Misconceptions and p-prims at ResearchED 2021

Many thanks to all those who attended my talk on “Dealing with Misconceptions: the p-prim and refining raw intuitions approach” and for the stimulating discussions afterwards!

And especially huge thanks to Bill Wilkinson for his help in sorting out some tech issues!

The PowerPoint can be downloaded below.

You can watch a short summary of the talk here.

The references are below with links to freely available copies (where I’ve been able to find them).

I think Redish and Kuo (2015) is an excellent introduction to the Resources Framework.


DiSessa, A. A. (1988). Knowledge in pieces. In G. Forman & P. B. Pufall (Eds.), Constructivism in the computer age (pp. 49–70). Lawrence Erlbaum Associates, Inc.

DiSessa, A. A. (1993). Toward an epistemology of physics. Cognition and instruction10(2-3), 105-225.

Hammer, D. (2000). Student resources for learning introductory physics. American Journal of Physics68(S1), S52-S59.

Nelmes, A. (2004). Putting conceptions in their place: using analogy to cue and strengthen scientifically correct conceptions.

Redish, E. F., & Kuo, E. (2015). Language of physics, language of math: Disciplinary culture and dynamic epistemology. Science & Education24(5), 561-590.

Booklet for teaching the Coulomb Train Model

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.

Extract from booklet

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….

Why we wrote ‘Cracking Key Concepts in Secondary Science’

From the Introduction

“We strongly believe that the central part of any science lesson or learning sequence is a well-crafted and executed explanation.

“But we are also aware that many – if not most – teachers have had very little training in how to actually go about crafting or executing their explanations. As advocates of evidence-informed teaching, we hope to bring a new perspective and set of skills to your teaching and empower you to take your place in the classroom as the imparter of knowledge.

“We do, however, wish to put paid to the suspicion that we advocate science lessons to be all chalk and talk: we strongly urge that teachers should use targeted and interactive questioning, model answers, practical work, guided practice and supported individual student practice in tandem with ‘teacher talk’. There is a time when the teacher should be a ‘guide on the side’ but the main focus of this book is to enable you to shine when you are called to be a science ‘sage on the stage’.

[…] “For many years, it seems that teacher explanation has been taken for granted. In a nation-wide focus on pedagogy, activity, student-led learning and social constructivism, the role of the teacher in taking challenging material and explaining it has been de-emphasised, with discovery, enquiry, peer-to-peer tuition and ‘figuring things out for yourself’ becoming ascendant. Not only that, but a significant number of influential organisations and individuals championed the cause of ‘talk-less teaching’ where the teacher was relegated to a near-voiceless ‘guide on the side’, sometimes enforced by observers with a stopwatch and an inflexible ‘teacher talk’ time limit.

“We earnestly hope that such egregious excesses are now a thing of the past; but we must admit that all too often, the mistakes engendered by well-meaning edu-initiatives live on, while whatever good they achieved lies composting with the CPD packs from ancient training days. Even if they are a thing of the past, there has been a collective deskilling when it comes to the crafting of a science explanation – there is little institutional wisdom and few, if any, resources for teachers to use as a reference.”

And that is one reason why we wrote the book.

What follows is an example of how we discuss a teaching sequence in the book.

Viewing waves through the lens of concrete to abstract progression

Many students have a concrete idea of a wave as something ‘wavy’ i.e. something with crests and troughs. However, in a normal teaching sequence we often shift from a wave profile representation to a wavefront representation to a ray diagram representation with little or no explanation — is it any wonder that some students get confused?

I have found it useful to consider the sequence from wave profile to wavefront to ray as representations that move from the concrete and familiar representation of waves as something that looks ‘wavy’ (wave profile) to something that looks less wavy (wavefront) to something more abstract that doesn’t look at all ‘wavy’ (ray diagram) as summarised in the table below.

