The Coulomb Train Model (CTM) is a straightforward, easily pictured representation that helps novice learners develop an initial “sense of mechanism” about how electric circuits work. You can read about it here and here (and, to some extent, track its development over time).
In this post, however, I want to focus on how effective the CTM is in helping students understand the energy and power formulas associated with electric circuits: notably E = QV,P = IV and P=I2R.
E=QV and the CTM
The E in E=QV stands for the energy transferred to the bulb by the electric current (in joules, J). The Q is the charge flow in coulombs, C. The V is the potential difference across the resistor in volts, V.
Using the Coulomb Train Model:
Each grey truck passing through the bulb represents one coulomb of charge flow. Q is therefore the number of grey trucks passing through the bulb in a certain time t. (We won’t specify what that time t is now but we will return to it shortly.)
The potential difference V is the energy transferred out of each coulomb as they pass through the bulb. If one joule (represented by the orange stuff in the truck) is transferred from each coulomb then the potential difference is one volt. If two joules then the potential difference is two volts, and so on.
How can we increase the energy transferred into the bulb? There are two ways:
Increase the total number of coulombs passing through the bulb. That is to say, increasing Q. We could do this by (a) waiting a longer time so that more coulombs pass through the bulb; or (b) increasing the current so that more coulombs pass through each second.
Increase the energy transferred from each coulomb into the bulb. That is to say, increasing V. We could do this by increasing the potential difference of the cell so that each coulomb is loaded up with more energy.
Or, of course, we could increase the values of Q and V simultaneously.
All you need is E = Q V
In other words, the energy transferred per second (or the power P in watts, W) is equal to the product of the current I in amperes (or coulombs per second) and the potential difference V in volts, V.
A higher current will increase the power transferred to the bulb: more coulombs will pass through the bulb per second so more energy is transferred to the bulb each second. This can be modelled using the Coulomb Train Model as shown:
Increasing the potential difference V (i.e. the energy carried by each coulomb) would also increase P.
Deriving P=I2R from P=IV
If we start with P=IV but remember that V=IR then P=I(IR) so P=I2R.
This can be represented on the Coulomb Train Model like this:
We can increase the power transferred to the resistor by:
Increasing the value of the resistor (and keeping I constant, which implies that V would have to be increased). Doubling the value of R would double the value of P.
Increasing the value of I. However, since the formula includes I squared then this would have a disproportionate effect on P. For example, if I was doubled then P would be quadrupled. A Hewitt representation can be useful for highlighting this to students; for example:
Linking the electrical power formulas
The electrical power equations when considered in isolation can seem random and unconnected. Making the links between them explicit can be not just a powerful aid to memory, but also hints at the power and coherence of that noble exploration of reality and possibility called physics.
As noted earlier, some students struggle with unit conversions. To take a simple example: if we need to convert 3.7 kilojoules (or ‘killer-joules’ as some insist on calling them *shudders*) into joules, then whilst many students know that the conversion involves applying a factor of one thousand, they do not know whether to multiply 3.7 by a thousand or divide 3.7 by a thousand.
Michael Porter shared a brilliant suggestion for helping students over this hurdle. He suggests that we break down the operation into two parts:
Consider if we are making the unit larger or smaller.
If making the unit larger, we must make the number smaller to compensate; and vice versa.
Let’s look at using the Porter system for the example shown above.
(Note: I have used kilojoules for our first example since, at least for GCSE Science calculation contexts, students are unlikely to have to convert kilograms into grams. This is because, of course, the kilogram (not the gram) is the base unit of mass in the SI System.)
By changing from kilojoules to joules we are making the unit smaller, since one kilojoule is larger than one joule.
To keep the measured quantity of energy the same magnitude, we must therefore make the number part of the measurement bigger to compensate for the reduction in size of the unit.
This leads us to the final answer.
