Sussing Out Solenoids With Dot and Cross

A solenoid is an electromagnet made of a wire in the form of a spiral whose length is larger than its diameter.

A solenoid

The word solenoid literally means ‘pipe-thing‘ since it comes from the Greek word ‘solen‘ for ‘pipe’ and ‘-oid‘ for ‘thing’.

An alternate universe version of the Troggs’ famous 1966 hit record

And they are such an all-embracingly useful bit of kit that one might imagine an alternate universe where The Troggs might have sang:

Pipe-thing! You make my heart sing!
You make everything groovy, pipe-thing!

And pipe-things do indeed make everything groovy: solenoids are at the heart of the magnetic pickups that capture the magnificent guitar riffs of The Troggs at their finest.

The Butterfly Field

Very few minerals are naturally magnetised. Lodestones are pieces of the ore magnetite that can attract iron. (The origin of the name is probably not what you think — it’s named after the region, Magnesia, where it was first found). In ancient times, lodestones were so rare and precious that they were worth more than their weight in gold.

Over many centuries, by patient trial-and-error, humans learned how to magnetise a piece of iron to make a permanent magnet. Permanent magnets now became as cheap as chips.

A permanent bar magnet is wrapped in an invisible evanescent magnetic field that, given sufficient poetic license, can remind one of the soft gossamery wings of a butterfly…

Orient a bar magnet vertically so that students can see the ‘butterfly field’ analogy…

The field lines seem to begin at the north pole and end at the south pole. ‘Seem to’ because magnetic field lines always form closed loops.

This is a consequence of Maxwell’s second equation of Electromagnetism (one of a system of four equations developed by James Clark Maxwell in 1873 that summarise our current understanding of electromagnetism).

Using the elegant differential notation, Maxwell’s second equation is written like this:

Maxwell’s second equation of electromagnetism.

It could be read aloud as ‘del dot B equals zero’ where B is the magnetic field and del (the inverted delta symbol) does not represent a quantity but is the differential operator which describes how the field lines curl in three dimensional space.

This also tells us that magnetic monopoles (that is to say, isolated N and S poles) are impossible. A north-seeking pole is always paired with a south-seeking pole.

Magnetising a solenoid

A current-carrying coil will create a magnetic field as shown below.

The magnetic field of a solenoid. (Note that the field lines in the centre are truncated to save space, but would form very large loops as mentioned above.)

The wire is usually insulated (often with a tough, transparent and nearly invisible enamel coating for commercial solenoids), but doesn’t have to be. Insulation prevents annoying ‘short circuits’ if the coils touch. At first sight, we see the familiar ‘butterfly field’ pattern, but we also see a very intense magnetic field in the centre of the solenoid,

For a typical air-cored solenoid used in a school laboratory carrying one ampere of current, the magnetic field in the centre would have a strength of about 84 microtesla. This is of the same order as the Earth’s magnetic field (which has a typical value of about 50 microtesla). This is just strong enough to deflect the needle of a magnetic compass placed a few centimetres away and (probably) make iron filings align to show the magnetic field pattern around the solenoid, but not strong enough to attract even a small steel paper clip. For reference, the strength of a typical school bar magnet is about 10 000 microtesla, so our solenoid is over one hundred times weaker than a bar magnet.

However, we can ‘boost’ the magnetic field by adding an iron core. The relative permeability of a material is a measurement of how ‘transparent’ it is to magnetic field lines. The relative permeability of pure iron is about 1500 (no units since it’s relative permeability and we are comparing its magnetic properties with that of empty space). However, the core material used in the school laboratory is more likely to be steel rather than iron, which has a much more modest relative permeability of 100.

So placing a steel nail in the centre of a solenoid boosts its magnetic field strength by a factor of 100 — which would make the solenoid roughly as strong as a typical bar magnet.

But which end is north…?

The N and S-poles of a solenoid can change depending on the direction of current flow and the geometry of the loops.

The typical methods used to identify the N and S poles are shown below.