Each row of the table shows the same situation represented by different conventions and it is important that students recognise this. You can quiz students to check they understand this idea. For example:

  • Top row: which part of the wave do the straight lines in the middle picture represent? (The crests of the waves.)
  • Top row: why are the rays in the last picture parallel? (To show that the waves are not spreading out.)
  • Middle row: compare the viewpoints in the first and middle picture. (The first is ‘from the side’, the middle is ‘from above, looking down.’)
  • Middle row: why are the rays in the last picture not parallel? (Because the waves are spreading out in a circular pattern.)

Once students are familiar with this shift in perspective, we can use to explain more complex phenomena such as refraction.

For example, we begin with the wave profile representation (most concrete and familiar to most students) and highlight the salient features.

Next, we move on to the same situation represented as wavefronts (more abstract).

Finally, we move on to the most abstract ray diagram representation.


‘Cracking Key Concepts in Secondary Science’ is available in multiple formats from Amazon and Sage Publishing. You can also order the paperback and hardback versions direct from your local bookshop 🙂

We hope you enjoy the book and find it useful.

STOP PRESS! 25% discount!

This is only available if you order directly from SAGE Publishing before 31/12/2021 and some terms and conditions apply (see SAGE website).

  1. Go to https://uk.sagepub.com/
  2. Search for ‘Cracking Key Concepts’
  3. Enter the discount code ‘UK21AUTHOR’ at the checkout.
  4. Wait for your copy to be delivered post-haste by Royal Mail.
  5. Enjoy!

From the Earth to the Sun in 270 BC

The sun is only 32 miles across and not more than 3000 miles from the Earth. It stands to reason it must be so. God made the sun to light the earth, and therefore must have placed it close to the task it was designed to do. What would you think of a man who built a house in Zion, Illinois and put the lamp to light it in Kenosha, Wisconsin?

Wilbur Glen Voliva c.1915 (quoted in Morgan and Langford 1982: 106)

Contrary to the above quote from noted ‘Flat Earther’ Wilbur Glen Voliva (1870-1942), we have very good reason to believe that the Sun is very far away from both the Earth and the Moon.

The argument was first put forward by Aristarchus (310 – 230 BC) and it relies on shadows and geometry.

The daylight moon

A surprisingly large proportion of people are unaware that the crescent or gibbous Moon is often visible in the daylight sky. (‘Gibbous’ = less than full, more than half.)

It’s actually only a completely Full Moon that is visible only at night since, almost by definition, it will rise at sunset and set at sunrise. (Which I find strange, because a common symbol for ‘night’ or ‘sleep’ is a stylised cresent Moon — but I digress…)

It’s only a phase . . .

A daylight Moon can provide a memorable demonstration of why the Moon has phases. Just stand in a patch of sunlight and hold up a ball when the Moon is in the sky…

The ping pong ball has the same ‘phase’ as the Moon. The Moon is also a ball partially lit by the Sun but much further away… [Image from https://skyandtelescope.org/astronomy-resources/what-are-the-phases-of-the-moon/]

By the light of the silvery (half) Moon…

Aristarchus realised that when the Moon was half-lit by the Sun as viewed from the Earth (the phases known as ‘First quarter’ and ‘Last quarter’) then a line drawn from the centre of the Earth to the Moon would be at 90 degrees to a line connecting the Moon to the Sun as shown below.

To an observer on Earth, the angular distance θ measured between the Moon and the Sun would be small if the Sun was close to the Earth; conversely, the angle θ would be large if the Sun was far away from the Earth.

Aristarchus realised that if he measured the angle θ between the Moon when it was half lit (i.e. during First Quarter or Last Quarter) and the Sun, then he would be able to find the ratio between the Earth-Moon distance and the Earth-Sun distance. Since he had previously worked out a method to measure the Earth-Moon distance, this meant that he could calculate the distance from the Earth to the Sun.

Modern measurements of the angle θ produce a mean value of 89 degrees and 51.2 minutes of arc (it does vary as the Moon has an elliptical rather than a circular orbit).

Using some trigonometry we calculate that the Earth-Sun distance (ES) is 400 times the Earth-Moon distance.