Now let’s look if we had to convert 830 microamps into amps:
The strange case of time
Obviously 1 minute is a very small quantity of time compared with a whole week. Indeed, our forefathers considered it small as compared with an hour, and called it “one minùte,” meaning a minute fraction — namely one sixtieth — of an hour. When they came to require still smaller subdivisions of time, they divided each minute into 60 still smaller parts, which, in Queen Elizabeth’s days, they called “second minùtes” (i.e., small quantities of the second order of minuteness).
Silvanus P. Thompson, “Calculus Made Easy” (1914)
It is probable that the division of units of time into sixtieths dates back many thousands of years to the ancient Babylonians(!) Is it any wonder that some students find it hard to convert units of time?
We can use the Porter system to help students with these conversions. For example, what is 7 hours in seconds?
This type of diagram is, I think, very useful for showing students explicitly what we are doing.
The S.I. System of Units is a thing of beauty: a lean, sinewy and utilitarian beauty that is the work of many committees, true; but in spite of that common saw about ‘a camel being a horse designed by a committee’, the S.I. System is truly a thing of rigorous beauty nonetheless.
Even the pedestrian Wikipedia entry on the 2019 Redefinition of the S.I. System reads like a lost episode from Homer’s Odyssey. As Odysseus tied himself to the mast of his ship to avoid the irresistible lure of the Sirens, so in 2019 the S.I, System tied itself to the values of a select number of universal physical constants to remove the last vestiges of merely human artifacts such as the now obsolete International Prototype Kilogram.
However, the austere beauty of the S.I. System is not always recognised by our students at GCSE or A-level. ‘Units, you nit!!!’ is a comment that physics teachers have scrawled on student work from time immemorial with varying degrees of disbelief, rage or despair at errors of omission (e.g. not including the unit with a final answer); errors of imprecision (e.g. writing ‘j’ instead of ‘J’ for ‘joule — unforgivable!); or errors of commission (e.g. changing kilograms into grams when the kilogram is the base unit, not the gram — barbarous!).
The saddest occasion for writing ‘Units, you nit!’ at least in my opinion, is when a student has incorrectly converted a prefix: for example, changing millijoules into joules by multiplying by one thousand rather than dividing by one thousand so that a student writes that 5.6 mJ = 5600 J.
This odd little issue can affect students from across the attainment range, so I have developed a procedure to deal with it which is loosely based on the Singapore Bar Model.
One millijoule is a teeny tiny amount of energy, so when we convert it joules it is only a small portion of one whole joule. So to convert mJ to J we divide by 1000.
One joule is a much larger quantity of energy than one millijoule, so when we convert joules to millijoules we multiply by one thousand because we need one thousand millijoules for each single joule.
In time, and if needed, you can move to a simplified version to remind students.
Strangely, one of the unit conversions that some students find most difficult in the context of calculations is time: for example, hours into seconds. A diagram similar to the one below can help students over this ‘hump’.
These diagrams may seem trivial, but we must beware of ‘the Curse of Knowledge’: just because we find these conversions easy (and, to be fair, so do many students) that does not mean that all students find them so.
The conversions that students may be asked to do from memory are listed below (in the context of amperes).
The burned hand teaches best. After that, advice about fire goes to the heart.
J. R. R. Tolkein, The Two Towers (1954)
As is often the case in an educational context, and with all due respect to Tolkein, I think Siegfried Engelman actually said it best.
The physical environment provides continuous and usually unambiguous feedback to the learner who is trying to learn physical operations . . .
Siegfried Engelmann and Douglas Carnine, Theory of Instruction (1982)
I am going to outline a practical approach that will help students understand that black objects are good emitters and good absorbers of infrared radiation.
What I propose is a simple, inexpensive and low risk procedure (similar to this one from the IoP) that won’t actually inflict any actual “burned hands” but will, hopefully, through a clever (imho) manipulation of the physical environment, speak directly to the heart — or at least to students’ “sense of mechanism” about how the world works.
Half human and half infrared detector
Obtain tubes of matt black and white facepaint. (These are typically £5 or less.) Choose a brand that is water based for easy removal and is compliant with EU and UK regulations.
