The ‘all-in-a-row’ circuit diagram convention for series and parallel circuits

Circuit diagrams can be seen either as pictures or abstractions but it is clear that pupils often find it hard to recognise the circuits in the practical situation of real equipment. Moreover, Caillot found that students retain from their work with diagrams strong images rather than the principles they are intended to establish. The topological arrangement of a diagram or a drawing presents problems for pupils which are easily overlooked. It seems that pupils’ spatial abilities affect their use of circuit diagrams: they sometimes do not regard as identical several circuits, which, though identical, have been rotated so as to have a different spatial arrangement. […] Niedderer found that pupils, when asked whether a circuit diagram would ‘work’ in practice, more often judged symmetrical diagrams to be functioning than non-symmetric ones.

Driver et al. (1994): 124 [Emphases added]

For the reasons outlined by Driver and others above, I think it’s a good idea to vary the way that we present circuit diagrams to students when teaching electric circuits. If students always see circuit diagrams presented so that (say) the cell is at the ‘top’ and ‘facing’ a certain way; or that they are drawn so that they are symmetrical (which is an aesthetic rather that a scientific choice), then they may well incorrectly infer that these and other ‘accidental’ features of our circuit diagrams are the essential aspects that they should pay the most attention to.

One ‘shake it up’ strategy is to redraw a circuit diagram using the ‘all-in-a-row’ convention.

If you arrange the real components in the ‘all-in-a-row’ arrangement, then a standard digital voltmeter has, what is in my opinion a regrettably underused functionality, that will show:

  • ‘positive’ potential differences: that is to say, the energy added to the coulombs as they pass through a cell or the electromotive force; and
  • ‘negative’ potential differences: that is to say, the energy removed from each coulomb as they pass through a resistor; these can be usefully referred to as ‘potential drops’

This can be shown on circuit diagrams as shown below/

In other words, the difference between the potential difference across the cell (energy being transferred into the circuit from the chemical energy store of the cell) is explicitly distinguished from the potential difference across the resistor (energy being transferred from the resistor into the thermal energy store of the surroundings). The all-in-a-row convention neatly sidesteps a common misconception that the potential difference across a cell is equal to the potential difference across a resistor: they are not. While they may be numerically equal, they are different in sign, as a consequence of Kirchoff’s Second Law. As I have suggested before, I think that this misconception is due to the ‘hidden rotation‘ built into standard circuit diagrams.

Potential divider circuits and the all-in-a-row convention

Although I am normally a strong proponent of the ‘parallel first heresy‘, I’ll go with the flow of ‘series circuit first’ in this post.

Diagrams 2 and 3 in the sequence show that the energy supplied to the coulombs (+1.5 V or 1.5 joules per coulomb) by the cell is transferred from the coulombs as they pass through the double resistor combination. Assuming that R1 = R2 then, as diagram 4 shows, 0.75 joules will be transferred out of each coulomb as they pass through R1; as diagram 5 shows, 0.75 joules will be transferred out of each coulomb as they pass through R2.

Parallel circuits and the all-in-a-row convention

I’ve written about using the all-in-a-row convention to help explain current flow in parallel circuits here, so I will focus on understanding potential difference in parallel circuit in this post.

Again, diagrams 2 and 3 in the sequence show that the positive 3.0 V potential difference supplied by the cell is numerically equal (but opposite in sign) to the negative 3.0 V potential drop across the double resistor combination. It is worth bearing in mind that each coulomb passing through the cell gains 3.0 joules of energy from the chemical energy store of the cell. Diagrams 4 and 5 show that each coulomb passing through either R1 or R1 loses its entire 3.0 joules of energy as it passes through that resistor. The all-in-a-row convention is useful, I think, for showing that each coulomb passes through just one resistor as it makes a single journey around the circuit.

