Make choo choo go faster

Captain Matthew Henry Phineas Riall Sankey sighed and shaded his tired eyes from the bright glare of the oil lamp. Its light reflected harshly from the jumbled mounds of papers that entirely covered the dark oak surface of his desk. He took a moment to roll down the wick and dim the light. The chaos of his work area hinted at the chaos currently roiling in his usually precise and meticulous engineer’s mind.

He leaned backward in his chair, his shoulders slumped in despair. This problem had defeated many other men before him, he reflected as he stroked his luxuriant moustache: there would be no shame in admitting defeat.

After all, he was only one man and he was attempting to face down the single most serious and most pressing scientific and engineering issue known to the world in the Victorian Era. And yet — he couldn’t help but feel that he was, somehow, close to solving it. The answer seemed to hover mirage-like in front of him, almost within his grasp but blurred and indistinct. It became as insubstantial as mist each time he reached for it. He needed a fresh perspective, a new way of looking simultaneously both at the whole and at the parts of the question. It was not so much a question of not seeing the wood for the trees, but rather seeing the wood, trees, twigs and leaves in sufficient detail at the same time.

And what was this problem that was occupying the finest scientific and technical minds at the close of the nineteenth century? It was simply this:

Make choo choo go faster

You think I jest. But no: in 1898 the world ran on the power of steam. Steam engines were the shining metal giants that laboured tirelessly where hundreds of millions of men and beasts had toiled in misery before. In less enlightened times, industry had rested on the backs of living things that strained and suffered under their load; now. however, it was built on the back of machines that felt no pain and could work day and night when fed with coal.

So much progress had been made over the years from the clanking primitive behemoths pioneered by Thomas Newcomen and James Watt. Those wasteful old engines had always teetered far too close to the edge of scalding catastrophe for comfort and demanded the tribute of a mountain of coal for a miserly hillock of work.

Modern steam engines were sleeker, safer and more efficient. But they still demanded too much coal for a given amount of work: somewhere, deep within their intricate web of moving parts, energy was wastefully haemorrhaging. No matter how much coal you loaded into the firebox or how hotly it burned, the dreadful law of diminishing returns worked its malevolent magic: the engine would accelerate to a certain speed, but no faster, no matter what you did. You always got less work out than you put in.

Captain Henry Phineas Sankey was searching for a tourniquet that would stem the malign loss of energy in the innards of these vital machines. He could not help but think of the wise words written by Jonathan Swift many long years ago:

Whoever could make two ears of corn, or two blades of grass, to grow upon a spot of ground where only one grew before, would deserve better of mankind, and do more essential service to his country, than the whole race of politicians put together.

What Captain Henry Phineas Sankey hoped to do was nothing less than reverse engineer the venerable Jonathan Swift: whereas previously a steam engine would burn two tons of coal to perform a task, he wanted to build an engine that would do the same work by burning only one ton of coal. That he hoped would be his enduring memorial both of his service to his country and to mankind.

But how to achieve this? How could one man hold in his head the myriad moving, spinning parts of a modern steam engine and ascertain how much loss there was here rather than there, and whether it was better to try and eliminate the loss here which might increase the weight of that particular part and hence lead to an unavoidably greater loss over there . . .

Captain Sankey’s restless eyes alighted on a framed drawing on the wall. It had been painstakingly drawn some years ago by his son, Crofton, and then delicately painted in watercolours by his daughter, Celia, when they were both still very young children. They had both been fascinated by the story of Napoleon’s ill-fated Russian Campaign of 1812. The drawing showed Charles Minard’s famous map of 1869.

It showed the initial progress of Napoleon’s huge army as a wide thick band as they proudly marched towards Moscow and its gradual whittling down by the vicissitudes of battle and disease; it also showed the army’s agonised retreat, harried by a resurgent Russian military, and fighting a constant losing battle against the merciless ‘General Winter’. Only a few — a paltry, unhappy few — Frenchmen had made it home, represented by the sad emaciated black line at journey’s end.

Mrs Eliza Sankey had questioned allowing their children to spend so much time studying such a ‘horrible history’ but Captain Sankey had encouraged them. Children should not only know the beauties of the world but also its cruelties, and everyone should attend to the lesson that ‘Pride goeth before a fall’.

The map showed all of that. It was not just a snapshot, but a dynamic model of the state of Napoleon’s army during the whole of the campaign: from the heady joys of its swift, initial victories to its inevitable destruction by cruel attrition. It was a technical document of genius, comparable to a great work of art, for it showed not only the wood but the trees and even the twigs all at one time . . .