We also need a good source of infrared radiation. Some suppliers such as Nicholl and Timstar can supply a radiant heat source that is safe to use in schools. Although these can be expensive to purchase, there may already be one hiding in a cupboard in your school. If you don’t have one, use a 60W filament light bulb mounted in desk lamp (do not use a fluorescent or LED lamp — they don’t produce enough IR!). Failing that, you could use a raybox with a 24W, 12V filament lamp to act as the infrared source. [UPDATE: Paul Bushen also recommends a more economical option — an infrared heat lamp.)
Use the facepaint to make 2 cm by 2 cm squares on the back of one hand in black and in white on the other. Hold each square up to the infrared source so they are a similar distance from it.
Hold the hands still in front of the source for a set time. This could be anywhere between five seconds and a few tens of seconds, depending on the intensity of the source. You should run through this experiment ahead of time to make sure that there is minimal risk of any serious burns for the time you intend to allocate. If you are using rayboxes then you might need a separate one for each hand.
The hand with the black paint becomes noticeably warmer when exposed to infrared radiation. We can deduce that this is because the colour black is better and absorbing the infrared than the white colour.
Energy is being transferred via light into the thermal energy store of the hand.
We can use a black painted hand as a rudimentary detector for infrared. The hotter it gets, the more infrared is being emitted.
Enter Leslie’s cube . . .
Fill a Leslie’s cube with hot water from a kettle and then get students to place the hand with the black square a couple of centimetres away from the black face of the cube. After a few seconds, ask them to place the same hand by the white face of the cube. (Although, for the best contrast, you should maybe try the polished silver side). Make sure the student’s hand does not actually touch the face of the Leslie’s cube, otherwise they may end up with an actual burned hand!
The fact that the black face emits more infrared radiation is immediately directly perceivable by the “infrared detector” hand which feels distinctly warmer than when it’s placed next to the black coloured face rather than the white face.
This procedure is, I think, more convincing to many students as opposed to merely using (say) a digital infrared detector and reading off a larger number from the dark side compared to the white side.
It is a thing plainly repugnant . . . to Minister the Sacraments in a Tongue not understanded of the People.
Gilbert, Bishop of Sarum. An exposition of the Thirty-nine articles of the Church of England (1700)
How can we help our students understand physics better? Or, in more poetic language, how can we make physics a thing that is more ‘understanded of the pupils’?
Redish and Kuo (2015: 573) suggest that the Resources Framework being developed by a number of physics education researchers can be immensely helpful.
In summary, the Resources Framework models a student’s reasoning as based on the activation of a subset of cognitive resources. These ‘thinking resources’ can be classified broadly as:
Embodied cognition: these are simple, irreducible cognitive resources sometimes referred to as ‘phenomenological primitives’ or p-prims such as ‘if-resistance-increases-then-the-output-decreases‘ and ‘two-opposing-effects-can-result-in-a-state-of-dynamic-balance‘. They are typically straightforward and ‘obvious’ generalisations of our lived, everyday experience as we move through the physical world. Embodied cognition is perhaps summarised as our ‘sense of mechanism’.
Encyclopedic (ancillary) knowledge: this is a complex cognitive resource made of a large number of highly interconnected elements: for example, the concept of ‘banana’ is linked dynamically with the concept of ‘fruit’, ‘yellow’, ‘curved’ and ‘banana-flavoured’ (Redish and Gupta 2009: 7). Encyclopedic knowledge can be thought of as the product of both informal and formal learning.
Contextualisation: meaning is constructed dynamically from contextual and other clues. For example, the phrase ‘the child is safe‘ cues the meaning of ‘safe‘ = ‘free from the risk of harm‘ whereas ‘the park is safe‘ cues an alternative meaning of ‘safe‘ = ‘unlikely to cause harm‘. However, a contextual clue such as the knowledge that a developer had wanted to but failed to purchase the park would make the statement ‘the park is safe‘ activate the ‘free from harm‘ meaning for ‘safe‘. Contextualisation is the process by which cognitive resources are selected and activated to engage with the issue.