Methods of locating N and S pole of a solenoid that you should NOT use…

To go in reverse order for no particular reason, I don’t like using the second method because it involves a tricky mental rotation of the plane of view by 90 degrees to imagine the current direction as viewed when looking directly at the ends of the magnet. Most students, understandably in my opinion, find this hard.

The first method I dislike because it creates confusion with the ‘proper’ right hand grip rule which tells us the direction of the magnetic field lines around a long straight conductor and which I’ve written about before . . .

The right hand grip rule illustrated: the field line curl in the same direction as the finger when the thumb is pointed in the direction of the current.

The direction of the current in the last diagram is shown using the ‘dot and cross’ convention which, by a strange coincidence, I have also written about before . . .

If King Harold had been more familiar with the dot and cross notation, history could have turned out very differently…

How a solenoid ‘makes’ its magnetic field . . .

To begin the analysis we imagine the solenoid cut in half: what biologists would call a longitudinal section. Then we show the current directions of each element using the dot and cross convention. Then we consider just two elements, say A and B as shown below.

Continuing this analysis below:

The region inside the solenoid has a very strong and nearly uniform magnetic field. By ‘uniform’ we mean that the field lines are nearly straight and equally spaced meaning that the magnetic field has the same strength at any point.

The region outside the solenoid has a magnetic field which gradually weakens as you move away from the solenoid (indicated by the increased spacing between the field lines); its shape is also nearly identical to the ‘butterfly field’ of a bar magnet as mentioned above.

Since the field lines are emerging from X, we can confidently assert that this is a north-seeking pole, while Y is a south-seeking pole.

Which end is north, using only the ‘proper’ right hand grip rule…

First, look very carefully at the geometry of current flow (1).

Secondly, isolate one current element, such as the one shown in picture (2) above.

Thirdly, establish the direction of the field lines using the standard right hand grip rule (3).

Since the field lines are heading into this end of the solenoid, we can conclude that the right hand side of this solenoid is, in fact, a south-seeking pole.

In my opinion, this is easier and more reliable than using any of the other alternative methods. I hope that readers that have read this far will (eventually) come to agree.

Put Not Your Trust In Pyramids

Put not your trust in princes.

Psalm 146, KJV

Triangles and pyramids do to teachers what catnip does to cats.

Translation from Lolcat: ‘More triangles, please!’

Put just about any idea in the form of a three-sided polygon and watch teachers adopt it en masse as an article of faith. And, boy, have we as a profession unquestionably and uncritically adopted some stinkers.

What follows is a countdown, from least-worst to worst (in my estimation), of what I would collectively call . . . [PLAYS SINISTER ORGAN NOTES, ACTIVATES VOICE-ECHO] . . . PERFIDIOUS PYRAMIDS!

Number 4: Maslow’s Hierarchy of Needs

Why bring this one up? Firstly, Maslow never put his hierarchy in the form of a pyramid. This implies that all of a student’s ‘Deficiency Needs’ must be met before the ‘Being (growth) Needs’ can be addressed; Maslow was more nuanced in his original writings.

The analogy of psychological needs to vitamins was drawn by Maslow. Like vitamins, each of the needs is individually required, just as having much of one vitamin does not negate the need for other vitamins. All needs should independently contribute to subjective well being.

Tay and Diener 2011

Secondly, the methodology by which Maslow arrived at the characteristics of a ‘self-actualized’ person was by looking at the writings and biographies of a number of people (including Albert Einstein and Mother Theresa) whom he considered to be ‘self-actualized’: this is a qualitative and subjective approach that would seem highly open to personal bias and hard to characterise as ‘scientific’ (McLeod 2018).

Number 3: Bloom’s Taxonomy

As with Maslow, there is little to argue with the intent behind Bloom’s Taxonomy, which was an attempt to classify educational objectives without — repeat, without — arranging them into a formal hierarchy.

Bloom’s Taxonomy is often missapplied in education because the ‘higher’ levels are deemed more desirable than the ‘lower’ levels.

As Sugrue (2002) notes:

It was developed before we understood the cognitive processes involved in learning and performance. The categories or ‘levels’ of Bloom’s taxonomy … are not supported by any research on learning.