Captain Sankey started suddenly. He had an idea. Unwilling to spare even an instant in case this will ‘o the wisp of an idea disappeared, he immediately clipped a blank sheet of paper to his drawing board. He slid the T-square into place and began to draw rapidly. This is the work that Captain wrought:

Later that evening, he wrote:

No portion of a steam plant is perfect, and each is the seat of losses more or less serious. If therefore it is desired to improve the steam plant as a whole, it is first of all necessary to ascertain separately the nature of the losses due to its various portions; and in this connection the diagrams in Plate 5 have been prepared, which it is hoped may assist to a clearer understanding of the nature and extent of the various losses.

The boiler; the engine; the condenser and air-pump; the feedpump and the economiser, are indicated by rectangles upon the diagram. The flow of heat is shown as a stream, the width of which gives the amount of heat entering and leaving each part of the plant per unit of time; the losses are shown by the many waste branches of the stream. Special attention is called to the one (unfortunately small) branch which represents the work done upon the pistons of the engine

Captain Sankey (1898)

The ubiquitous Sankey diagram had been born . . .

How NOT to draw a Sankey diagram for a filament lamp

Although this diagram draws attention to the ‘unfortunately small’ useful output of a filament lamp, and it is still presented in many textbooks and online resources, it is not consistent with the IoP’s Energy Stores and Pathways model since it shows the now defunct ‘electrical energy’ and ‘light energy’.

Note that I use the ‘block’ approach which is far easier to draw on graph paper as opposed to the smooth, aesthetically pleasing curves on the original Sankey diagram.

How to draw a Sankey diagram for a filament lamp

We can, however, draw a similar Sankey diagram for a filament lamp that is completely consistent with the IoP’s Energy Stores and Pathways model if we focus on the pathways by which energy is transferred, rather than on the forms of energy.

The second diagram, in my opinion, provides a much more secure foothold for understanding the emission spectrum of an incandescent filament lamp.

And, as the Science National Curriculum reminds us, we should seek to use ‘physical processes and mechanisms, rather than energy, to explain’ how systems behave. Energy is a useful concept for placing a limit on what can happen, but at the school level I think it is sometimes overused as an explanation of why things happen.

Closing thought

Stephen Hawking surmised that humanity had perhaps 100 years left on a habitable Earth. We are in a race to make a less destructive impact on our environment. ‘Reverse engineering’ Swift’s ‘two ears of corn where one grew before’ so that one joule of energy would do the same work as two joules did previously would be a huge step forward.

And for that goal, the humble Sankey diagram might prove to be an invaluable tool.

Helping Students With Extended Writing Questions in Science

Part one: general principles

He knew all the tricks: dramatic irony, metaphor, pathos, puns, parody, litotes* and . . . satire. He was vicious.

Monty Python, The Tale of the Pirhana Brothers

As we all know, students really struggle with questions in science exams which require answers written ‘at paragraph length’ (dread words!). What follows are some tips that I have found useful when coaching students to improve performance.

Many teachers of English enjoy great success with acronyms such as PEEL (Point. Example. Explain. Link). However, I think these have limited applicability in Science as the required output of extended writing questions (EWQs) varies too much for even a loose one-size-fits-all approach.

What I encourage students to do is:

1. Write in bullet points

The bullet points (BPs) should be short but fully grammatical sentences (and not single words or part sentences).

The reason for this is twofold:

  • Focus: it stops an attempted answer spiralling out of control. Without organising my answer using BPs, I find myself running out of space. I start with the best of intentions but realise, as I fill in the last remaining line of the allocated space, that I haven’t reached the end of the first sentence yet!!!
  • Organisation: it discourages students from repeating the same thing again and again. I have sometimes marked extended writing answers that repeat the same point multiple times. Yes, they have filled the space and yes, they have written in complete sentences. But there is no additional information except the first section rewritten using different words!

2. Use correct scientific vocabulary

Students often make the incorrect assumption that ‘Explain‘ means ‘Explain to a non-specialist using jargon-free everyday language‘.

In fact nothing could be further from the truth. The expectation of EWQs in general is that students should be able to communicate to a scientist-peer using technical language appropriate for GCSE or A-level.

Partly, this misconception is our own fault. When students ask for an explanation from their teachers, we often — with the best of intentions! — try to express it in non-threatening, jargon-free language.

This is the model that many students follow when responding to EWQs. For example, I remember groaning in frustration when marking an A-level Physics script where the student has repeatedly written the word ‘move’ when the terms ‘accelerate’ or ‘constant velocity’ would have communicated her understanding with far more clarity.