Using the Resources Framework for teaching
I have previously used aspects of the Resources Framework in my teaching and have found it thought provoking and helpful to my practice. However, the ideas are novel and complex — at least to me — so I have been trying to think of a way of conveniently organising them.
What follows in my ‘first draft’ . . . comments and suggestions are welcome!
The RGB Model of the Resources Framework
The red circle (the longest wavelength of visible light) represents Embodied Cognition: the foundation of all understanding. As Kuo and Redish (2015: 569) put it:
The idea is that (a) our close sensorimotor interactions with the external world strongly influence the structure and development of higher cognitive facilities, and (b) the cognitive routines involved in performing basic physical actions are involved in even in higher-order abstract reasoning.
The green circle (shorter wavelength than red, of course) represents the finer-grained and highly-interconnected Encyclopedic Knowledge cognitive structures.
At any given moment, only part of the [Encyclopedic Knowledge] network is active, depending on the present context and the history of that particular network
Redish and Kuo (2015: 571)
The blue circle (shortest wavelength) represents the subset of cognitive resources that are (or should be) activated for productive understanding of the context under consideration.
A human mind contains a vast amount of knowledge about many things but has limited ability to access that knowledge at any given time. As cognitive semanticists point out, context matters significantly in how stimuli are interpreted and this is as true in a physics class as in everyday life.
Redish and Kuo (2015: 577)
Suboptimal Understanding Zone 1
A common preconception held by students is that the summer months are warmer because the Earth is closer to the Sun during this time of year.
The combination of cognitive resources that lead students to this conclusion could be summarised as follows:
Encyclopedic knowledge: the Earth’s orbit is elliptical
Embodied cognition: The closer to a heat source you are the warmer it is.
Both of these cognitive resources, considered individually, are true. It is their inappropriate selection and combination that leads to the incorrect or ‘Suboptimal Understanding Zone 1’.
To address this, the RF(RGB) suggests a two pronged approach to refine the contextualisation process.
Firstly, we should address the incorrect selection of encyclopedic knowledge. The Earth’s orbit is elliptical but the changing Earth-Sun distance cannot explain the seasons because (1) the point of closest approach is around Jan 4th (perihelion) which is winter in the northern hemisphere; (2) seasons in the northern and southern hemispheres do not match; and (3) the Earth orbit is very nearly circular with an eccentricity e of 0.0167 where a perfect circle has e = 0.
Secondly, the closer-is-warmer p-prim is not the best embodied cognition resource to activate. Rather, we should seek to activate the spread-out-is-less-intense ‘sense of mechanism’ as far as we are able to (for example by using this suggestion from the IoP).
Suboptimal Understanding Zone 2
Another common preconception held by students is all waves have similar properties to the ‘breaking’ waves on a beach and this means that the water moves with the wave.
The structure of this preconception could be broken down into:
Embodied cognition: if I stand close to the water on a beach, then the waves move forward to wash over my feet.
Encyclopaedic knowledge: the waves observed on a beach are water waves
Considered in isolation, both of these cognitive resources are unproblematic: they accurately describes our everyday, lived experience. It is the contextualisation process that leads us to apply the resources inappropriately and places us squarely in Suboptimal Understanding Zone 2.
The RF(RGB) Model suggests that we can address this issue in two ways.
Firstly, we could seek to activate a more useful embodied cognition resource by re-contextualising. For example, we could ask students to imagine themselves floating in deep water far from the shore: do the waves carry them in any particular direction or simply move them up or down as they pass by?
Secondly, we could seek to augment their encyclopaedic knowledge: yes, the waves on a beach are water waves but they are not typical water waves. The slope of the beach slows down the bottom part of the wave so the top part moves faster and ‘topples over’ — in other words, the water waves ‘break’ leading to what appears to be a rhythmic back-and-forth flow of the waves rather than a wave train of crests and troughs arriving a constant wave speed. (This analysis is over a short period of time where the effect of any tidal effects is negligible.)
Both processes try to ‘tug’ student understanding into the central, optimal zone.