Quibbles and Caveats

Aristarchus measured an angle of 87 degrees for θ which meant that he calculated that the Sun was only 20 times further away from the Earth than the Moon. Also, trigonometrical techniques were not available to him which meant he had to use a geometrical method to calculate the Earth-Sun distance. However, Aristarchus achievement is still worth celebrating!


This is part 3 of a series exploring how humans ‘measured the size of the sky’.

Part 1: How Eratosthenes measured the size of the Earth

Part 2: How Aristarchus measured the distance from the Earth to the Moon

REFERENCE

Langford D & Morgan C.. (1982), Facts and Fallacies: A book of definitive mistakes and misguided predictions. Corgi Books.

From the Earth to the Moon in 270 BC

The brain is wider than the sky,

For, put them side by side,

The one the other will include

With ease, and you beside.

Emily Dickinson

How did human beings first work out the distance from the Earth to the Moon?

Aristrarchus of Samos (310 BC – 230 BC) figured out a way to do so in terms of the radius of the Earth in 270 BC. Combined with Eratosthenes’ measurement of the radius of the Earth (c. 240 BC) it enabled people to calculate the actual distance to the Moon. The ancient Greeks used a measurement of distance called stadia (singular: stadium) but we will present the measurements here in terms of kilometres.

Magic with a shadow, not with mirrors

Aristarchus used the fact that the Moon passes through the Earth’s shadow during a total lunar eclipse, which happen once every two to three years on average.

What does a total lunar eclipse look like? Watch this amazing 33 second time lapse video from astrophotographer Bartosz Wojczyński.

https://www.youtube.com/watch?v=LK_44AbfH2Q Note that Mr Wojczyński altered the exposure time of each shot to compensate for the reduced brightness of the Moon as it crossed into the shadow. For reference, the exposure time for the brightly lit Moon was 1/2500 second, and for the dim ‘Blood Moon’ (turned red by sunlight refracted by the Earth’s atmosphere) it was 6 seconds.

The video is sped up so that 1 second of video represents 8 minutes of real time. In the video, the Moon is in shadow for 24 seconds which equates to 8 x 24 = 192 minutes or 3 hours 12 minutes. We will use this later to model Aristarchus’ original calculation.

It’s always Aristarchus before the dawn…

Aristarchus began with the assumption that the Earth of radius r creates a cylinder of shadow that is 2r wide as shown in the diagram below.

The Moon orbits the Earth on a roughly circular path of radius R so it cover a total distance of 2πR. This means that its average speed over its whole journey is 2πR/T where T is the orbital period of the Moon, which is 27.3 days or 27.3 x 24 = 655.2 hours.

The average speed of the Moon as it passes through the Earth’s shadow is 2r / t where t is the time for a lunar eclipse (3 hours 12 minutes, in our example).

The average speed of the speed of the Moon is the same in both instances so we can write:

We can simplify by cancelling out the common factor of two:

Then we can rearrange to make R the subject:

Putting in values for t = 3 hours 12 minutes or 3.2 hours, T = 655.2 hours and Eratosthenes’ value for the radius of the Earth r = 6371 km (which was established a few years later):

So now they do it with mirrors…

Aristarchus’ value is just a shade over 7% too large compared with the modern value of the Earth-Moon distance of 384 400 km, but is impressive for a first approximation carried out in antiquity!

The modern value is measured in part by directing laser beams on to special reflectors left on the Moon’s surface by the Apollo astronauts and also the automated Lunokhod missions. Under ideal conditions, this method can measure the Earth-Moon distance to the nearest millimetre.

Quibbles, Caveats and Apologies

Aristarchus’ estimate was too large in part because of his assumption that Earth’s shadow was a cylinder with a uniform diameter. The Sun is an extended light source so Earth’s shadow forms a cone as shown below.

The value of t is smaller than it would if the shadow was 2r wide, leading to a too-large value of R using Aristarchus’ method.