In Science, what is often derided as ‘jargon’ isn’t an actual barrier to understanding. In truth, a shared, specialist language is an essential pathway to concision and clarity and a guard-rail against inadvertent miscommunication.

3. Write as many BPs as there are marks

For example, students should aim to write 3 BPs in response to a 3-mark EWQ.

4. Read all your BPs. Taken as a whole — do they *answer* the damn question?

If yes, move on. If no, then add another BP.


Part two: modelling the EWQ response-process

‘What does “quantum” mean, anyway?’

‘It means “add another nought.”‘

Terry Pratchett, Pyramids

This EWQ has 3 marks, so we should aim for 3 BPs.

I use the analogy of crossing a river using stepping stones. One stepping stone won’t be enough but three will let us get across — hopefully without us getting our feet wet.

Let’s write our first BP. I suggest that students begin by stating what they may think is obvious.

Next, we think about what we could write as our second BP. But — and this is essential! — we consider it from the vantage point of our first BP.

Our second BP is the next-most-obvious-BP: what happens to the solenoid when an electric current goes through it? Remember that we are supposed to use technical language, so we will call a solenoid a solenoid, so to speak.

Next, we consider what to write for our third (and maybe final) BP. Again, we should be thinking of this from the viewpoint of what we have already written.

Finally, and this point is not to be missed, we should look back at all the BPs we have written and ask ourselves the all-important ‘Have I actually answered the question that was asked originally?

In this case, the answer is YES, we have explained why the door unlocks when the switch is closed.

This means that we can stop here and move on to the next question.


*Litotes (LIE-tote-ees): an ironic understatement in which an affirmative is expressed as a negative e.g. I won’t be sorry to get to the end of this not-at-all-overlong blog post.

The Apparatus of Golgi: Science As It Should Be

Carl Sagan said that science is unique in having it’s own built-in error correcting machinery:

The scientific way of thinking is at once imaginative and disciplined. This is central to its success. Science invites us to let the facts in, even when they don’t conform to our preconceptions [ . . .] One of the reasons for its success is that science has built-in, error correcting machinery at its very heart. Some may consider this an overbroad characterization, but to me every time we exercise self-criticism, every time we test our ideas against the outside world, we are doing science. When we are self-indulgent and uncritical, when we confuse hopes and facts, we slide into pseudoscience and superstition.

Sagan 1997: 35 [emphasis added]

Of course, scientists are only human, and are sometimes as susceptible to self-indulgence and reluctance to criticise their own “pet” theories as the next person. But not always.

Richard Dawkins (2006) shares the following story about the reaction of a highly respected “elder statesman” of science to evidence countering his long-held opinion about a structure inside living cells called the Apparatus of Golgi (GOL-jee).

An animal cell. The Apparatus of Golgi is labelled 6. For other labels, click on the link https://en.wikipedia.org/wiki/Golgi_apparatus

I have previously told the story of a respected elder statesman of the Zoology Department at Oxford when I was an undergraduate [c.1960]. For years he had passionately believed, and taught, that the Golgi Apparatus (a microscopic feature of the interior of cells) was not real: an artefact, an illusion. Every Monday afternoon it was the custom for the whole department to listen to a research talk by a visiting lecturer. One Monday, the visitor was an American cell biologist who presented completely convincing evidence that the Golgi Apparatus was real. At the end of the lecture, the old man strode to the front of the hall, shook the American by the hand and said — with passion — “My dear fellow, I wish to thank you. I have been wrong these fifteen years.” We clapped our hands red.

Dawkins 2006: 283

References

Dawkins, R. (2006). The God Delusion. Bantam Press.

Sagan, C. (1997). The Demon-Haunted World: Science As A Candle In The Dark. Random House Digital, Inc

Using dimensional analysis to estimate the energy released by an atomic bomb

Legend has it that in the early 1950s, British physicist G. I. Taylor was visited by some very serious men from the military authorities. His crime? He had apparently secured unauthorised access to worryingly accurate and top secret information about the energy released by the first atom bomb.

Sir G. I. Taylor (1896-1965)

Taylor explained that, actually, he hadn’t: he had estimated the energy yield from a series of photographs of the first atomic test explosion published by Life magazine. Taylor had used the standard physics technique known as dimensional analysis.

Part of the sequence of photographs of the Trinity atomic weapon test (16/7/45) published by Life magazine in 1950

The published pictures had helpfully included a scale to indicate the size of the atomic fireball in each photograph and Taylor had been able to complete a back-of-the-envelope calculation which gave a surprisingly accurate value for what was then the still highly classified energy yield of an atomic weapon.