Suboptimal Understanding Zone 3
Redish and Kuo (2015: 585) recount trying to help a student understand the varying brightness of bulbs in the circuit shown.
The student said that they had spent nearly an hour trying to set up and solve the Kirchoff’s Law loop equations to address this problem but had been unsuccessful in accounting for the varying brightnesses.
Redish suggested to the student that they try an analysis ‘without the equations’ and just look at the problems in simpler physical terms using just the concept of electric current. Since current is conserved it must split up to pass through bulbs B and C. Since the brightness is dependent on the current, the smaller currents in B and C compared with A and D accounts for their reduced brightness.
When he was introduced to [this] approach to using the basic principles, he lit up and was able to solve the problem quickly and easily, saying, ‘‘Why weren’t we shown this way to do it?’’ He would still need to bring his conceptual understanding into line with the mathematical reasoning needed to set up more complex problems, but the conceptual base made sense to him as a starting point in a way that the algorithmic math did not.
Analysing this issue using the RF(RGB) it is plausible to suppose that the student was trapped in Suboptimal Understanding Zone 3. They had correctly selected the Kirchoff’s Law resources from their encyclopedic knowledge base, but lacked a ‘sense of mechanism’ to correctly apply them.
What Redish did was suggest using an embodied cognition resource (the idea of a ‘material flow’) to analyse the problem more productively. As Redish notes, this wouldn’t necessarily be helpful for more advanced and complex problems, but is probably pedagogically indispensable for developing a secure understanding of Kirchoff’s Laws in the first place.
The RGB Model is not a necessary part of the Resources Framework and is simply my own contrivance for applying the RF in the context of physics education at the high school level. However, I do think the RF(RGB) has the potential to be useful for both physics and science teachers.
Hopefully, it will help us to make all of our subject content more ‘understanded of the pupils’.
Redish, E. F., & Gupta, A. (2009). Making meaning with math in physics: A semantic analysis. GIREP-EPEC & PHEC 2009, 244.
Redish, E. F., & Kuo, E. (2015). Language of physics, language of math: Disciplinary culture and dynamic epistemology. Science & Education, 24(5), 561-590.
In this post, the question I want to address in more detail is: why do I recommend introducing waves using dynamics trolleys rather than a slinky spring or elastic rope?
I think I have a couple of persuasive reasons; but first a digression.
A thing doing a thing…
Robert Graves wrote his charming comedic poem Welsh Incident in 1929. It describes (at least as I read it) the visitation of disparate group of space aliens to a small town on the North Wales coast in the 1920s.
I will paraphrase one line where the Narrator says:
Then one of things did a thing that was finally recognisable as a thing.
The first example of a thing doing a thing…
Now don’t get me wrong: slinky springs and elastic ropes are brilliant examples of mechanical waves and I do use them (a lot!) — I just don’t use them as the first example.
The problem as I see it is that they could lead students to file wave mechanics in an entirely separate and independent category from mechanics.
I want students to see that wave behaviour isn’t distinct from forces and particles but rather is a direct (and fairly straightforward) consequence of a particular arrangement of particles with a specific pattern of forces between them.
Since the first example, is often the ‘loudest’ (metaphorically speaking), it’s not a bad idea to start with longitudinal waves.
I use standard wooden dynamics trolleys. Dowel rods or metal posts can be used to link the trolleys together. The system is more stable if a pair of springs is used at the front and back of each trolley. The springs used are the ones we typically use for the Hooke’s Law experiment.
A compression carrying energy along a line of trolleys linked by springs can be easily modelled:
So can a rarefaction:
Transverse waves can be modelled like this:
Amongst the advantages of this approach are:
Students are introduced to an unknown thing (wave behaviour) by means of more familiar things (trolleys and springs)
The idea that there is no net movement of the ‘particles’ as energy is transferred is much more directly observable using this arrangement rather than the slinky or elastic rope.
The frequency of a wave (which in some ways is a more fundamental measurement than wavelength) can be associated with the repeating motion of a single ‘particle’ and extended outwards to the whole system, rather than vice versa.
