## Is Lenz’s Law Just An Instance Of Newton’s Third Law?

The longer answer exemplifies one of the lovely things about physics: its internal unity, and the fact that a few basic principles manifest in a plethora of circumstances. And one of the typical shortcomings of our textbooks (and by extension our lectures).

For me, this was one of the attractions of the subject when I was a young student … if one understood a few basic principles, and one had reasonably good mathematical technique, one could go far in understanding nature, at least in simple situations. But ay, there’s the rub: Physics is simple, but subtle. Even in the simple situations that physics is concerned with, deep and careful thought is often required to arrive at simple explanations, simple understandings, simple results.

Back to Lenz’s law. The textbook we’re currently using in the course I’m teaching states the law as follows (page 823):

There is an induced current in a closed, conducting loop if and only if the magnetic flux through the loop is changing. The direction of the induced current is such that the induced magnetic field opposes the change in the flux.

There follows several pages of very clear explanations and examples, including helpful diagrams.

The textbook misses an opportunity (as does every other textbook on my shelf) to make an important connection: Lenz’s law is just a version of Newton’s third law applied to a particular kind of situation; that is, a situation where a changing magnetic flux induces a current in a loop of wire.

Some of the books I consulted make the point that if Lenz’s law predicted the opposite orientation of induced current, then perpetual-motion machines would be possible, which would violate the principle of conservation of energy. Good. But going further to make the connection between Lenz’s law and Newton’s third law of motion would be better.

This is the purpose of teaching, which includes writing textbooks. All the information is already out there in cyberspace, readily available to anyone with an internet connection. Yet if the availability of information were all that were needed for good education, then everyone would be a genius by now. But, as Jacques Barzun stated, we think with ideas, not with information.

But this is what is largely missing in cyberspace: thinking tools; and making connections is one of the most important thinking tools. The giant textbooks we use are full of technical details, full of drill … we need more guidance for students in how to think.

I have taught mathematics and physics since the mid 1980s. I have also been a textbook writer/editor since then. Currently I am working independently on a number of writing and education projects while teaching physics at my local university. I love math and physics, and love teaching and writing about them. My blog also discusses education, science, environment, etc. https://qedinsight.wordpress.com Further resources, and online tutoring, can be found at my other site http://www.qedinfinity.com
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### 14 Responses to Is Lenz’s Law Just An Instance Of Newton’s Third Law?

1. Guan Kheng Sze says:

Dear Prof Santo D’Agostino, I am puzzled by your assertion that Len’s Law is linked to Newton’s Third Law. To me, they are not related. Take the following standard demonstration of EM Induction as an example. Suppose I push a magent with its North pole approaching a wire coil. Len’s Law tells us that the induced current will flow in a way so as to produce a force that repels the magnet. But suppose that we live in a strange world in which the opposite of Lenz’s law happens – the approaching magent is attracted by the induced current instead. This violates Lenz’s Law (and hence conservation of energy), but it does not violate Newton’s Third Law as long as the wire coil is also attracted by the magent. So my point is, if Lenz’s Law can be violated without violating Newton’s thrid law, then the former cannot be an instance of the latter.
I would greatly appreciate it if you could point out any flaws that my arguments above might have. Thank you.

• Dear Guan Kheng Sze,
Your question is a very interesting one. Let us translate your question into a purely mechanical setting. Suppose that you pushed on a wall. Suppose also that you lived in a strange world in which, instead of pushing back on you, the wall actually pulled you in, but in such a way that you also attracted the wall. The question is whether in this strange world Newton’s third law is always satisfied; the answer is no. For initially you pushed on the wall, and because the wall did not push back on you, Newton’s third law was initially violated, even if it was satisfied later in the experiment. So it is a very strange world indeed, in which sometimes Newton’s third law is satisfied and sometimes not.

Now let’s return to the Lenz’s law experiment you describe in your comment. It’s the same situation as the purely mechanical experiment I just described. You push the magnet towards the coil of wire; you do the pushing, exerting a purely mechanical force. If the coil thereby attracts the magnet, then Newton’s third law is violated from the start, because the coil of wire should respond to your push by pushing back on you, not by pulling the magnet in. If later the coil of wire and the magnet mutually attract each other, then we have the same strange situation as in the previous paragraph, a world where sometimes Newton’s third law is satisfied and sometimes not.

