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.

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.