Also, the plane of the Moon’s orbit is tilted with respect to the plane of the Earth’s orbit. This means that the path of the Moon during an eclipse might not pass through the ‘thickest’ part of the shadow. Aristarchus used the average time t calculated from a number of lunar eclipses.

When timing the lunar eclipse shown in Mr Wojczyński’s excellent video, I started the clock when the leading edge of the Moon entered the shadow, but I confess that I ‘cheated’ a little bit by not stopping the clock when the leading edge of the Moon left the shadow — the error is entirely mine and was deliberate in order to arrive at a reasonable value of R for pedagogic impact.

UPDATE: You could also watch this stunning visualisation of a lunar eclipse from Andrew McCarthy where the shadow of the Earth is tracked rather than the Moon.


This is part 2 of a series exploring how humans ‘measured the size of the sky’.

Part 1: How Eratosthenes measured the size of the Earth

Part 3: How Aristarchus measured the distance from the Earth to the Sun

Measuring the radius of the Earth in 240 BC

The brain is wider than the sky,
For, put them side by side,
The one the other will include
With ease, and you beside.

Emily Dickinson, ‘The Brain’

Most science teachers find that ‘Space’ is one of the most enduringly fascinating topics for many students: the sense of wonder engendered as our home planet becomes lost in the empty vastness of the Solar System, which then becomes lost in the trackless star-studded immensity of the Milky Way galaxy, is a joy to behold.

But a common question asked by students is: How do we know all this? How do we know the distance to the nearest star to the Sun is 4 light-years? Or how do we know the distance to the Sun? Or the Moon?

I admit, with embarrassment, that I used to answer with a casual and unintentionally-dismissive ‘Oh well, scientists have measured them!’ which (though true) must have sounded more like a confession of faith rather than a sober recounting of empirical fact. Which, to be fair, it probably was; simply because I had not yet made the effort to find out how these measurements were first taken.

The technological resources available to our ancestors would seem primitive and rudimentary to our eyes but, coupled with the deep well of human ingenuity that I like to think is a hallmark of our species, it proved not just ‘world-beating’ but ‘universe-beating’.

I hope you enjoy this whistle stop tour of this little-visited corner of the scientific hinterland, and choose to share some these stories with your students. It is good to know that the brain is indeed ‘wider than the sky’.

I have presented this in a style and format suitable for sharing and discussing with KS3/KS4 students (11-16 year olds).

Mad dogs and Eratosthenes go out in the midday Sun…

To begin at the beginning: the first reliable measurement of the size of the Earth was made in 240 BC and it all began (at least in this re-telling) with the fact that Eratosthenes liked talking to tourists. (‘Err-at-oss-THen-ees’ with the ‘TH’ said as in ‘thermometer’ — never forget that students of all ages often welcome help in learning how to pronounce unfamiliar words)

Alexandria (in present day Egypt) was a thriving city and a tourist magnet. Eratosthenes made a point of speaking to as many visitors as he could. Their stories, taken with a pinch of salt, were an invaluable source of information about the wider world. Eratosthenes was chief librarian of the Library of Alexandria, regarded as one of the Seven Wonders of the World at the time, and considered it his duty to collect, catalogue and classify as much information as he could.

One visitor, present in Alexandria on the longest day of the year (June 21st by our calendar), mentioned something in passing to Eratosthenes that the Librarian found hard to forget: ‘You know,’ said the visitor, ‘at noon on this day, in my home town there are no shadows.’

How could that be? pondered Eratosthenes. There was only one explanation: the Sun was directly overhead at noon on that day in Syene (the tourist’s home town, now known as Aswan).

The same was not true of Alexandria. At noon, there was a small but noticeable shadow. Eratosthenes measured the angle of the shadow at midday on the longest day. It was seven degrees.

No shadows at Syene, but a 7 degree shadow at Alexandria at the exact same time. Again, there was only one explanation: Alexandria was ’tilted’ by 7 degrees with respect to Syene.