• Guan Kheng Sze says:

Dear Professor Santo D’Agostino,
Thank you for taking time to reply my question. However, I still have some doubts over your explanation. Firstly, if I understand you correctly, you said that if I push the magent toward the coil, the coil should, by Newton’s Third Law, repels the magent. My understanding is that the reaction force (of Newton’s Third Law) to my push on the magent should be the magnet’s push back on me (e.g. the frictional force that the magnet exerts on my palm if I am grabbing it). On the other hand, the repulsion that the coil exerts on the magnet has its own reaction force, which is the repulsion of the magent on the coil. The two forces that you mentioned in your reply – the force exerted by me on the magent, and the repulsion that the coil exerts on the magnet, both act on the magnet. Hence, they can’t be an action-reaction pair of Newton’s Third Law.
However, let’s focus on our main question, which is whether Lenz’s Law is indeed an instance of Newton’s Third Law. To avoid the complication mentioned in your reply, let us modify the experiment slightly. Let us suppose that the magnet is given a tap at its end and starts moving on a horizontal frictionless surface in uniform velocity toward the coil. The coil is originally incomplete (i.e. open circuit) and hence no induced current flows in it initially. It is only closed only when the magnet has started moving. This avoids the complication due to the force exerted by a third party (that’s me) on the magnet. We thus have a situation of a magnet moving toward a coil and they form an isolated system. My argument is that Lenz’s Law will predict that the induced current in the coil once it is closed will repel the approaching magnet, whereas Newton’s Third Law makes no such prediction, in the sense that even if the induced current flows in the other direction so as to attract the magnet, Newton’s third law is not violated as long as the magnet also attracts the coil. Hence, the Lenz’s Law is not an instance of Newton’s Third Law. Once again, I would greatly appreciate it if you can take some time to answer my question. Thank you.

• Dear Guan Kheng Sze,

You wrote: However, I still have some doubts over your explanation. Firstly, if I understand you correctly, you said that if I push the magent toward the coil, the coil should, by Newton’s Third Law, repels the magent. My understanding is that the reaction force (of Newton’s Third Law) to my push on the magent should be the magnet’s push back on me (e.g. the frictional force that the magnet exerts on my palm if I am grabbing it). On the other hand, the repulsion that the coil exerts on the magnet has its own reaction force, which is the repulsion of the magent on the coil. The two forces that you mentioned in your reply – the force exerted by me on the magent, and the repulsion that the coil exerts on the magnet, both act on the magnet. Hence, they can’t be an action-reaction pair of Newton’s Third Law.

You are correct about this, I was sloppy in my language. I was treating your hand/the magnet as one system. So let us be precise and think only about the magnet and the coil, as you suggest. Your new thought-experiment is wonderfully precise and focusses our thinking excellently.

OK, so the magnet is in motion towards the coil, which has not been closed yet. The magnet exerts no force on the coil, and the coil exerts no force on the magnet. Now the coil is closed. Here is the key question: Does the magnet exert a force on the coil?

How do you answer this question? Well, you do the experiment and see what happens. And indeed, if you do the experiment, you will observe that the magnet indeed exerts a force on the closed coil. After having done many, many experiments, we now have a well-developed theory that we can use to describe and explain what happens without actually doing the experiment, which makes conversations like this possible. So how can we explain the force that the magnet exerts on the coil without doing the experiment? Well, you use the relevant force law, which is that the force of the magnetic field on an individual charged particle in the coil is equal to the charge of the particle times the cross product of the relative velocity vector (of the particle relative to the magnet) and the magnetic field vector. Taking the sum of the forces on all of the charged particles in the coil (the forces on the negatively charged particles are in one sense, and the forces on the positively charged particles are in the opposite sense, which results in a current flowing in the coil) gives you the net force of the magnet on the coil. Then, by Newton’s third law, the force that the coil exerts on the magnet is equal and opposite to the force that the magnet exerts on the coil.

Here is the point: Lenz’s law is just a short-cut that allows you to determine the direction of the current in the closed coil without having to do all of the long analysis (which includes Newton’s third law) in the previous paragraph. Is this explanation that Lenz’s law is a special case of Newton’s third law convincing?