Acknowledgements

Circuit simulations from the excellent https://phet.colorado.edu/sims/html/circuit-construction-kit-dc/latest/circuit-construction-kit-dc_en.html

Circuit diagrams drawn using https://www.circuit-diagram.org/editor/

Reference

Driver, R., Squires, A., Rushworth, P., & Wood-Robinson, V. (1994). Making sense of secondary science: Research into children’s ideas. Routledge.

Just a moment . . .

Many years ago, I was taught this compact and intuitive convention to show turning moments. I think it should be more widely known, as it not only is concise and powerful, but also meets the criterion of being an effective form of dual coding which is helpful for both GCSE and A-level Physics students.

Let’s look at an example question.

Let’s start by ‘annotating the hell’ out of the diagram.

We could take moments around any of the marked points A-E on the diagram. However, we’re going to take moments around B as it enables us to ignore the upward reaction force acting on the rule at B. (This force is not shown on the diagram.)

To indicate that we’re going to be considering the sum of the clockwise moments about point B, we use this intuitive notation:

If we consider the sum of anticlockwise moments about point B, we use this:

We lay out our calculations of the total clockwise and anticlockwise moments about B as follows.

We show that we are going to apply the Principle of Moments (the sum of clockwise moments is equal to the sum of anticlockwise moments for an object in equilibrium) like this:

The rest, as they say, is not history but algebra:

I hope you find this ‘momentary’ convention useful(!)

Showing complex energy transfers using bar models

Real life energy transfers can be messy. That is to say, they are complicated and difficult to understand. I think many students get lost in the dense forest of verbiage that has to be deployed to describe their detail and nuance. Bar models are, I think, an effective teaching tool to avoid cognitive overload, especially for GCSE Physics and Combined Science students.

Windmills of our minds

As an example, let’s consider a wind turbine used to generate electricity. As a starting point, let’s think about how much of the kinetic energy ‘harvested’ by the blades is transferred to the generator. The answer is, of course. not as much as we would hope. The majority is, hopefully, but a significant proportion is unavoidably lost via work done by friction to the thermal energy store of the gears.

This can be shown in a visually impactful way using the Bar Model approach:

Note that in this style of energy transfer diagram, the Principle of Conservation of Energy is communicated visually via the width of the bars. The bottom ‘End’ bar has to be exactly the same width as the top ‘Start’ bar.

What happens if a helpful maintenance engineer tops up the oil reservoir of the wind turbine? Well, we have a much happier situation, as shown below.

As we can see, a much greater proportion of the total energy is transferred usefully (and can be used to generate electrical power) in a well-maintained wind turbine.

Using energy efficient appliances in the home

How can we explain the advantages of using more efficient appliances in the home?

A diagram like this can help. The household that uses less efficient appliances has to buy more energy from their energy supplier to achieve exactly the same outcomes as the first. This is both more costly for the household as well as demanding that more resources are needed to generate electricity for no good reason.

Parachute vs. no parachute

Exactly the same amount of energy is transferred from the gravitational energy store of a parachutist whether their parachute deploys successfully or not. However, in the case of a successful deployment, much more energy is transferred into the thermal energy store of the surroundings than into their kinetic energy store. This helps ensure a safe landing!

Introducing vectors (part 1)

I think that teaching vectors to 14-16 year olds is a bit like teaching them to play the flute; that is to say, it’s a bit like teaching them to play the flute as presented by Monty Python (!)

Monty Python (1972), ‘How to play the flute’

Part of the trouble is that the definition of a vector is so deceptively and seductively easy: a vector is a quantity that has both magnitude and direction.

There — how difficult can the rest of it be? Sadly, there’s a good deal more to vectors than that, just as there’s much more to playing the flute than ‘moving your fingers up and down the outside'(!)

What follows is a suggested outline teaching schema, with some selected resources.

Resultant vector = total vector: the ‘I’ phase

‘2 + 2 = 4’ is often touted as a statement that is always obviously and self-evidently true. And so it is — arithmetically and for mere scalar quantities. In fact, it would be more precisely rendered as ‘scalar 2 + scalar 2 = scalar 4’.