You can read more in Chapter 25 of Cracking Key Concepts in Secondary Science.
I hope readers will try this demonstation: hopefully introducing students to a thing which is already recognisable as a thing will make wave behaviour more comprehensible and less like an unwelcome diversion into terra incognita.
Readers who are ‘rich in years’ like myself will recognise this demonstration as being adapted from the old Nuffield linear A-level Physics course.
You can listen to Richard Burton’s great reading of Robert Graves’ Welsh Incident here.
Indeed, one might argue with some very sound justification that both measurements are actually of the same potential difference and that there is no real difference between what we chose to call ‘the potential difference across the cell’ and ‘the potential difference across the resistor’.
Try another way…
But let’s consider drawing the circuit a different (but operationally identical) way:
What would happen if we measured the potential difference across the cell and the resistor as before…
This time, we get a reading (same assumptions as before) of [positive] +1.5 volts of potential difference for the potential difference across the cell and [negative] -1.5 volts for the potential difference across the resistor.
This, at least to me, is a far more conceptually helpful result for student understanding. It implies that the charge carriers are gaining energy as they pass through the cell, but losing energy as they pass through the resistor.
We can, of course, obtain a similar result for the conventional layout, but only at the cost of ‘crossing the leads’ — a sin as heinous as ‘crossing the beams’ for some students (assuming they have seen the original Ghostbusters movie).
A Hidden Rotation?
The argument I am making is that the conventional way of drawing simple circuits involves an implicit and hidden rotation of 180 degrees in terms of which end of the resistor is at a more positive potential.
Of course, experienced physics learners and instructors take this ‘hidden rotation’ in their stride. It is an example of the ‘curse of knowledge’: because we feel that it is not confusing we fail to anticipate that novice learners could find it confusing. Wherever possible, we should seek to make whatever is implicit as explicit as we can.
A translation is, of course, a sliding transformation, rather than a circumrotation. Hence, I had to dispense with this post’s original title of ‘Circuit Diagrams: Lost in Translation’.
However, I do genuinely feel that some students understanding of circuits could be inadvertently ‘lost in rotation’ as argued above.
I hope my fellow physics teachers try introducing potential difference using the ‘all-in-row’ orientation shown.
I would be fascinated to know if they feel its a helpful contribition to their teaching repetoire!
At the bottom of the post are some links to a student booklet for teaching part of the electricity content for AQA GCSE Physics / AQA GCSE Combined Science using the Coulomb Train Model.
I have believed for a long time that the electricity content is often ‘under-explained’ at GCSE: in other words, not all of the content is explicitly taught. I have deliberately have gone to the opposite extreme here — indeed, some teachers may feel that I have ‘over-explained’ too much of the content. However, the booklets are editable so feel free to adapt!
I think the booklet is suitable for teacher-led instruction as well as independent study — I would love to hear how your students have responded to it.
The animations will be ‘live’ for the Google Docs and MS Word versions, but will be frozen for the PDF version. They can be cut and pasted into Powerpoint or other teaching packages (but please note that in some versions of PPT, the animations will appear frozen until you go into presenter mode).
Please feel free to download, use and adapt as you see fit. It is released under the terms of the Creative Commons Attribution License CC BY-SA 4.0 (details here), so please flag if you see versions being sold on TES or similar websites.
The remaining content for AQA electricity will be released (fingers crossed) over the next couple of months.
Feedback and comments (hopefully mainly positive) always welcome….
‘Transformers’ is one of the trickier topics to teach for GCSE Physics and GCSE Combined Science.
I am not going to dive into the scientific principles underlying electromagnetic induction here (although you could read this post if you wanted to), but just give a brief overview suitable for a GCSE-level understanding of:
The basic principle of a transformer; and
How step down and step up transformers work.
One of the PowerPoints I have used for teaching transformers is here. This is best viewed in presenter mode to access the animations.
The basic principle of a transformer
The primary and secondary coils of a transformer are electrically isolated from each other. There is no charge flow between them.