This story was shared by the excellent David Cotton (@NewmanPhysics) on Twitter, and included a link to a useful summary which forms the basis of what follows. (NB Any errors or omissions are my own.)

It is presented here for A-level Physics teachers to consider using as an example of the power of dimensional analysis beyond the usual “predicting the form of the equation for the period of a simple pendulum”(!)

Taylor’s method: step one

Taylor began by assuming that the radius R of the fireball would depend on:

  • The energy E released by the bomb. The larger the energy released then the larger the fireball.
  • The density of the air ρ. The greater the density of the air then the smaller the fireball since more work would have to be done to push the air out of the path of the fireball.
  • The time elapsed t from the explosion. The longer the time then the larger the size of the fireball (until the moment when it began to collapse).

These three factors can be combined into a single relationship:

k is an unknown arbitrary constant. Note that we would expect the exponent y to be negative since R is expected to decrease as ρ increases. We would, however, expect x and z to be positive.

Taylor’s method: step two

Next we think of the dimensions of each of the values in terms of the basic dimensions or measurements of length [L], mass [M] and time [T].

  • R has the dimension of length, so R = [L].
  • E is in joules or newton metres (since work done = force x distance). From F=ma we can conclude that the dimensions of newtons are [M] [L] [T]-2. This makes the dimensions of energy [M] [L]2 [T]-2.
  • ρ is in kilograms per cubic metre so it has the dimensions [M] [L]-3.
  • t has the dimension of time [T].

Taylor’s method: step three

Next we write equation 1 in terms of the dimensions of each of the quantities. We can ignore k as we assume that this is a purely numerical value with no units. This gives us:

Simplifying this expression, we get:

Taylor’s method: step four

Next, let’s look at the exponents of [M], [L] and [T].

Firstly, we can see that x + y = 0 since there is no [M] term on the left hand side.

Secondly, we can see that 2x – 3y = 1 since there is an [L] term on the left hand side.

Thirdly, we can see that z – 2x = 0 since there is no [T] term on the left hand side.

Taylor’s method: step five

We now have a system of three equations detailing three unknowns.

We can solve for x, y and z using simultaneous equations. This gives us x=(1/5), y=(-1/5) and z=(2/5).

Taylor’s method: step six

Let’s rewrite equation 1 using these values. This gives us:

Rearranging for E gives us:

Taylor’s method: step seven

Next we read off the value of t=0.006 s and estimate R=75 m from the photograph. The density of air ρ at normal atmospheric pressure is ρ=1.2 kg/m3.

If we substitute these values into equation 6 (assuming that k=1) we get E= 7.9 x 1013 joules.

Conclusion

Modern sources estimate the yield of the Trinity test as being equivalent to between 18-20 kilotons of TNT. Let’s take the mean value of 19 kilotons. One kiloton is equivalent to 4.184 terajoules. This means that, according to declassified sources that were not available to Taylor, the energy released by the Trinity test was 7.9 x 1013 joules.

As you can see, Taylor’s “guesstimated” value using the dimensional analysis technique was remarkably close to the actual value. No wonder that the military authorities were concerned about this apparent “leak” of classified information.

Explaining current flow in conductors (part two)

Do we delve deeply enough into the actual physical mechanism of current flow through electrical conductors using the concepts of charge carriers and electric fields in our treatments for GCSE and A-level Physics? I must reluctantly admit that I am increasingly of the opinion that the answer is no.

In part one we discussed two common misconceptions about the physical mechanism of current flow, namely:

  1. The all-the-electrons-in-a-conductor-repel-each-other misconception; and
  2. The electric-field-of-the-battery-makes-all-the-charge-carriers-in-the-circuit-move misconception.

What, then, does produce the internal electric field that drives charge carriers through a conductor?

Let’s begin by looking at the properties that such a field should have.

Current and electric field in an ohmic conductor

(You can see a more rigorous derivation of this result in Duffin 1980: 161.)

We can see that if we consider an ohmic conductor then for a current flow of uniform current density J we need a uniform electric field E acting in the same direction as J.

What produces the electric field inside a current-carrying conductor?

The electric field that drives charge carriers through a conductor is produced by a gradient of surface charge on the outside of the conductor.

Rings of equal charge density (and the same sign) contribute zero electric field at a location midway between the two rings, whereas rings of unequal charge density (or different sign) contribute a non-zero field at that location.

Sherwood and Chabay (1999): 9

These rings of surface charge produce not only an internal field Enet as shown, but also external fields than can, under the right circumstances, be detected.