Seven degrees of separation

The sphericity of the Earth had been recognised by astronomers from c. 500 BC so this difference was no surprise to Eratosthenes, but what he realised that since he was comparing the length of shadows at two places on the Earth’s surface at the same time then the 7o wasn’t just the angle of the shadow: 7o was the angle subtended at the centre of the Earth by radial lines drawn from both locations.

Eratosthenes paid a person to pace out the distance between Alexandria and Syene. (This was not such an odd request as it sounds to our ears: in the ancient world there were professionals called bematists who were trained to measure distances by counting their steps.)

It took the bematist nearly a month to walk that distance and it turned out to be 5000 stadia or 780 km by our measurements.

Eratosthenes then used a simple ratio method to calculate the circumference of the Earth, C:

Then:

The modern value for the radius of the Earth is 6371 km.

Ifs and buts…

There is still some debate as to the actual length of one Greek stadium but Eratosthenes’ measurement is generally agreed to within 1-2% of the modern value.

Sadly, none of the copies of the book where Eratosthenes explained his method called On the measure of the earth have survived from antiquity so the version presented here is a simplified one outlined by Cleomedes in a later book. For further details, readers are directed to the excellent Wikipedia article on Eratosthenes.

Astronomer Carl Sagan also memorably explained this method in his 1980 TV documentary series Cosmos.

You might want to read…

This is part of a series exploring how humans ‘measured the size of the sky’:

Part 2: How Aristarchus measured the distance between the Earth and the Moon

Part 3: How Aristarchus measured the distance between the Earth and the Sun

Binding energy: the pool table analogy

Nuclear binding energy and binding energy per nucleon are difficult concepts for A-level physics students to grasp. I have found the ‘pool table analogy’ that follows helpful for students to wrap their heads around these concepts.

Background

Since mass and energy are not independent entities, their separate conservation principles are properly a single one — the principle of conservation of mass-energy. Mass can be created or destroyed , but when this happens, an equivalent amount of energy simultaneously vanishes or comes into being, and vice versa. Mass and energy are different aspects of the same thing.

Beiser 1987: 29

E = mc2

There, I’ve said it. This is the first time I have directly referred to this equation since starting this blog in 2013. I suppose I have been more concerned with the ‘andallthat‘-ness side of things rather than E=mc2. Well, no more! E=mc2 will form the very centre of this post. (And about time too!)

The E is for ‘rest energy’: that is to say, the energy an object of mass m has simply by virtue of being. It is half the energy that would be liberated if it met its antimatter doppelganger and particles and antiparticles annihilated each other. A scientist in a popular novel sternly advised a person witnessing an annihilation event to ‘Shield your eyes!’ because of the flash of electromagnetic radiation that would be produced.

Well, you could if you wanted to, but it wouldn’t do much good since the radiation would be in the form of gamma rays which are to human eyes what the sound waves from a silent dog whistle are to human ears: beyond the frequency range that we can detect.

The main problem is likely to be the amount of energy released since the conversion factor is c2: that is to say, the velocity of light squared. For perspective, it is estimated that the atomic bomb detonated over Hiroshima achieved its devastation by directly converting only 0.0007 kg of matter into energy. (That would be 0.002% of the 38.5 kg of enriched uranium in the bomb.)

Matter contains a lot of energy locked away as ‘rest energy’. But these processes which liberate rest energy are mercifully rare, aren’t they?

No, they’re not. As Arthur Beiser put it in his classic Concepts of Modern Physics:

In fact, processes in which rest energy is liberated are very familiar. It is simply that we do not usually think of them in such terms. In every chemical reaction that evolves energy, a certain amount of matter disappears, but the lost mass is so small a fraction of the total mass of the reacting substances that it is imperceptible. Hence the ‘law’ of conservation of mass in chemistry.

Beiser 1987: 29

Building a helium atom

The constituents of a helium nucleus have a greater mass when separated than they do when they’re joined together.

Here, I’ll prove it to you:

The change in mass due to the loss of energy as the constituents come together is appreciable as a significant fraction of its original mass. Although 0.0293/4.0319*100% = 0.7% may not seem like a lot, it’s enough of a difference to keep the Sun shining.