However, for vector quantities, things are a wee bit different. For vectors, it is better to say that ‘vector 2 + vector 2 = a vector quantity with a magnitude somewhere between 0 and 4’.

For example, if you take two steps north and then a further two steps north then you end up four steps away from where you started. Also, if you take two steps north and then two steps south, then you end up . . . zero steps from where you started.

So much for the ‘zero’ and ‘four’ magnitudes. But where do the ‘inbetween’ values come from?

Simples! Imagine taking two steps north and then two steps east — where would you end up? In other words, what distance and (since we’re talking about vectors) in what direction would you be from your starting point?

This is most easily answered using a scale diagram.

To calculate the vector distance (aka displacement) we draw a line from the Start to the End and measure its length.

The length of the line is 2.8 cm which means that if we walk 2 steps north and 2 steps east then we up a total vector distance of 2.8 steps away from the Start.

But what about direction? Because we are dealing with vector quantities, direction just as important as magnitude. We draw an arrowhead on the purple line to emphasise this.

Students may guess that the direction of the purple ‘resultant’ vector (that is to say, it is the result of adding two vectors) is precisely north-east, but this can be a vague description so let’s use a protractor so that we can find the compass bearing.

And thus we find that the total resultant vector — the result of adding 2 steps north and 2 steps east — is a displacement of 2.8 steps on a compass bearing of 045 degrees.

Resultant vector = total vector: the ‘We’ phase

How would we go about finding the resultant vector if we moved 3 metres north and 4 metres east? If you have access to an interactive whiteboard, you could choose to use this Jamboard for this phase. (One minor inconvenience: you would have to draw the straight lines freehand but you can use the moveable and rotatable ruler and protractor to make measurements ‘live’ with your class.)

We go through a process similar to the one outlined above.

  • What would be a suitable scale?
  • How long should the vertical arrow be?
  • How long should the horizontal arrow be?
  • Where should we place the ‘End’ point?
  • How do we draw the ‘resultant’ vector?
  • What do we mean by ‘resultant vector’?
  • How should we show the direction of the resultant vector?
  • How do we find its length?
  • How do we convert the length of the arrow on the scale diagram into the magnitude of the displacement in real life?

The resultant vector is, of course, 5.0 m at a compass bearing of 053 degrees.

Resultant vector = total vector: the ‘You’ phase

Students can complete the questions on the worksheets which can be printed from the PowerPoint below.

Answers are shown on this second PowerPoint, plus an optional digital ruler and protractor are included on the third slide if you wish to use them.

Enjoy!

Electromagnetic induction — using the LEFT hand rule…?

They do observe I grow to infinite purchase,
The left hand way;

John Webster, The Duchess of Malfi

Electromagnetic induction — the fact that moving a conductor inside a magnetic field in a certain direction will generate (or induce) a potential difference across its ends — is one of those rare-in-everyday-life phenomena that students very likely will never have come across before. In their experience, potential differences have heretofore been produced by chemical cells or by power supply units that have to be plugged into the mains supply. Because of this, many of them struggle to integrate electromagnetic induction (EMI) into their physical schema. It just seems such a random, free floating and unconnected fact.

What follows is a suggested teaching sequence that can help GCSE-level students accept the physical reality of EMI without outraging their physical intuition or appealing to a sketchily-explained idea of ‘cutting the field lines’.

‘Look, Ma! No electrical cell!’

I think it is immensely helpful for students to see a real example of EMI in the school laboratory, using something like the arrangement shown below.

A length of copper wire used to cut the magnetic field between two Magnadur magnets on a yoke will induce (generate) a small potential difference of about 5 millivolts. What is particularly noteworthy about doing this as a class experiment is how many students ask ‘How can there be a potential difference without a cell or a power supply?’