Relationship between surface charge densities and the internal electric field

Picture a large capacity parallel plate capacitor discharging through a length of high resistance wire of uniform cross section so that the capacitor takes a long time to discharge. We will consider a significant period of time (a small fraction of RC) when the circuit is in a quasi-steady state with a current density of constant magnitude J. Since E = J / σ then the internal electric field Enet produced by the rings of surface charge must be as shown below.

Schematic diagram showing the relationship between the surface charge density and the internal electric field

In essence, the electric field of the battery polarises the conducting material of the circuit producing a non-uniform arrangement of surface charges. The pattern of surface charges produces an electric field of constant magnitude Enet which drives a current density of constant magnitude J through the circuit.

As Duffin (1980: 167) puts it:

Granted that the currents flowing in wires containing no electromotances [EMFs] are produced by electric fields due to charges, how is it that such a field can follow the tortuous meanderings of typical networks? […] Figure 6.19 shows diagrammatically (1) how a charge density which decreases slowly along the surface of a wire produces an internal E-field along the wire and (2) how a slight excess charge on one side can bend the field into the new direction. Rosser (1970) has shown that no more than an odd electron is needed to bend E around a ninety degree corner in a typical wire.

Rosser suggests that for a current of one amp flowing in a copper wire of cross sectional area of one square millimetre the required charge distribution for a 90 degree turn is 6 x 10-3 positive ions per cm3 which they call a “minute charge distribution”.

Observing the internal and external electric fields of a current carrying conductor

Jefimenko (1962) commented that at the time

no generally known methods for demonstrating the structure of the electric field of the current-carrying conductors appear to exist, and the diagrams of these fields can usually be found only in the highly specialized literature. This […] frequently causes the student to remain virtually ignorant of the structure and properties of the electric field inside and, especially, outside the current-carrying conductors of even the simplest geometry.

Jefimenko developed a technique involving transparent conductive ink on glass plates and grass seeds (similar to the classic linear Nuffield A-level Physics electrostatic practical!) to show the internal and external electric field lines associated with current-carrying conductors. Dry grass seeds “line up” with electric field lines in a manner analogous to iron filings and magnetic field lines.

Photograph from Jefimenko (1962: 20). Annotations added

Coming soon . . .

In part 3, we will analyse the transient processes by which these surface charge distributions are set up.

References

Duffin, W. J. (1980). Electricity and magnetism (3rd ed.). McGraw Hill Book Co.

Jefimenko, O. (1962). Demonstration of the electric fields of current-carrying conductorsAmerican Journal of Physics30(1), 19-21.

Rosser, W. G. V. (1970). Magnitudes of surface charge distributions associated with electric current flow. American Journal of Physics38(2), 265-266.

Sherwood, B. A., & Chabay, R. W. (1999). A unified treatment of electrostatics and circuits. URL http://cil. andrew. cmu. edu/emi. (Note: this article is dated as 2009 on Google Scholar but the text is internally dated as 1999)

Explaining current flow in conductors (part one)

Do we delve deeply enough into the actual physical mechanism of current flow through electrical conductors (in terms of charge carriers and electric fields) in our treatments for GCSE and A-level Physics? I must reluctantly admit that I am increasingly of the opinion that the answer is no.

Of course, as physics teachers we talk with seeming confidence of current, potential difference and resistance but — when push comes to shove — can we (say) explain why a bulb lights up almost instantaneously when a switch several kilometres away is closed when the charge carriers can be shown to be move at a speed comparable to that of a sedate jogger? This would imply a time delay of some tens of minutes between closing the switch and energy being transferred from the power source (via the charge carriers) to the bulb.

When students asked me about this, I tended to suggest one of the following:

  • “The electrons in the wire are repelling each other so when one close to the power source moves, then they all move”; or
  • “Energy is being transferred to each charge carrier via the electric field from the power source.”

However, to be brutally honest, I think such explanations are too tentative and “hand wavy” to be satisfactory. And I also dislike being that well-meaning but unintentionally oh-so-condescending physics teacher who puts a stop to interesting discussions with a twinkly-eyed “Oh you’ll understand that when you study physics at degree level.” (Confession: yes, I have been that teacher too often for comfort. Mea culpa.)

Sherwood and Chabay (1999) argue that an approach to circuit analysis in terms of a predominately classical model of electrostatic charges interacting with electric fields is very helpful:

Students’ tendency to reason locally and sequentially about electric circuits is directly addressed in this new approach. One analyzes dynamically the behaviour of the *whole* circuit, and there is a concrete physical mechanism for how different parts of the circuit interact globally with each other, including the way in which a downstream resistor can affect conditions upstream.

