very simple one at that. To make a fuse, place a short piece of wire that melts at a low temperature in the circuit. Once the wire has melted, the circuit is broken. The switch is irreversible, and the fuse must be thrown away and replaced. This is a cheap price to pay in return for enhanced safety. A correctly chosen fuse, which doesn't melt until the current exceeds the normal working current for the appliance, is a one-time switch. In practice there are many different working currents for difference appliances, and fuses are made with only a limited number of safe working (rated) currents. Once these rated currents are exceeded, the fuse melts. So to choose a fuse, calculate the normal working current for the appliance and then select the smallest rating of fuse that exceeds this value. Fuses are typically available with ratings of 3, 5, 10 and 13 ampere.

Although fuses insert an infinite resistance into any loop with a fault, they are a bit slow to react. There is a current in the loop whilst the fuse is melting. That loop might well include a human, and that's not a good thing. So now far faster and more senisitive current-operated switches are used (earth leakage circuit breakers), but these have more complex machanisms.

Building a model of an earthing circuit

Many UK domestic circuits have three wires: live, neutral and earth. Live and neutral do the same jobs as the positive and negative terminals of the battery, setting charge in motion round a loop, of which these terminals are a part. But what is the function of the earth wire? It is again a safety feature, and specifically one for appliances that have conducting surfaces that a user can touch. These include: toasters, kettles with metal casings, microwave ovens, conventional ovens, hobs, and many fridges and freezers. The danger that the earth wire is designed to protect against is that a fault develops inside the appliance in a way that a human could become a part of a circuit, and so have a lethal current in their body. Such faults can be of two kinds: one where a wire directly connected to the live terminal becomes connected to the conducting surface open to the human, and one where a wire directly connected to the neutral terminal touches that surface. In between the live and neutral terminals is the device at the core of the appliance, where the energy is dissipated when it is working correctly. So you might guess that the first condition, where the human is effectively touching the live wire from the socket, is likely to be more dangerous. You'd be right. But why is this?

Parallel connections are the key to understanding earthing circuits

To find out more, you have to think about circuits with parallel connections. These have several loops. When working normally, there is only one loop between live and neutral. If a wire makes a connection with the outer metal case, an extra loop is created between live and earth, or between neutral and earth, depending again on which side of the dissipating core the fault happens. If, in addition to this fault, someone touches the exposed conductor, there is another loop between live and earth, or neutral and earth, again depending on which side of the core the fault happens. There is only one key question to establish the safety of the appliance under these conditions: How much current is there in the individual? To find the answer, consider each loop in isolation, as with analysing any collection of parallel connections in a circuit.

However, to tidy up these loops further, first you need to know that the neutral wire is connected to earth at the power station, and so a direct current circuit model that connects the earth and neutral terminals represents what happens rather well. It doesn't work so well in providing safety features in real appliances, although for a while the earth and neutral wires were combined in the USA (to save copper, which was required for the war effort). In our model there shouldn't be a battery between the earth and neutral terminals – perhaps only some rather poorly conducting wire.

Now to explore and exploit the constructed model

Use the interactive to explore the model of earthing circuits that we have developed. See how adding the extra earthing loop makes appliances safe. Note that a fuse is drawn as the safety device here, but that it could equally well be a faster and more complex device, better able to cut off the current quickly enough so that the human does not get a shock. The earth wire fulfils the same function in the case of this substitution.

Appliances that do and do not use earthing circuits to enhance safety

Some appliances don't have the earth wire connected. These same appliances are often referred to as being double-insulated. The reason is that the surfaces that the user can touch are all insulators, so there is no accessible conductor that could provide a link from the live or neutral wire to them. Typically, the outer surface is moulded plastic (e.g. a kettle). They do still have fuses, so that only the designed amount of power is switched in the device: more power than the deisgned value and you might get a fire.

Miniature circuit breakers are used to replace fuses,as they are faster, so they too are current-limiting switches – once the current in the circuit protected by the circuit breaker rises above a certain preset value, the switch is opened. However, unlike a fuse, the switch can be closed again because the circuit breaker can be reset. There are usually two active parts in the circuit breaker: an electromagnetic switch that responds to sudden and large increases in current and a thermal switch that responds to smaller and more slowly changing excess currents.