Now, if you imagine that the same experiment takes place in a strange world where Lenz’s law is different, it’s unclear whether this strange world has consistent laws of physics or not, as I mentioned in my previous comment. To clarify this point, think at the level of forces, and ask yourself whether the force law mentioned earlier is still valid, and if not, specify what the new force law is. Then, is Newton’s third law valid in this strange world? If so, then you can analyze the situation and decide what happens to the magnet and coil, whether they attract or repel under certain conditions, and so on. After all this, you will be able to derive your own version of Lenz’s law that is valid for the strange world. So even in this case, there will be a Lenz’s law that follows from Newton’s third law.

It would have been good if I had mentioned this explanation in the original post, as I think it would have been much clearer. I thank you very much for asking such excellent questions, and I will update the post to reflect our conversation.

Feel free to continue the discussion if you have further questions.

All the best!

• Guan Kheng Sze says:

Dear Professor Santo D’Agostino,
you wrote in your reply about the fictitious strange world where the Lenz’s Law that we are familiar with do not hold. You asked a few guiding questions after that, and I would like to answer them one at a time:
Q: it’s unclear whether this strange world has consistent laws of physics or not
A; Based on our earlier discussion, the Principle of Conservation of Energy will certainly be violated.

Q: ask yourself whether the force law mentioned earlier is still valid
A: In my opinion the force law in this strange world can still be the same as ours. We only need to modify Faraday’s Law.

Q: and if not, specify what the new force law is
A: My guess is probably superficial – to have the induced current attracting the apporaching magent, we just need to remove the minus sign from Faraday’s Law. This does not violate Newton’s Third Law.

You also wrote : “So even in this case, there will be a Lenz’s law that follows from Newton’s third law.” If in our fictitious world there is a NEW Lenz’s Law that follows from Newton’s Third Law, then doesn’t that support my view that the Lenz’s Law (our version) is not an example of Newton’s Third Law? Thank you.

Cheers,
Guan Kheng Sze

• Dear Guan Kheng Sze,

Q: it’s unclear whether this strange world has consistent laws of physics or not
A: Based on our earlier discussion, the Principle of Conservation of Energy will certainly be violated.

Q: ask yourself whether the force law mentioned earlier is still valid
A: In my opinion the force law in this strange world can still be the same as ours. We only need to modify Faraday’s Law.

Q: and if not, specify what the new force law is
A: My guess is probably superficial – to have the induced current attracting the apporaching magent, we just need to remove the minus sign from Faraday’s Law. This does not violate Newton’s Third Law.

My point is that you cannot arbitrarily change a law of physics without also changing others, because they are connected. Faraday’s law is not independent of the force law, so you can’t just change one without changing the other.

Another point is that the negative sign in Faraday’s law is Lenz’s law. (That is, the absolute value of the right side of Faraday’s law tells us the magnitude of the effect, and Lenz’s law (the negative sign) tells us the direction.) Removing the negative sign is equivalent to changing Lenz’s law. One possible aspect of your argument (I am not sure if you are actually saying this) is thus a tautology: If we change Faraday’s law by removing the negative sign, then we change Lenz’s law. But this is equivalent to saying, “If we change Lenz’s law, then we change Lenz’s law.” I am not sure this is part of your argument, but I point it out to illustrate the difficulties that may arise.

Let me try once more to clarify the main point, that Lenz’s law is an instance of Newton’s third law:

Suppose we do not know Lenz’s law, but know all of the rest of the relevant physics, and we are asked to analyze your thought experiment of the magnet tapped and moving with uniform speed towards a coil that is then closed. Here is a way to do this:

1. Use the force law F = Q v*B, or an equivalent way, to determine the force exerted by the moving magnet on the charged particles in the coil, which then indicates the direction of the induced current, and allows us to calculate the magnitude of the induced current.

2. By Newton’s third law of motion, the induced current then exerts an equal and opposite force on the magnet.

Given the fact of #1 (which is the way things work in our universe), then Lenz’s law is equivalent to #2.

Although it is possible to determine the direction of the induced current without using Newton’s third law (i.e., just using #1), the way Lenz’s law is phrased embodies #2.

You also wrote : “So even in this case, there will be a Lenz’s law that follows from Newton’s third law.” If in our fictitious world there is a NEW Lenz’s Law that follows from Newton’s Third Law, then doesn’t that support my view that the Lenz’s Law (our version) is not an example of Newton’s Third Law? Thank you.