(Side note: I think the Coulomb Train Model — although highly simplified and applicable only to a limited set of “steady state” situations — is consistent with Sherwood and Chabay’s approach, but more on that later.)

Misconception 1: “The electrons in a conductor push each other forwards.”

On this model, the flowing electrons push each other forwards like water molecules pushing neighbouring water molecules through a hose. Each negatively charged electron repels every other negatively charged electron so if one free electron within the conductor moves, then the neighbouring free electrons will also move. Hence, by a chain reaction of mutual repulsion, all the electrons within the conductor will move in lockstep more or less simultaneously.

The problem with this model is that it ignores the presence of the positively charged ions within the metallic conductor. A conveniently arranged chorus-line of isolated electrons would, perhaps, behave analogously to the neighbouring water molecules in a hose pipe. However, as Sherwood and Chabay argue:

Averaged over a few atomic diameters, the interior of the metal is everywhere neutral, and on average the repulsion between flowing electrons is canceled by attraction to positive atomic cores. This is one of the reasons why an analogy between electric current and the flow of water can be misleading.

The flowing electrons inside a wire cannot push each other through the wire, because on average the repulsion by any electron is canceled by the attraction of a nearby positive atomic core (Diagram from Sherwood and Chabay 1999: 4)

Misconception 2: “The charge carriers move because of the electric field from the battery.”

Let’s model the battery as a high-capacity parallel plate capacitor. This will avoid the complexities of having to consider chemical interactions within the cells. Think of a “quasi-steady state” where the current drawn from the capacitor is small so that electric charge on the plates remains approximately constant; alternatively, think of a mechanical charge transfer mechanism similar to the conveyor belt in a Van de Graaff generator which would be able to keep the charge on each plate constant and hence the potential difference across the plates constant (see Sherwood and Chabay 1999: 5).

A representation of the electric field around a single cell battery (modelled as a parallel plate capacitor)

This is not consistent with what we observe. For example, if the charge-carriers-move-due-to-electric-field-of the-battery model was correct then we would expect a bulb closer to the battery to be brighter than a more distant bulb; this would happen because the bulb closer to the battery would be subject to a stronger electric field and so we would expect a larger current.

A bulb closer to the battery is NOT brighter than a bulb further away from the battery (assuming negligible resistance in the connecting wires)

There is the additional argument if we orient the bulb so that the current flow is perpendicular to the electric field line, then there should be no current flow. Instead, we find that the orientation of the bulb relative to the electric field of the battery has zero effect on the brightness of the bulb.

There is no change of brightness as the orientation of the bulb is changed with respect to the electric field lines from the battery

Since we do not observe these effects, we can conclude that the electric field lines from the battery are not solely responsible for the current flow in the circuit.

Understanding the cause of current flow

If the electric field of the battery is not responsible on its own for the potential difference that causes a current to flow, where does the electric field come from?

Interviews reveal that students find the concept of voltage difficult or incomprehensible. It is not known how many students lose interest in physics because they fail to understand basic concepts. This number may be quite high. It is therefore astonishing that this unsatisfactory situation is accepted by most physics teachers and authors of textbooks since an alternative explanation has been known for well over one hundred years. The solution […] was in principle discovered over 150 years ago. In 1852 Wilhelm Weber pointed out that although a current-carrying conductor is overall neutral, it carries different densities of charges on its surface. Recognizing that a potential difference between two points along an electric circuit is related to a difference in surface charges [is the answer].

Härtel (2021): 21

We’ll look at these interesting ideas in part two.

[Note: this post edited 10/7/22 because of a rewritten part two]

References

Härtel, H. (2021). Voltage and Surface ChargesEuropean Journal of Physics Education12(3), 19-31.

Sherwood, B. A., & Chabay, R. W. (1999). A unified treatment of electrostatics and circuits. URL http://cil. andrew. cmu. edu/emi. (Note: this article is dated as 2009 on Google Scholar but the text is internally dated as 1999)

Acknowledgements

The circuit representations were produced using the excellent PhET Sims circuit simulator.

I was “awoken from my dogmatic slumbers” on this topic (and alerted to Sherwood and Chabay’s treatment) by Youtuber Veritasium‘s provocative videos (see here and here).

Speed of sound using Phyphox

You can get encouragingly accurate values for the speed of sound in the school laboratory using a tape measure and two smartphones (or tablets) running the Phyphox app.

Phyphox (pronounced FEE-fox) is an award-winning free app that was developed by physicists at Aachen University who wanted to give users direct access to the many sensors (e.g. accelerometers and magnetometers) which are standard features on many smartphones. In effect, it turns even the humblest smartphone or tablet into a multifunctional measuring instrument comparable to one of Star Trek’s famous ‘tricorders’.

