## 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.

## 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.

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

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…

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:

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 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.

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 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 . . .

### 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.

## Magnetism? THERE IS NO MAGNETISM!!!!

Has a school physics experiment or demonstration ever changed the course of human history?

On 21 April 1820, one such demonstration most definitely did. According to physics lore, Hans Christian Øersted was attempting to demonstrate to his students that, according to the scientific understanding of the day, there was in fact no connection between magnetism and electricity.

To this laudable end, he placed a compass needle near to a wire to show that when the current was switched on, the needle would not be affected.

Except that it was affected. Frequently. Each and every time Øersted switched on the electric current, the needle was deflected from pointing North.

Everybody has heard that wise old saw that ‘If it doesn’t work, it’s physics…” except that in this case ‘It did actually work as it was supposed to but in an unexpected way due to a hitherto-unknown-completely-new-branch-of-physics.’

Øersted, to his eternal credit, did not let it lie there and was a pioneer of the new science of electromagnetism.

### Push-me-pull-you: or, two current-carrying conductors

One curious consequence of Øersted’s new science was the realisation that, since electric currents create magnetic fields, two wires carrying electric currents will exert a force on each other.

Let’s consider two long, straight conductors placed parallel to each other as shown.

In the diagram above, the magnetic field produced by the current in A is shown by the green lines. Applying Fleming’s Left Hand Rule* to conductor B, we find that a force is produced on B which acts towards conductor A. We could go through a similar process to find the force acting on B, but it’s far easier to apply Newton’s Third Law instead: if body A exerts a force on body B, then body B exerts an equal and opposite force on body A. Hence, conductor A experiences a force which pulls it towards conductor B.

So, two long, straight conductors carrying currents in the same direction will be attracted to each other. By a similar analysis, we find that two long, straight conductors carrying currents in opposite directions will be repelled from each other.

In the past, this phenomenon was used to define the ampere as the unit of current: ‘The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 m apart in vacuum, would produce between these conductors a force equal to 2×10−7 newton per metre of length.‘ However, the 2019 redefinition of the SI system has ditched this and adopted a new definition in terms of the transfer of the elementary charge, e.

### Enter Albert Einstein, pursuing an enigma

What is the connection between magnetism and electricity? It was precisely this puzzle that started Albert Einstein on the road to special relativity. It is one of the unsung triumphs of this theory that it lays bare the connection between magnetism and electricity.

In what follows, we’re going to apply Einstein’s analysis to the situation of two long, straight current-carrying conductors. Acknowledgment: I’m going to following a line of argument laid out in Beiser 1988: 19-22.

### It’s gotta be perfect (or ‘idealised’, if you prefer)

Let’s consider two idealised conductors A and B both at rest in the inertial reference frame of the laboratory. The flow of charge in both conductors is made up of positive and negative charge carriers moving in opposite directions with a speed v.

None of the charges in A interact with the other charges in A because we are considering an idealised conductor. However, the charges in A will interact with the charges in B.

### Flip the inertial reference frame

Now let’s look at the situation from the inertial reference frame of one of the positive charges in A. For simplicity, we can focus on a single positive charge in A since it does not interact with any of the other charges in A.

With reference to this inertial frame, the positive charge in A is stationary and the positive charges in B are also stationary.

However, the inertial frame of the laboratory is moving right-to-left with a speed v and the negative charges are moving right-to-left with a speed of 2v.

Since the positive charges in B are stationary with respect to the positive charge in A, the distance between them is the same as it was in the laboratory inertial frame. However, since the negative charges in B are moving with speed 2v with respect to positive charge in A, the spacing between is contracted due to relativistic length contraction (see Lottie and Lorentzian Length Contraction).

Because of this, the negative charge density of B increases since they are closer together. However, the positive charge density of B remains the same since they are stationary relative to the positive charge in A so there is no length contraction.

This means that, as far as the positive charge in A is concerned, conductor B has a net negative charge which means the positive charge experiences an attractive Coulomb’s Law electrical force towards B.

A similar analysis applied to electric currents in opposite directions would show that the positive charge in A would experience a repulsive Coulomb’s Law electrical force. The spacing between the positive charges in B would be contracted but the spacing between the negative charges remains unchanged, so conductor B has a net positive charge because the positive charge density has increased but the negative charge density is unchanged.

### Magnetism? THERE IS NO MAGNETISM!!!!

So what we normally think of as a ‘magnetic’ force in the inertial frame of the laboratory can be explained as a consequence of special relativity altering the charge densities in conductors. Although we have just considered a special case, all magnetic phenomena can be interpreted on the basis of Coulomb’s Law, charge invariance** and special relativity.

For the interested reader, Duffin (1980: 388-390) offers a quantitative analysis where he uses a similar argument to derive the expression for the magnetic field due to a long straight conductor.

Update: I’m also indebted to @sbdugdale who points out the there’s a good treatment of this in the Feynman Lectures on Physics, section 13.6.