The point of this experiment is that in this instance the student is the power supply: the faster they plunge the wire between the magnets then the larger the potential difference that will be induced. Their kinetic energy store is being used to generate electrical power instead of the more usual chemical energy store of a cell.

But how to explain this to students?

A common option at this point is to start talking about the conductor cutting magnetic field lines: this is hugely valuable, but I recommend holding fire on this picture for now — at least for novice learners.

What I suggest is that we explain EMI in terms of a topic that students will have recently covered: the motor effect.

This has two big ‘wins’:

  • It gives a further opportunity for students to practice and apply their knowledge of the motor effect.
  • Students get the chance to explain an initially unknown phenomenon (EMI) in terms of better understood phenomenon (motor effect). The motor effect will hopefully act as the footing (to use a term from the construction industry) for their future understanding of EMI.

Explaining EMI using the motor effect

The copper conductor contains many free conduction electrons. When the conductor is moved sharply downwards, the electrons are carried downwards as well. In effect, the downward moving conductor can be thought of as a flow of charge; or, more to the point, as an electrical current. However, since electrons are negatively charged, this downward flow of negative charge is equivalent to an upward flow of positive charge. That is to say, the conventional current direction on this diagram is upwards.

Applying Fleming Left Hand Rule (FLHR) to this instance, we find that each electron experiences a small force tugging it to the left — but only while the conductor is being moved downwards.

This results in the left hand side of the conductor becoming negatively charged and the right hand side becoming positively charged: in short, a potential difference builds up across the conductor. This potential difference only happens when the conductor is moving through the magnetic field in such a way that the electrons are tugged towards one end of the conductor. (There is, of course, the Hall Effect in some other instances, but we won’t go into that here.)

As soon as the conductor stops moving, the potential difference is no longer induced as there is no ‘charge flow’ through the magnetic field and, hence, no current and no FLHR motor effect force acting on the electrons.

Faraday’s model of electromagnetic induction

Michael Faraday (1791-1867) discovered the phenomenon of electromagnetic induction in 1831 and explained it using the idea of a conductor cutting magnetic field lines. This is an immensely valuable model which not only explains EMI but can also generate quantitative predictions and, yes, it should definitely be taught to students — but perhaps the approach outlined above is better to introduce EMI to students.

The left hand rule not knowing what the right hand rule is doing . . .

We usually apply Fleming’s Right Hand Rule (FRHW) to cases of EMI, Can we replace its use with FLHR? Perhaps, if you wanted to. However, FRHR is a more direct and straightforward shortcut to predicting the direction of conventional current in this type of situation.

Split ring commutator? More like split ring commuHATER!

Students find learning about electric motors difficult because:

  1. They find it hard to predict the direction of the force produced on a conductor in a magnetic field, either with or without Fleming’s Left Hand Rule.
  2. They find it hard to understand how a split ring commutator works.

In this post, I want to focus on a suggested teaching sequence for the action of a split ring commutator, since I’ve covered the first point in previous posts.

Who needs a ‘split ring commutator’ anyway?

We all do, if we are going to build electric motors that produce a continuous turning motion.

If we naively connected the ends of a coil to power supply, then the coil would make a partial turn and then lock in place, as shown below. When the coil is in the vertical position, then neither of the Fleming’s Left Hand Rule (FLHR) forces will produce a turning moment around the axis of rotation.

When the coil moves into this vertical position, two things would need to happen in order to keep the coil rotating continuously in the same direction.

  • The current to the coil needs to be stopped at this point, because the FLHR forces acting at this moment would tend to hold the coil stationary in a vertical position. If the current was cut at this time, then the momentum of the moving coil would tend to keep it moving past this ‘sticking point’.
  • The direction of the current needs to be reversed at this point so that we get a downward FLHR force acting on side X and an upward FLHR force acting on side Y. This combination of forces would keep the coil rotating clockwise.