Star Trek Tricorder (not Phyphox)

To measure the speed of sound, we need two smartphones or tablets running Phyphox. We will be using the ‘Acoustic Stopwatch’ which measures the time between two acoustic events.

Phyphox main screen

Step 1: Place two devices a measured distance s apart. Typically about 2 or 3 m should be OK otherwise the sound made by A will not be loud enough to control stopwatch B — this can be established through trial and error and depends on many factors including the background noise level.

Step 2: Person A makes a loud sound (a clap or a single syllable shout like ‘Hey!’ is good).

Step 3: Person B and stopwatch B wait for sound created by A to reach them.
Step 4: The sound reaches stopwatch B and starts it running and B hears the sound.

Step 5: B makes a loud sound in response.

Step 6: The loud sound made by B reaches stopwatch B and makes it stop. Let’s call the time displayed tB. This measures the delay between the sound from A reaching stopwatch B and B reacting to the sound and stopping the clock. It includes the time taken for the initial sound travelling from the device to B, B’s reaction time, and the time taken for the sound made by B to travel to stopwatch B. B does not have to be particularly ‘quick off the mark’ to respond to A’s sound — although the shorter the time then the less likely it is then a background noise will interrupt the experiment.

Step 7: The sound made by B travels toward stopwatch A.

Step 8: The sound made by B reaches stopwatch A and makes it stop. Let’s call the time recorded on stopwatch A tA.

If we break down the events included in tA and tB, we find that tA is always larger than tB:

If we subtract tAtB we find that this is the time it takes sound to travel a distance of 2s.

Step 9: We can therefore use this formula to find the v the speed of sound.

We have found that this method works well giving mean values of about 350 m/s for the speed of sound (which will vary with air temperature). This video models the method.

And so we have a reasonably practicable method of measuring the speed of that doesn’t involve complex equipment that is unfamiliar to most students; or a method that involves finding a large and featureless wall that produces a detectable echo when a loud sound is made from a point several metres in front of it.

I don’t know about you, but as a physics teacher, I feel cheated. If it doesn’t involve a double beam oscilloscope, a signal generator, two microphones and two power amplifiers then I simply don’t want to know about it . . .

The Rite of AshkEnte, quite simply, summons and binds Death.  Students of the occult will be aware that it can be performed with a simple incantation, three small bits of wood and 4cc of mouse blood, but no wizard worth his pointy hat would dream of doing anything so unimpressive; they knew in their hearts that if a spell didn’t involve big yellow candles, lots of rare incense, circles drawn on the floor with eight different colours of chalk and a few cauldrons around the place then it simply wasn’t worth contemplating.

Terry Pratchett, ‘Mort’ (1987)

Modelling Electrical Power Equations using the CTM

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

An animated version of the Coulomb Train Model

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:

  1. 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.
  2. 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.
Hewitt representation of changing the values of Q and V while keeping the other fixed

Or, of course, we could increase the values of Q and V simultaneously.

Hewitt representation of increasing 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:

Using the CTM to show the relationship between current I and power P

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:
A Hewitt representation of the effect of doubling I on P

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.

Through a glass, lightly

Any sufficiently advanced technology is indistinguishable from magic.

Profiles of the Future (1962)

So wrote Arthur C. Clarke, science fiction author and the man who invented the geosynchronous communications satellite.

Clarke later joked of his regret at the billions of dollars he had lost by not patenting the idea, but one gets the impression that what really rankled him was another (and, to be fair, uncharacteristic) failure of imagination: he did not foresee how small and powerful solid state electronics would become. He had pictured swarms of astronauts crewing vast orbital structures, having their work cut out as they strove to maintain and replace the thousands of thermionic valves burning out under the weight of the radio traffic from Earth . . .

A communications nexus that could fit into a volume the size of a minibus, then (as the technology developed) a suitcase and then a pocket seemed implausible to him — and to pretty much everyone else as well.

And yet, here we are, living in the world that microelectronics has wrought. We are truly in an age of Clarkeian magic: our technology has become so powerful, so reliable and so ubiquitous — and so few of us have a full understanding of how it actually works — that it is very nearly indistinguishable from magic.

I want to outline a fictional vignette on the same theme which Clarke wrote in 1961 which has stayed with me since I read it. Please bear with me while I sketch out the story — it will lend some perspective to what follows.