The loss of energy is called the binding energy and for a helium atom it corresponds to a release of 27 MeV (mega electron volts) or 4.4 x 10-12 joules. Since there are four nucleons (particles that make up a nucleus) then the binding energy per nucleon (which is a guide to the stability of the new nucleus) is some 7 MeV.

But why must systems lose energy in order to become more stable?

The Pool Table Analogy for binding energy

Imagine four balls on a pool table as shown.

The balls have the freedom to move anywhere on the table in their ‘unbound’ configuration.

However, what if they were knocked into the corner pocket?

To enter the ‘bound’ configuration they must lose energy: in the case of the pool balls we are talking about gravitational potential energy, a matter of some 0.30 J per ball or a total energy loss of 4 x 0.30 = 1.2 joules.

The binding energy of a pool table ‘helium nucleus’ is thus some 1.2 joules while the ‘binding energy per nucleon’ is 0.30 J. In other words, we would have to supply 1.2 J of energy to the ‘helium nucleus’ to break the forces binding the particles together so they can move freely apart from each other.

Just as a real helium nucleus, the pool table system becomes more stable when some of its constituents lose energy and less stable when they gain energy.


Reference

Beiser, A. (1987). Concepts of modern physics. McGraw-Hill Companies.

Visualising How Transformers Work

‘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

A GIF showing the basic principle of a transformer.
(BTW This can be copied and pasted into a presentation if you wish,)

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.)

Stepping Down

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.

A GIF showing the basic principle of a step down transformer.
(BTW This can be copied and pasted into a presentation if you wish,)

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.

Stepping up

A GIF showing the basic principle of a step up transformer.
(BTW This can be copied and pasted into a presentation if you wish,)

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)

A Gnome-inal Value for ‘g’

The Gnome Experiment Kit from precision scale manufacturers Kern and Sohn.

. . . setting storms and billows at defiance, and visiting the remotest parts of the terraqueous globe.

Samuel Johnson, The Rambler, 17 April 1750

That an object in free fall will accelerate towards the centre of our terraqueous globe at a rate of 9.81 metres per second per second is, at best, only a partial and parochial truth. It is 9.81 metres per second per second in the United Kingdom, yes; but the value of both acceleration due to free fall and the gravitational field strength vary from place to place across the globe (and in the SI System of measurement, the two quantities are numerically equal and dimensionally equivalent).

For example, according to Hirt et al. (2013) the lowest value for g on the Earth’s surface is atop Mount Huascarán in Peru where g = 9.7639 m s-2 and the highest is at the surface of the Arctic Ocean where g = 9.8337 m s-2.

Why does g vary?

There are three factors which can affect the local value of g.

Firstly, the distribution of mass within the volume of the Earth. The Earth is not of uniform density and volumes of rock within the crust of especially high or low density could affect g at the surface. The density of the rocks comprising the Earth’s crust varies between 2.6 – 2.9 g/cm3 (according to Jones 2007). This is a variation of 10% but the crust only comprises about 1.6% of the Earth’s mass since the density of material in the mantle and core is far higher so the variation in g due this factor is probably of the order of 0.2%.

Secondly, the Earth is not a perfect sphere but rather an oblate spheroid that bulges at the equator so that the equatorial radius is 6378 km but the polar radius is 6357 km. This is a variation of 0.33% but since the gravitational force is proportional to 1/r2 let’s assume that this accounts for a possible variation of the order of 0.7% in the value of g.

Thirdly, the acceleration due to the rotation of the Earth. We will look in detail at the theory underlying this in a moment, but from our rough and ready calculations above, it would seem that this is the major factor accounting for any variation in g: that is to say, g is a minimum at the equator and a maximum at the poles because of the Earth’s rotation.


The Gnome Experiment

In 2012, precision scale manufacturers Kern and Sohn used this well-known variation in the value of g to embark on a highly successful advertising campaign they called the ‘Gnome Experiment’ (see link 1 and link 2).