This sounds like a tall order, but a little device known as a split ring commutator can help here.

One (split) ring to rotate them all

The word commutator shares the same root as commute and comes from the Latin commutare (‘com-‘ = all and ‘-mutare‘ = change) and essentially means ‘everything changes’. In the 1840s it was adopted as the name for an apparatus that ‘reverses the direction of electrical current from a battery without changing the arrangement of the conductors’.

In the context of this post, commutator refers to a rotary switch that periodically reverses the current between the coil and the external circuit. This rotary switch takes the form of a conductive ring with two gaps: hence split ring.

Tracking the rotation of a coil through a whole rotation

In this picture below, we show the coil connected to a dc power supply via two ‘brushes’ which rest against the split ring commutator (SRC). Current is flowing towards us through side X of the coil and away from us through side Y of the coil (as shown by the dot and cross 2D version of the diagram. This produces an upward FLHR force on side X and a downward FLHR force on side Y which makes the coil rotate clockwise.

Now let’s look at the coil when it has turned 45 degrees. We note that the SRC has also turned by 45 degrees. However, it is still in contact with the brushes that supply the current. The forces on side X and side Y are as noted before so the coil continues to turn clockwise.

Next, we look at the situation when the coil has turned by another 45 degrees. The coil is now in a vertical position. However, we see that the gaps in the SRC are now opposite the brushes. This means that no current is being supplied to the coil at this point, so there are no FLHR forces acting on sides X and side Y. The coil is free to continue rotating clockwise because of momentum.

Let’s now look at the situation when the coil has rotated a further 45 degrees to the orientation shown below. Note that the side of the SRC connected to X is now touching the brush connected to the positive side of the power supply. This means that current is now flowing away from us through side X (whereas previously it was flowing towards us). The current has reversed direction. This creates a downward FLHR force on side X and an upward FLHR force on side Y (since the current in Y has also reversed direction).

And a short time later when the coil has moved a total of180 degrees from its starting point, we can observe:

And later:

And later still:

And then:

And then eventually we get back to:

Summary

In short, a split ring commutator is a rotary switch in a dc electric motor that reverses the current direction through the coil each half turn to keep it rotating continuously.

A powerpoint of the images used is here:

And a worksheet that students can annotate (and draw the 2D versions of the diagrams!) is here:

I hope that this teaching sequence will allow more students to be comfortable with the concept of a split ring commutator — anything that results in a fewer split ring commuHATERS would be a win for me 😉

Explaining current flow in conductors (part three)

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.

In part two we looked at how the distribution of surface charges on electrical conductors produces the internal electric fields that guide and push charges around electric circuits and highlighted the published evidence that supports this model.

A representation of a surface charge distribution giving rise to the internal electric field (purple arrows) of a current-carrying conductor

In part three, we are going to look at the transient processes that produce the required distribution of surface charges. In this treatment, I am going to lean very heavily on the analysis presented in Duffin (1980: 167-8).

Connecting wires to a chemical cell

Let’s connect up a simple circuit using a chemical cell as our source of EMF ℰ.

The first diagram shows the cell and wires before they are connected.

When the wires are connected there is a momentary current flow from the cell that creates the surface charge distribution shown below.

The current will stop when the ends of the wire at a potential difference V which is equal to the EMF ℰ of the cell. The ends of the wire act as a small capacitor (∼10-15 F or less). The wires act as equipotential volumes so the very small charge must be distributed over the surface of the wires with a slight concentration of charge at the ends.

Making the circuit

If the ends of the wire are now connected, then the capacitance drops to zero and the ends of the wires become discharged. This leads to very low concentration of surface charge in this region.

However, just enough surface charge remains to produce the internal electric field as shown below. The field lines of the internal electric field are parallel to the wire.

The potential diagram is after Figure 6.17 (Duffin 1980: 160). The ‘dip’ between C and A is due to the effect of the internal resistance of the cell. As we can see in this instance, when there is a steady flow of current then V is slightly smaller than ℰ.