### Notes and references

* Although you could use a non-FLHR catapult field analysis, of course

** ‘A current-carrying conductor that is electrically neutral in one frame of reference might not be neutral in another frame. How can this observation be reconciled with charge invariance? The answer is that we must consider the entire circuit of which the conductor is a part. Because a circuit must be closed for a current to occur in it, for every current element in one direction that a moving observer find to have, say, a positive charge, there must be another current element in the opposite direction which the same observer finds to have a negative charge. Hence, magnetic forces always act between different parts of the same circuit, even though the circuit as a whole appears electrically neutral to all observers.’ Beiser 1988: 21

Beiser, A. (1988). Concepts of modern physics. Tata McGraw-Hill Education

Duffin, W. J. (1980). Electricity and magnetism. McGraw-Hill.

## Electric Motors Without The Left Hand Rule

There is little doubt that students find understanding how an electric motor works hard.

What follows is an approach that neatly sidesteps the need for applying Fleming’s Left Hand Rule (FLHR) by using the idea of the catapult field.

The catapult field is a neat bit of Physics pedagogy that appears to have fallen out of favour in recent years for some unknown reason. I hope to rehabilitate and publicise this valuable approach so that more teachers may try out this electromagnetic ‘road less travelled’.

(Incidentally, if you are teaching FLHR, the mnemonic shown above is not the best way to remember it: try using this approach instead.)

### The magnetic field produced by a long straight conductor

Moving electric charges produce magnetic fields. When a current flows through a conductor, it produces a magnetic field in the form of a series of cylinders centred on the wire. This is usually shown on a diagram like this:

If we imagine looking down from a point directly above the centre of the conductor (as indicated by the disembodied eye), we would see a plan view like this:

We are using the ‘dot and cross‘ convention (where an X represents an arrow heading away from us and a dot represents an arrow heading towards us) to easily render a 3D situation as a 2D diagram.

The direction of the magnetic field lines is found by using the right hand grip rule.

The thumb is pointed in the direction of the current. The field lines ‘point’ in the same direction as the fingers on the right hand curl.

### 3D to 2D

Now let’s think about the interaction between the magnetic field of a current carrying conductor and the uniform magnetic field produced by a pair of magnets.

In the diagrams below, I have tried to make the transition between a 3D and a 2D representation explicit, something that as science teachers I think we skip over too quickly — another example of the ‘curse of knowledge’, I believe.

### Magnetic Field on Magnetic Field

If we place the current carrying conductor inside the magnetic field produced by the permanent magnets, we can show the magnetic fields like this:

Note that, in the area shaded green, the both sets of magnetic field lines are in the same direction. This leads a to stronger magnetic field here. However, the opposite is true in the region shaded pink, which leads to a weaker magnetic field in this region.

The resultant magnetic field produced by the interaction between the two magnetic fields shown above looks like this.

Note that the regions where the magnetic field is strong have the magnetic field lines close together, and the regions where it is weak have the field lines far apart.

### The Catapult Field

This arrangement of magnetic field lines shown above is unstable and is called a catapult field.

Essentially, the bunched up field lines will push the conductor out of the permanent magnetic field.

If I may wax poetic for a moment: as an oyster will form a opalescent pearl around an irritant, the permanent magnets form a catapult field to expel the symmetry-destroying current-carrying conductor.

The conductor is pushed in the direction of the weakened magnetic field. In a highly non-rigorous sense, we can think of the conductor being pushed out of the enfeebled ‘crack’ produced in the magnetic field of the permanent magnets by the magnetic field of the current carrying conductor…

Also, the force shown by the green arrow above is in exactly the same direction as the force predicted by Fleming’s Left Hand Rule, but we have established its direction using only the right hand grip rule and a consideration of the interaction between two magnetic field.

### The Catapult Field for an electric motor

First, let’s make sure that students can relate the 3D arrangement for an electric motor to a 2D diagram.

The pink highlighted regions show where the field lines due to the current in the conductor (red) are in the opposite direction to the field line produced by the permanent magnet (purple). These regions are where the purple field lines will be weakened, and the clear inference is that the left hand side of the coil will experience an upward force and the right hand side of the coil will experience a downward force. As suggested (perhaps a little fancifully) above, the conductors are being forced into the weakened ‘cracks’ produced in the purple field lines.

The catapult field for the electric motor would look, perhaps, like this:

### And finally…

On a practical teaching note, I wouldn’t advise dispensing with Fleming’s Left Hand Rule altogether, but hopefully the idea of a catapult field adds another string to your pedagogical bow as far as teaching electric motors is concerned (!)

I have certainly found it useful when teaching students who struggle with applying Fleming’s Left Hand Rule, and it is also useful when introducing the Rule to supply an understandable justification why a force is generated by a current in a magnetic field in the first place.

The catapult field is a ‘road less travelled’ in terms of teaching electromagnetism, but I would urge you to try it nonetheless. It may — just may — make all the difference.