In the novel A Fall of Moondust (1961) set in the year 2100, the lunar surface transport Selene has become trapped under fifteen metres of moondust. Rescue teams on the surface have managed to drill down to the stricken vessel with a metal pipe to supply the unfortunate passengers and crew with oxygen from the rescue ship. Communication would seem to be impossible as the Selene’s radio has been destroyed, but luckily Chief Engineer (Earthside) Lawrence has a plan:

They would hear his probe, but there was no way in which they could communicate with him. But of course there was. The easiest and most primitive means of all, which could be so readily overlooked after a century and a half of electronics.

A few hours later, the rescue team’s pipe drills through the roof of the sunken vessel.

The brief rush of air gave everyone a moment of unnecessary panic as the pressure equalised. Then the pipe was open to the upper world, and twenty two anxious men and women waited for the first breath of oxygen to come gushing down it.

Instead, the tube spoke.

Out of the open orifice came a voice, hollow and sepulchral, but perfectly clear. It was so loud, and so utterly unexpected, that a gasp of surprise came from the company. Probably not more than half a dozen of these men and women had ever heard of a ‘speaking tube’; they had grown up in the belief that only through electronics could the voice be sent across space. This antique revival was as much a novelty to them as a telephone would have been to an ancient Greek.

The humble converging lens as an ‘antique revival’

If you hold a magnifying glass (or any converging lens) in front of a white screen, then it will produce a real, inverted image of any bright objects in front of it. This simple act can, believe it or not, draw gasps of surprise from groups of our ‘digital native’ students: they assume that images can only be captured electronically. The fact that a shaped piece of glass can do so is as much a novelty to them as an LCD screen would have been to Galileo.

Think about it for a moment: how often has one of our students seen an image projected by a lens onto a passive screen? The answer is: possibly never.

The cinema? Not necessarily — many cinemas use large electronic screens now; there is no projection room, no projector painting the action on the screen from behind us with ghostly, dancing fingers of light. School? In the past, we had overhead projectors and even interactive whiteboards had lens systems, but these have largely been replaced by LED and LCD screens.

I believe that if you do not take the time to show the phenomenon of a single converging lens projecting a real image on to a passive white screen to your students, they are likely to have no familiar point of reference on which to build their understanding and lens diagrams will remain a puzzling set of lines that have little or no connection to their world.

Teaching Ray Diagrams

Start with a slide that looks something like this:

What represents the lens? The answer is not the blue oval. On ray diagrams, the lens is represented by the vertical dotted line. F1 and F2 are the focal points of this converging lens and they are each a distance f from the centre of the lens, where f is the focal length.

Now what happens to a light ray from the object that passes through the optical centre of the lens?

The answer, of course, is a big fat nothing. Light rays which pass through the optical centre of a thin lens are undeviated.

Now let’s track what happens to a light ray that travels parallel to the principal axis as shown?

Make sure that your students are aware that this light ray hasn’t ‘missed’ the lens. The lens is the vertical dotted line, not the blue oval. What will happen is that it will be deviated so that it passes through F1 (this is because this is a converging lens; if it had been a diverging lens then it would be be bent so that it appeared to come from F2).

The image is formed where the two light rays cross, as shown below.

We can see that the image is inverted and reduced.

The image is formed close to F1 but not precisely at F1. This is because, although the object is distant from the lens (‘distant’ in this case being ‘further than 2f away’) it is not infinitely far away. However, the further we move the object away from the lens, then the closer to F1 the image is formed. The image will be formed a distance f from the screen when the object in very, very, very large distance away — or an ‘infinite distance’ away, if you prefer.

One of my physics teachers liked to say that ‘Infinity starts at the window sill’. In the context of thinking about lenses, I think he was right . . .

Method for finding the focal length of a converging lens (image from https://www.youtube.com/watch?v=AElLVGW9kxQ)

Some free stuff . . .

The PowerPoint that I used to produce the ray diagrams above is here. It is imperfect in a lot of ways but. truth be told, it has served me well over a number of years. It also features some other slides and animations that you may find useful — enjoy!

Postscript: ‘Through a glass, darkly’

The phrase ‘Through a glass, darkly’ comes from the writings of the apostle Paul:

For now we see through a glass, darkly; but then face to face: now I know in part; but then shall I know even as also I am known.

KJV 1 Corinthians 13:12

The New International Version translates the phrase less poetically as ‘Now we see but a poor reflection as in a mirror.’

It has been argued that the ‘glass’ Paul was referring to were pieces of naturally-occurring semi-transparent mineral that were used in the ancient world as lenses or windows. They tended to produce a recognisable but distorted view of the world — hence, ‘darkly’.

Better technology means that there is much less distortion produced by our glasses — hence, ‘through a glass, lightly’.