Reference

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

Whoa, black body (bam-ba-lam): part two

In part one, we looked at the fact that the hotter an object then the greater the intensity of electromagnetic radiation that will be emitted. For simplicity, we looked at so-called ‘blackbodies’ — that is say, objects which are perfect absorbers (hence ‘blackbodies’) and more importantly, perfect emitters of electromagnetic radiation.

To human eyes, things look very dull in the visible part of the electromagnetic spectrum until we reach temperatures of several hundreds of degrees — however, objects at room temperature (or just above) glow brightly in the infrared part of the electromagnetic spectrum, as we can see easily if we have access to an infrared camera.

By ‘intensity’ of course, we mean the power (‘energy per second’) emitted per unit area.

This links in neatly with 4.6.3.2 of the 2015 AQA GCSE Physics specification:

Stretch and challenge for students (1): Is the intensity of emitted radiation directly proportional to the temperature of the object?

The short answer is no. If you doubled the temperature (measured in kelvins!) of an object then the intensity of radiation would increase by a factor of 16. In other words, the intensity I of radiation emitted by an object is directly proportional to the absolute temperature T raised to the power of 4.

This is a consequence of the Stefan-Boltzmann radiation law (covered in A-level Physics):

In part 1 we estimated the intensity of radiation emitted by two blackbodies by ‘counting squares’ to find the area underneath a graph. We can show that the values obtained are consistent with the Stefan-Boltzmann radiation law.

Since we have dealt comprehensively with the relationship between intensity of radiation and temperature, I propose to move along and look at how the wavelength distribution changes with the temperature of the body.

How does the temperature of a blackbody affect the distribution of emitted wavelengths?

Let’s consider an object that approximates to a blackbody: the filament of an old school incandescent lamp.

The graph of the radiation produced by both objects is shown below.

First, let’s look at the visible wavelengths produced by both bulbs.

  • The 1700 degree Celsius bulb produces only a very small amount of visible light and the vast majority of that is towards the red end of the spectrum: you can see the section where the left hand edge of the 1700 curve just nicks the visible light wavelengths. This means that the 1700 degree filament emits a barely perceptible reddish glow to our eyes with its peak output still firmly in the infrared.
  • The 2200 degree Celsius bulb produces a much larger amount of visible light: look at the left hand side of the curve. What is more, it appears as white light to our eyes since it includes all the colours of the rainbow. However, it’s still a very reddish-tinged white. Photographs taken in artificial light with chemical films (very old school!) had to be taken using special colour balanced film stock otherwise this bias was very evident in the final print(!) Modern digital cameras have software that automatically compensates for artificial vs. daylight colour balance issues.

Second, let’s look at the position of the peak wavelength.

  • The 1700 degree Celsius bulb has its peak output at a wavelength of 1.5 x 10-6 m (shown by the blue dotted line on the graph).
  • The 2200 degree Celsius bulb has its peak output at a wavelength of 1.2 x 10-6 m (shown by the red dotted line on the graph.)

Assuming that you wanted to, these findings could be summarised in song (sung to the tune of ‘Black Betty’ by Ram Jam):

Whoa, black body (Bam-ba-lam)
Whoa, black body (Bam-ba-lam)
More heat, peak shifts left
Waves out with more zest
Wavelengths out not alike
Some hues power spike!
Whoa, black body (Bam-ba-lam)
Whoa, black body 
Bam-ba-laaam, yeah yeah
 

Stretch and challenge for students (2): predicting the position of the peak output wavelength

The position of the peak output wavelength can be predicted using Wien’s Displacement Law (studied in A-level Physics:

As we can see, the peak output wavelength on the graph agrees well with the position as calculated by Wien’s Displacement Law.