The coils are also electrically isolated from the core that links them. The material of the core — iron — is chosen not for its electrical properties but rather for its magnetic properties. Iron is roughly 100 times more permeable (or transparent) to magnetic fields than air.
The coils of a transformer are linked, but they are linked magnetically rather than electrically. This is most noticeable when alternating current is supplied to the primary coil (green on the diagram above).
The current flowing in the primary coil sets up a magnetic field as shown by the purple lines on the diagram. Since the current is an alternating current it periodically changes size and direction 50 times per second (in the UK at least; other countries may use different frequencies). This means that the magnetic field also changes size and direction at a frequency of 50 hertz.
The magnetic field lines from the primary coil periodically intersect the secondary coil (red on the diagram). This changes the magnetic flux through the secondary coil and produces an alternating potential difference across its ends. This effect is called electromagnetic induction and was discovered by Michael Faraday in 1831.
Energy is transmitted — magnetically, not electrically — from the primary coil to the secondary coil.
As a matter of fact, a transformer core is carefully engineered so to limit the flow of electrical current. The changing magnetic field can induce circular patterns of current flow (called eddy currents) within the material of the core. These are usually bad news as they heat up the core and make the transformer less efficient. (Eddy currents are good news, however, when they are created in the base of a saucepan on an induction hob.)
One of the great things about transformers is that they can transform any alternating potential difference. For example, a step down transformer will reduce the potential difference.
The secondary coil (red) has half the number of turns of the primary coil (green). This halves the amount of electromagnetic induction happening which produces a reduced output voltage: you put in 10 V but get out 5 V.
And why would you want to do this? One reason might be to step down the potential difference to a safer level. The output potential difference can be adjusted by altering the ratio of secondary turns to primary turns.
One other reason might be to boost the current output: for a perfectly efficient transformer (a reasonable assumption as their efficiencies are typically 90% or better) the output power will equal the input power. We can calculate this using the familiar P=VI formula (you can call this the ‘pervy equation’ if you wish to make it more memorable for your students).
Thus: Vp Ip = Vs Is so if Vs is reduced then Is must be increased. This is a consequence of the Principle of Conservation of Energy.
There are more turns on the secondary coil (red) than the primary (green) for a step up transformer. This means that there is an increased amount of electromagnetic induction at the secondary leading to an increased output potential difference.
Remember that the universe rarely gives us something for nothing as a result of that damned inconvenient Principle of Conservation of Energy. Since Vp Ip = Vs Is so if the output Vs is increased then Is must be reduced.
If the potential difference is stepped up then the current is stepped down, and vice versa.
Last nail in the coffin of the formula triangle…
Although many have tried, you cannot construct a formula triangle to help students with transformer calculations.
Now is your chance to introduce students to a far more sensible and versatile procedure like FIFA (more details on the PowerPoint linked to above)
In this post, we are going to look at series circuits using the Coulomb Train Model.
The Coulomb Train Model (CTM) is a helpful model for both explaining and predicting the behaviour of real electric circuits which I think is useful for KS3 and KS4 students.
Without further ado, here is a a summary.
A circuit with one resistor
Let’s look at a very simple circuit to begin with:
This can be represented on the CTM like this:
The ammeter counts 5 coulombs passing every 10 seconds, so the current I = charge flow Q / time t = 5 coulombs / 10 seconds = 0.5 amperes.
We assume that the cell has a potential difference of 1.5 V so there is a potential difference of 1.5 V across the resistor R1 (that is to say, each coulomb loses 1.5 J of energy as it passes through R1).
The resistor R1 = potential difference V / current I = 1.5 / 0.5 = 3.0 ohms.
A circuit with two resistors in series
Now let’s add a second identical resistor R2 into the circuit.
This can be shown using the CTM like this:
Notice that the current in this example is smaller than in the first circuit; that is to say, fewer coulombs go through the ammeter in the same time. This is because we have added a second resistor and remember that resistance is a property that reduces the current. (Try and avoid talking about a high resistance ‘slowing down’ the current because in many instances such as two conductors in parallel a high current can be modelled with no change in the speed of the coulombs.)