Whatever units their lying LCD displays show, electronic scales don’t measure mass or even weight: they actually measure the reaction force the scales exert on the item in their top pan. The reading will be affected if the scales are accelerating.

In diagram A, the apple is not accelerating so the resultant upward force on the apple is exactly 0.981 N. The scales show a reading of 0.981/9.81 = 0.100 000 kg = 100.000 g (assuming, of course, that they are calibrated for use in the UK).

In diagram B, the apple and scales are in an elevator that is accelerating upward at 1.00 metres per second per second. The resultant upward force must therefore be larger than the downward weight as shown in the free body diagram. The scales show a reading of 1.081/9.81 – 0.110 194 kg = 110.194 g.

In diagram C, the the apple and scales are in an elevator that is accelerating downwards at 1.00 metres per second per second. The resultant upward force must therefore be smaller than the downward weight as shown in the free body diagram. The scales show a reading of 0.881/9.81 – 0.089 806 kg = 89.806 g.


Never mind the weight, feel the acceleration

Now let’s look at the situation the Kern gnome mentioned above. The gnome was measured to have a ‘mass’ (or ‘reaction force’ calibrated in grams, really) of 309.82 g at the South Pole.

Showing this situation on a diagram:

Looking at the free body diagram for Kern the Gnome at the equator, we see that his reaction force must be less than his weight in order to produce the required centripetal acceleration towards the centre of the Earth. Assuming the scales are calibrated for the UK this would predict a reading on the scales of 3.029/9.81= 0.30875 kg = 308.75 g.

The actual value recorded at the equator during the Gnome Experiment was 307.86 g, a discrepancy of 0.3% which would suggest a contribution from one or both of the first two factors affecting g as discussed at the beginning of this post.

Although the work of Hirt et al. (2013) may seem the definitive scientific word on the gravitational environment close to the Earth’s surface, there is great value in taking measurements that are perhaps more directly understandable to check our comprehension: and that I think explains the emotional resonance that many felt in response to the Kern Gnome Experiment. There is a role for the ‘artificer’ as well as the ‘philosopher’ in the scientific enterprise on which humanity has embarked, but perhaps Samuel Johnson put it more eloquently:

The philosopher may very justly be delighted with the extent of his views, the artificer with the readiness of his hands; but let the one remember, that, without mechanical performances, refined speculation is an empty dream, and the other, that, without theoretical reasoning, dexterity is little more than a brute instinct.

Samuel Johnson, The Rambler, 17 April 1750

References

Hirt, C., Claessens, S., Fecher, T., Kuhn, M., Pail, R., & Rexer, M. (2013). New ultrahigh‐resolution picture of Earth’s gravity fieldGeophysical research letters40(16), 4279-4283.

Jones, F. (2007). Geophysics Foundations: Physical Properties: Density. University of British Columbia website, accessed on 2/5/21.

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Nature Abhors A Change In Flux

Aristotle memorably said that Nature abhors a vacuum: in other words. he thought that a region of space entirely devoid of matter, including air, was logically impossible.

Aristotle turned out to be wrong in that regard, as he was in numerous others (but not quite as many as we – secure and perhaps a little complacent and arrogant as we look down our noses at him from our modern scientific perspective – often like to pretend).

An amusing version which is perhaps more consistent with our current scientific understanding was penned by D. J. Griffiths (2013) when he wrote: Nature abhors a change in flux.

Magnetic flux (represented by the Greek letter phi, Φ) is a useful quantity that takes account of both the strength of the magnetic field and its extent. It is the total ‘magnetic flow’ passing through a given area. You can also think of it as the number of magnetic field lines multiplied by the area they pass through so a strong magnetic field confined to a small area might have the same flux (or ‘effect’) as weaker field spread out over a large area.