An unannotated pdf of the graph can be downloaded here:

Whoa, black body (bam-ba-lam)

The 2015 AQA GCSE Physics specification (4.6.3.1) asks that students understand that:

In my experience, students find it natural to accept that bodies absorb electromagnetic radiation — but surely only extremely hot objects (like the filament of a light bulb) emit electromagnetic waves?

This is a consequence of the fact that our eyes can only detect the tiny slice of the vast electromagnetic spectrum.

That tiny sliver known as ‘visible light’ looks insultingly small even on a diagram with a logarithmic scale as shown above. On a linear scale, it’s even worse: if we represented the em spectrum by a line stretching from London to New York, then the range of wavelengths that human eyes can detect would be a strip two centimetres wide.

This calls to my mind some lines quoted many years ago by Arthur C. Clark in his wonderful essay ‘Things We Cannot See’: A being who hears me tapping / The five-sensed cane of mind / Amid such greater glories / That I am worse than blind.

Seeing the unseeable

A Leslie’s cube is a cuboid with black and silver coloured faces that can filled with hot water.

In visible light, there is no difference between its appearance when at room temperature (say 15 degrees Celsius) and when filled with hot water (say 70 degrees Celsius).

However, seen through an infrared camera, things look very different: the hot sides glow brightly, emitting huge amounts of infrared em waves.

There is another effect: the black coloured side throws out more infrared than the silver side. Why? Because any object which is good at absorbing em radiation is also good at emitting radiation.

As the AQA GCSE spec puts it:

By this definition, the Sun is a good approximation of a black body since it absorbs nearly all of the radiation falling on it (from other stars! — as well as the odd photon bounced back from minuscule specks like the Earth) as well being highly effective at emitting em radiation.

Black body radiation curves

4.6.3.2 of the AQA GCSE Physics spec says:

One of the ways to cover this is to look at the radiation curves of two black bodies at different temperatures (pdf here). Both of these objects are at relatively low temperatures, so they emit most of their energy in the infrared part of the em spectrum. The visible light range is shown by the coloured bar just to the right of the y-axis.

Because it is a perfect emitter — as well as absorber — of radiation, the intensity (power per unit area) of emitted radiation from a black body depends only on the temperature of the black body.

Estimating the total power emitted per unit area

We can estimate the total power emitted per unit area by approximating the area underneath the curve. We’re going to count any square which is larger than a half square as one whole square and ignore any part squares which are smaller than a half square.

The area under the blue curve is 325 W/m^2.

The intensity of radiation emitted by the hot object (red curve) is larger than the intensity emitted by the cold object (blue curve).

You can download an unannotated pdf copy of the graph by clicking on the link below.

We will look at the distribution of wavelengths in a later post.

Whoa, black body (bam-ba-lam)

Physics can occasionally, go better with a song.

Whoa, black body (Bam-ba-lam)
Whoa, black body (Bam-ba-lam)
Black body e.m. waves (Bam-ba-lam)
Come out with peak shapes (Bam-ba-lam)
Planck said, “I’m worryin’ outta mind (Bam-ba-lam)
UV won’t align!” (Bam-ba-lam)
He said, oh black body (Bam-ba-lam)
Whoa, black body (Bam-ba-lam)

Whoa, black body (Bam-ba-lam)
Whoa, black body (Bam-ba-lam)
Planck set out to Quantise (Bam-ba-lam)
UV don’t catastrophize! (Bam-ba-lam)
Theory rock steady (Bam-ba-lam)
“No prob now,” said he (Bam-ba-lam)
Whoa, black body (Bam-ba-lam)
Whoa, black body (Bam-ba-lam)

Get it!

Whoa, black body (Bam-ba-lam)
Whoa, black body (Bam-ba-lam)
More heat, peak shifts left
Waves out with more zest
Wavelengths out not alike
Some hues power spike!
Whoa, black body (Bam-ba-lam)
Whoa, black body 
Bam-ba-laaam, yeah yeah

(with apologies to Huddie Ledbetter)

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.