Notice also that the voltmeter is making identical measurements on both the circuit diagram and the CTM animation. It is measuring the total energy change of the coulombs as they pass through both R1 and R2.
The current I = charge flow Q / time t = 5 coulombs / 20 seconds = 0.25 amps. This is half the value of the current in the first circuit.
We have an identical cell of potential difference 1.5 V the voltmeter would measure 1.5 V. We can calculate the total resistance using R = V / I = 1.5 / 0.25 = 6.0 ohms.
This is to be expected since the total resistance R = R1 + R2 and R1 = 3.0 ohms and R2 = 3.0 ohms.
Looking at the resistors individually
The above circuit can be represented using the CTM as follows:
Between A and B, the coulombs are each gaining 1.5 joules since the cell has a potential difference of 1.5 V. (Remember that V = E energy transferred (in joules) / Q charge flow (in coulombs.)
Between B and C the coulombs lose no energy; that is to say, we are assuming that the connecting wires have negligible resistance.
Between C and D the coulombs lose some energy. We can use the familar V = I x R to calculate how much energy is lost from each coulomb, since we know that R1 is 3.0 ohms and I is 0.25 amperes (see previous section).
V = I x R = 0.25 x 3.0 = 0.75 volts.
That is to say, 0.75 joules are removed from each coulomb as they pass through R1 which means that (since 1.5 joules were added to each coulomb by the cell) that 0.75 joules are left in each coulomb.
The coulombs do not lose any energy travelling between D and E because, again, we are assuming negligible resistance in the connecting wire.
0.75 joules is removed from each coulomb between E and F making the potential difference across R2 to be 0.75 volts.
Thus we find that the familiar V = V1 + V2 is a direct consequence of the Principle of Conservation of Energy.
FAQ: ‘How do the coulombs know to drop off only half their energy in R1?’
Simple answer: they don’t.
This may be a valid objection for some donation models of electric circuits (such as the pizza delivery van model) but it doesn’t apply to the CTM because it is a continuous chain model (with the caveat that the CTM applies only to ‘steady state’ circuits where the current is constant).
Let’s look at a numerical argument to support this:
The magnitude of the current is controlled by only two factors: the potential difference of the cell and the total resistance of the circuit.
In other words, if we increased the value of R1 to (say) 4 ohms and reduced the value of R2 to 2 ohms so that the total resistance was still 6 ohms, the current would still be 0.25 amps.
However, in this case the energy dissipated by each coulomb passing through R1 would V = I x R = 0.25 x 4 = 1 volt (or 1 joule per coulomb) and similarly the potential difference across R2 would now be 0.5 volts.
The coulombs do not ‘know’ to drop off 1 joule at R1 and 0.5 joules at R2: rather, it is a purely mechanical interaction between the moving coulombs and each resistor.
R1 has a bigger proportion of the total resistance of the circuit than R2 so it seems self-evident (at least to me) that the coulombs will lose a larger proportion of their total energy passing through R1.
A similar analysis would apply if we made R2 = 4 ohms and R1 = 2 ohms: the coulombs would now lose 0.5 joules passing through R1 and 1 joule passing through R2.
Thus, we see that the current in a series circuit is affected by the ‘global’ or ‘whole circuit’ properties such as the potential difference of the cell and the total resistance of the circuit. The CTM models this property of real circuits by being a continuous chain of mechanically-linked ‘trucks’ so that a change in any one part of the circuit affects the movement of all the coulombs.
However, the proportion of the energy lost by a coulomb travelling through one part of the circuit is affected — not by ‘magic’ or a weird form of ‘coulomb telepathy’ — but only by the ‘local’ properties of that section of the circuit i.e. the electrical resistance of that section.
(PS You can read more about the CTM and potential divider circuits here.)
You may be relieved to hear that this is the last post in my series on ‘The CTM revisited’. My thanks to the readers who have stayed with me through the series (!)
I will close by saying that I have appreciated both the expressions of enthusiasm about CTM and the thoughtful criticisms of it.