And, sadly,

the popular misinterpretation of the taxonomy has led to a multi-generational loss of learning opportunities. It is a triumph of philosophy over science, of populism over rigour, of politics over responsibility.

James Murphy, The False Dichotomy

Number 2: Formula Triangles

The main issue with formula triangles is that they are a replacement for algebra rather than a system or scaffold for supporting students in learning how to manipulate equations.

Koenig (2015) makes some trenchant criticisms of forumula triangles, as does Southall (2016) who argues that they are a form of ‘procedural’ teaching rather than the demonstrably more effective ‘conceptual’ teaching. Conceptual teaching encourages students to understand why a particular technique is used rather than applying it as a ‘magic’ formula. Borij, Radmehr and Font (2019) also have an interesting and nuanced discussion on these types of teaching (in the context of learning calculus).

Workable alternatives to formula triangles are the FIFA and EVERY systems.

But the winner for the educationally worst pyramid or triangle is . . . [DRUM ROLL]

NUMBER 1: The Learning Pyramid

To put it bluntly, there is no research to support the percentage retention rates claimed on any version of this pyramid.

Modern versions of the learning pyramid seem to be based on Edgar Dale’s ‘Cone of Experience’ first published in 1946 in his influential book Audio-Visual Teaching Techniques.

Dale’s Cone of Experience as presented in the 1954 edition (with ‘Television’ added from previous versions). From Lalley and Miller 2007

Dale’s main argument was to encourage

the use of audio-visual materials in teaching – materials that do not depend primarily upon reading to convey their meaning. It is based upon the principle that all teaching can be greatly improved by the use of such materials because they can help make the learning experience memorable…this central idea has, of course, certain limits. We do not mean that sensory materials must be introduced into every teaching situation. Nor do we suggest that teachers scrap all procedures that do not involve a variety of audio-visual methods

Dale 1954 quoted by Lalley and Miller 2007

The peculiarly neat percentage increments in retention rates on the learning pyramid are first found in Treichler (1967). As Letrud and Hernes (2018) note:

Treichler asserted that these numbers came from studies, but he did not say where they could be found. […] A set of learning modalities similar to those distributed by Treichler were at some point fused with a misreading of Edgar Dale’s Cone of experience as a hierarchy of learning modalities, and these early categories were supplemented and partly replaced with categories of presentation modalities like “audiovisual”, “demonstrations”, and “discussion groups”.

The final word is perhaps best left to Lalley and Miller:

The research reviewed here demonstrates that use of each of the methods identified by the pyramid resulted in retention, with none being consistently superior to the others and all being effective in certain contexts. A paramount concern, given conventional wisdom and the research cited, is the effectiveness and importance of reading and direct instruction, which in many ways are undermined by their positions on the pyramid. Reading is not only an effective teaching/learning method, it is also the main foundation for becoming a “life-long learner”


Borji, V., Radmehr, F., & Font, V. (2019). The impact of procedural and conceptual teaching on students’ mathematical performance over time. International Journal of Mathematical Education in Science and Technology, 1-23.

Dale, E. (1954). Audio-visual methods in teaching (2 ed.). New York: The Dryden Press.

Koenig, J. (2015). Why Are Formula Triangles Bad? Education In Chemistry, Royal Society of Chemistry.

Lalley, J., & Miller, R. (2007). The learning pyramid: Does it point teachers in the right direction. Education128(1), 16.

Letrud, K., & Hernes, S. (2018). Excavating the origins of the learning pyramid myths. Cogent Education5(1), 1518638.

McLeod, S. (2018). Maslow’s hierarchy of needs. Simply psychology1, 1-8.

Southall, E. (2016). The formula triangle and other problems with procedural teaching in mathematics. School Science Review97(360), 49-53.

Sugrue, B. Problems with Bloom’s Taxonomy. Presented at the International Society for Performance Improvement Conference 2002

Tay, L., & Diener, E. (2011). Needs and subjective well-being around the world. Journal of personality and social psychology101(2), 354.

Treichler, D. G. (1967). Are you missing the boat in training aids? Film and Audio-Visual Communication, 1(1), 14–16, 28–30,48.