Electromagnetism

Applying Lenz’s Law

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Nature abhors a change of flux.

D. J. Griffiths’ (2013) genius re-statement of Lenz’s Law, modelled on Aristotle’s historically influential but now debunked aphorism that ‘Nature abhors a vacuum’

A student recently asked for help with this AQA A-level Physics multiple choice question:

AQA A-level Physics question from 2019 Paper 2

This question is, of course, about Lenz’s Law of Electromagnetic Induction. The law can be stated easily enough: ‘An induced current will flow in a direction so that it opposes the change producing it.’ However, it can be hard for students to learn how to apply it.

What follows is my suggested explanatory sequence.

Step 1: simplify the diagram using the ‘dot and cross’ convention

When the switch is closed, a current I begins to flow in coil P. We can assume that I starts at zero and increases to a maximum value in a very small but not negligible period of time.

Simplified 2D representation of the top diagram. The current directions I are arbitrary based on my ‘best guess’ interpretation of the 3D diagram and could be reversed if desired.

Step 2: consider the magnetic field produced by P

You can read more about a simple method of deducing the direction of the magnetic field produced by a coil or a solenoid here.

Step 3: apply Faraday’s Law to coil Q

Since Q is experiencing a change in magnetic flux, then an induced current will flow through it.

Step 4: apply Lenz’s Law to coil Q

The current in coil Q must flow in such a direction so that it opposes the change producing it.

Since P is producing an increasing magnetic flux through Q, then the current in Q must flow in such a way so that it tries to prevent the increase in magnetic flux which is inducing it. The direction of the magnetic field BQ produced by Q must therefore be opposite to the direction of the magnetic field produced by P.

Step 5: consider the polarity of the magnetic fields of P and Q

We can see the magnetic field lines of coil P produce a north magnetic field on its right hand side. The magnetic field of Q will produce a north magnetic field on its left hand side. Coil P will therefore push coil Q to the right.

It follows that we can eliminate options A and C from the question.

Step 6: What happens when the magnetic field of P reaches its steady value?

Because the magnetic field produced by coil P has how reached its steady maximum value, this means that the magnetic flux through coil Q also has a constant, unchanging value. Since there is no change in magnetic flux, then this means that no emf is induced across the coil so no induced current flows. Since Q does not have a magnetic field it follows that there is no magnetic interaction between them.

The answer to the question must therefore be D.

Step 7: check student understanding

For the alternative question, the correct answer of C can be explained by going through a process similar to the one outlined above.

  • When the switch is opened, the magnetic flux through Y begins to decrease.
  • A changing magnetic flux through Y induces current flow.
  • Lenz’s Law predicts that the direction of this current is such that it opposes the change producing it.
  • The current through Y will therefore be in the same direction as the current through X to produce a magnetic field in the same direction.
  • The coils will attract each other.
  • Eventually, the magnetic flux produced by coil X drops to a constant value of zero.
  • Since there is no change in magnetic flux through Y, there is no induced current flow through Y and hence no magnetic field.
  • There is no magnetic interaction between X and Y and therefore the force on Y is zero.

Conclusion

I hope teachers find this detailed analysis of a Lenz’s Law question useful! As in much of A-level Physics, the devil is not in the detail but rather in the application of the detail. Students who encounter more examples will have a more secure understanding.

Reference

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

Split ring commutator? More like split ring commuHATER!

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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:

split-ring-commutator-or-split-ring-commuhaterDownload

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

split-ring-commutator-split-ring-commuhaterDownload

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 😉

Nature Abhors A Change In Flux

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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.

Graph adapted from https://www.a-levelphysicstutor.com/field-electro-mag-ind.php

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 Chemie, 107 (31), pp. 483–494

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

Sussing Out Solenoids With Dot and Cross

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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.