My point here is that if you can imagine a strange universe in which #1 is different from our universe, but Newton’s third law is still valid, then there will be an analogue of Lenz’s law in the strange universe that embodies Newton’s third law. This emphasizes the close connection between Lenz’s law and Newton’s third law; of course if you change the force law, there will be consequent changes all over, as I mentioned previously.

All the best!

2. Guan Kheng Sze says:

Dear Professor Santo D’Agostino,
I certainly agree that physical laws are inter-connected. This is in fact the root of our discussion. You opined that Lenz’s Law is related to both Principle of Conservation of Energy and Newton’s Third Law, while I feel that it is only related to the former.
Yes, I am fully aware that the negative sign in Faraday’s Law is equivalent to Lenz’s Law. Removing it is akin to changing Lenz’s Law, which means I am just stating how Faraday’s Law will look like in the fictitious world I have proposed. I am just explaining how the fictitious universe can be ‘realized’ in response to your question (and if not, specify what the new force law is).
My replies to your arguments are embedded in yours below in brackets.
1. Use the force law F = Q v*B, or an equivalent way, to determine the force exerted by the moving magnet on the charged particles in the coil, which then indicates the direction of the induced current, and allows us to calculate the magnitude of the induced current. (When using F = Q v*B, I believe you are referring to the equivalent situation of the coil moving toward the magnet instead of the magnet approaching the coil. Some textbook authors, such as David Griffiths, refer to this situation as different from Electromagnetic Induction. But that is not important to our discussion and I will work along this new version. In our fictitious universe where Lenz’s Law does not hold, the force formula law will become F = – Q v*B so that induced current flows in opposite direction to what it should have.
2. By Newton’s third law of motion, the induced current then exerts an equal and opposite force on the magnet. (The repulsive force that you mention here should be the reaction of the repulsive force that the magnet exerts on the entire coil, which is not the same as the sum of the Q v*B force experienced by all the charged particles in the coil. But I understand what you are saying. However, since F = – Q v*B in the fictitious universe, the force exerted by the induced current on the magnet will be an attractive one.
Given the fact of #1 (which is the way things work in our universe), then Lenz’s law is equivalent to #2. (This is where the crux is. In the fictitious world, #1 is not true. That is F = – Q v*B instead. So Newton’s Third Law combined with F = -Qv*B will give you the opposite of the Lenz’s Law that we are familiar with. So you can’t say that the way Lenz’s law is phrased embodies #2.
I have presented a fictitious situation in which Lenz’s law can be violated (either removing the negative sign from Faraday’s Law or changing F = Q v*B to F = – Q v*B) without violating Newton’s Third Law. To prove that I am wrong to say that these two laws are not related, you need to show to me that this is impossible not matter how the force laws are modified.
I am thinking that there are three physical laws that we can play around with:
1. Newton’s Third Law
2. Lenz’s Law
3. F = Q v*B
Not sure if I am over-simplifying – it seems that we can negate any of the two above and the third one can be left intact. In my fictitious world, I negated 2 and 3, leaving #1 intact. Hence, #2 alone is not equivalent to #1.
(Sorry for the late reply. I was unfamiliar with the wordpress interface and when I was typing my reply earlier, I accidentally hit an arrow key or the backspace key and I lost all that I have typed then. A real disater…)

Thanks.

• Dear Guan Kheng Sze,

Sorry for the delay in replying, but it has been unusually busy lately.

I am thinking that there are three physical laws that we can play around with:
1. Newton’s Third Law
2. Lenz’s Law
3. F = Q v*B
Not sure if I am over-simplifying – it seems that we can negate any of the two above and the third one can be left intact. In my fictitious world, I negated 2 and 3, leaving #1 intact. Hence, #2 alone is not equivalent to #1.

As I mentioned in a previous comment, Lenz’s law is a short cut; that is, Lenz’s law is not a fundamental law of physics, because you can ignore it completely and still perform all calculations and understand all situations just as successfully using other laws of physics. It is convenient, however. In the case of your magnet and coil example, it is quite possible to analyze the situation completely without using Lenz’s law.

Therefore, it is not true that you can vary any two of the three laws in your list and leave the third one intact, as you suggest, because the laws are not independent.