Lenz’s Law

Emil Lenz formulated an earlier statement of the Nature abhors a change of flux principle when he stated what I think is the most consistently underrated laws of electromagnetism, at least in terms of developing students’ understanding:

The current induced in a circuit due to a change in a magnetic field is directed to oppose the change in flux and to exert a mechanical force which opposes the motion.

Lenz’s Law (1834)

This is a qualitative rather than a quantitive law since it is about the direction, not the magnitude, of an induced current. Let’s look at its application in the familiar A-level Physics context of dropping a bar magnet through a coil of wire.


Dropping a magnet through a coil in pictures

Picture 1

In picture 1 above, the magnet is approaching the coil with a small velocity v. The magnet is too far away from the coil to produce any magnetic flux in the centre of the coil. (For more on the handy convention I have used to draw the coils and show the current flow, please click on this link.) Since there is no magnetic flux, or more to the point, no change in magnetic flux, then by Faraday’s Law of Electromagnetic Induction there is no induced current in the coil.

Picture 2

in picture 2, the magnet has accelerated to a higher velocity v due to the effect of gravitational force. The magnet is now close enough so that it produces a magnetic flux inside the coil. More to the point, there is an increase in the magnetic flux as the magnet gets closer to the coil: by Faraday’s Law, this produces an induced current in the coil (shown using the dot and cross convention).

To ascertain the direction of the current flow in the coil we can use Lenz’s Law which states that the current will flow in such a way so as to oppose the change in flux producing it. The red circles show the magnetic field lines produced by the induced current. These are in the opposite direction to the purple field lines produced by the bar magnet (highlighted yellow on diagram 2): in effect, they are attempting to cancel out the magnetic flux which produce them!

The direction of current flow in the coil will produce a temporary north magnetic pole at the top of the coil which, of course, will attempt to repel the falling magnet; this is ‘mechanical force which opposes the motion’ mentioned in Lenz’s Law. The upward magnetic force on the falling magnet will make it accelerate downward at a rate less than g as it approaches the coil.

Picture 3

In picture 3, the purple magnetic field lines within the volume of the coil are approximately parallel so that there will be no change of flux while the magnet is in this approximate position. In other words, the number of field lines passing through the cross-sectional area of the coil will be approximately constant. Using Faraday’s Law, there will be no flow of induced current. Since there is no change in flux to oppose, Lenz’s Law does not apply. The magnet will be accelerating downwards at g.

Picture 4

As the magnet emerges from the bottom of the coil, the magnetic flux through the coil decreases. This results in a flow of induced current as per Faraday’s Law. The direction of induced current flow will be as shown so that the red field lines are in the same direction as the purple field lines; Lenz’s Law is now working to oppose the reduction of magnetic flux through the coil!

A temporary north magnetic pole is generated by the induced current at the lower end of the coil. This will produce an upward magnetic force on the falling magnet so that it accelerates downward at a rate less than g. This, again, is the ‘mechanical force which opposes the motion’ mentioned in Lenz’s Law.


Dropping a magnet through a coil in graphical form

This would be one of my desert island graphs since it is such a powerfully concise summary of some beautiful physics.

The graph shows the reversal in the direction of the current as discussed above. Also, the maximum induced emf in region 2 (blue line) is less than that in region 4 (red line) since the magnet is moving more slowly.

What is more, from Faraday’s Law (where ℇ is the induced emf and N is total number of turns of the coil), the blue area is equal to the red area since:

and N and ∆Φ are fixed values for a given coil and bar magnet.

As I said previously, there is so much fascinating physics in this graph that I think it worth exploring in depth with your A level Physics students 🙂

Other news

If you have enjoyed this post, then you may be interested to know that I’ve written a book! Cracking Key Concepts in Secondary Science (co-authored with Adam Boxer and Heena Dave) is due to be published by Corwin in July 2021.

References

Lenz, E. (1834), “Ueber die Bestimmung der Richtung der durch elektodynamische Vertheilung erregten galvanischen Ströme”, Annalen der Physik und Chemie107 (31), pp. 483–494

Griffiths, David (2013). Introduction to Electrodynamics. p. 315.