Earth leakage circuit breakers again function as switches, but they work a little differently. The current in the live wire is compared with the current in the neutral wire, by comparing the magnetic fields produced by coils wound using both of these wires (more on this in the SPT: Magnetism topic). If the electrical loop is complete and there are no additional loops (say via earth), then these two magnetic fields and currents will be the same. The sameness of the currents is just elementary circuit theory, as emphasised in the SPT: Electric circuits topic. If the currents are not the same then the live–neutral circuit is not a single complete electrical loop and so the circuit is cut. The difference in current can be very small – that is what makes these devices more sensitive than fuses, and is another reason that they are better at protecting humans.

Teach physics – not algebra

The physics relied on in the arguments here depends on thinking about the flows resulting from the resistors impeding the flow of charge and the batteries driving the flow of charge.

This understanding is very physical: the effectiveness of the rope model exploits these relationships. Children are more likely to develop a feel for how current, voltage and resistance interact in a circuit if they are both challenged to think in these physical terms and shown how to think in this way.

To understand circuits is to imagine, without pain, how the currents and potential differences will behave. The currents and potential differences are seen to behave according to their characters. Being able to get inside these is the key to making friends with circuits.

Don't rely on memorised rules and algebraic tricks to jump to an analysis of a circuit. These shortcuts are for experts, not for beginners. Keep all reasoning as physical as possible.

Teacher Tip: Reason physically – use algebraic relationships only where these support this physical reasoning.

Shortcuts – for teacher's private pleasure?

There are only a few quantities in these three element circuits containing either only series or only parallel connections:

V: potential difference across the battery

R: effective resistance of the circuit

I: current in the battery

R₁: resistance of first resistor

I₁: current in first resistor

V₁: potential difference across first resistor

R₂: resistance of second resistor

I₂: current in second resistor

V₂: potential difference across second resistor

But you can still do some rather involved algebra to give yourself shortcuts to finding any one of these quantities, given a number of others. These shortcuts, once committed to memory, enable you to avoid working through all of the steps necessary to analyse the circuit, starting from the physical behaviours.

Deriving the shortcuts might give you pleasure, which is why they are included here, but these shortcuts are only useful if you know the lie of the land and where you are going. Children are not in this position, so it's probably better to keep working through all of the steps. This has the further advantage of keeping the classroom discussions close to the physical behaviours.

How energy is shifted in circuits with different connections

Two lamps connected in parallel each glow as brightly as one connected by itself. By now you might expect this, because lamps are often used as informal ammeters in classrooms, and the potential difference is the same across both lamps. So twice as much energy is being shifted in the circuit with parallel connections compared with the simple circuit with only one lamp. The consequence is that the chemical store of energy associated with the battery is depleted twice as much after the circuits have been running for identical lengths of time.

Two lamps connected in series deplete the battery much less, because the current in the circuit is smaller than in the circuit with only a single lamp and battery, and the potential difference is the same for both circuits. Less energy is shifted by the two lamps compared with the single lamp.

Power in pathway and time determine energy shifted to store

The same patterns of dissipation of energy apply to resistors, only these don't glow (at least visibly, but they may radiate at frequencies below those that we can see, perhaps in the infrared). Resistors shift some energy to a thermal store using the heating by radiation pathway, and typically much more energy using the heating by particles pathway.

To calculate exactly how much energy is shifted during 1 second in each lamp in all of the situations, you need to determine how much electrical working is happening: the power in the electrical pathway. That is the focus of episode 02.

Pulling the model together

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

Modelling with loops

To analyse a circuit containing several resistances, reduce it to a single equivalent loop containing only one resistor. A circuit is equivalent when the current in the battery is the same.

For series connections, add the resistances to find the equivalent resistor. Then find the current. This current can then be related to the potential difference across each resistor by using one of the two constraining relationships: V₁ = R₁ × I; V₂ = R₂ × I.

For parallel connections, treat the two loops separately and then add the current in each loop to find the current. From the current and potential difference you can calculate the required single equivalent resistor.

The current in and the potential difference across each element of the circuit set how much energy is shifted by that element over any fixed duration.

Circuits with both kinds of connections – nothing new, just loops

Compound circuits, containing both parallel and series connections, require more sophisticated analysis. This analysis still uses loops, which is part of the reason why they are emphasised here.

In particular, a resistor (or lamp) that is part of two separate loops will have two contributions to how its resistance constrains the current and potential difference. The complexities that arise make this more suited to post-16 study because more difficult algebraic manipulations are essential.