I have presented a fictitious situation in which Lenz’s law can be violated (either removing the negative sign from Faraday’s Law or changing F = Q v*B to F = – Q v*B) without violating Newton’s Third Law. To prove that I am wrong to say that these two laws are not related, you need to show to me that this is impossible not matter how the force laws are modified.

Once again, my point is that Lenz’s law is not a fundamental law of physics, just a convenient short cut that is equivalent to some other laws of physics. You cannot arbitrarily change some of the laws of physics and then draw conclusions about Lenz’s law because Lenz’s law is not independent of the other laws.

Let me restate your argument: You create a fictitious world, propose that in this world the magnet and coil attract each other (consistent with Newton’s third law), and therefore Lenz’s law has nothing to do with Newton’s third law.

Here are my counter arguments:

1. Your fictitious world violates the principle of conservation of energy, and if we dig deeper we may find that other laws of physics are also different from our world. How do you know that your fictitious world is logically consistent? Because laws of physics are inter-related, you can’t just change one or two laws of physics and expect no other problems.

2. Lenz’s law is not independent of other laws of electrodynamics, so if you change Faraday’s law of induction or the force law, you must expect that Lenz’s law will also be changed. You say this proves your point that Lenz’s law is not equivalent to Newton’s third law. But I did not say that Lenz’s law is equivalent to Newton’s third law; I said Lenz’s law is an instance of Newton’s third law.

The magnet moves towards the coil, and in doing so exerts a force on the coil. By Newton’s third law the coil exerts an equal and opposite force on the magnet, opposing its motion. But this is what Lenz’s law says, that the current in the coil is in such a direction that the force it exerts on the magnet opposes its motion.

All the best!

• Guan Kheng Sze says:

 Dear Guan Kheng Sze,
Dear Professor Santo D’Agostino,
(As I mentioned in a previous comment, Lenz’s law is a short cut; that is, Lenz’s law is not a fundamental law of physics, because you can ignore it completely and still perform all calculations and understand all situations just as successfully using other laws of physics.)
I might be wrong on this but I am thinking one cannot speak in absolute terms that Lenz’s Law is not a fundamental law of physics but just a short-cut to help us determine the direction of the induced emf. Certainly you are right that given the other laws of electromagnetism, such as F= Bqv, one can still determine the direction of the induced emf without Lenz’s Law. However, is it not possible that we deduce that the force acting on a charge particle is F = Bqv from observing that the induced emf is always in a direction so as to oppose the change that is producing it? That is, we regard Lenz’s Law as fundamental (after all, it is consistent with the principle of conservation of energy) and the force law a natural outcome of it. I guess which view (whether Len’s Law is a fundamental law or otherwise) one takes depends on personal taste.
As I mentioned in a previous comment, Lenz’s law is a short cut; that is, Lenz’s law is not a fundamental law of physics, because you can ignore it completely and still perform all calculations and understand all situations just as successfully using other laws of physics. It is convenient, however. In the case of your magnet and coil example, it is quite possible to analyze the situation completely without using Lenz’s law.
(1. Your fictitious world violates the principle of conservation of energy, and if we dig deeper we may find that other laws of physics are also different from our world. How do you know that your fictitious world is logically consistent? Because laws of physics are inter-related, you can’t just change one or two laws of physics and expect no other problems.)
Yes, my fictitious world violates the principle of conservation of energy because I have stated right from the start that Lenz’s Law is related to principle of conservation of energy and if the former fails, the latter will fail too. I agree that changing one physical law may require modification of other physical laws for the entire system to remain consistent. If I change law 1 and law 2 is immediately affected (e.g. changing Lenz’s Law will immediately cause principle of conservation of energy to become invalid), then I will say law 1 and law 2 are related. If immediately clear that a certain law 3 is also affected (such as Newton’s third law in this case), then law 3 cannot be considered an instance of law 1. After all, if we argue that Lenz’s Law is an instance of Newton’s third law based solely on possible inconsistency of physical laws and not direct linkage, then there is no reason why one cannot say that Lenz’s Law is an instance of say, second law of thermodynamics, or Newton’s second law of motion.
(2. Lenz’s law is not independent of other laws of electrodynamics, so if you change Faraday’s law of induction or the force law, you must expect that Lenz’s law will also be changed. You say this proves your point that Lenz’s law is not equivalent to Newton’s third law. But I did not say that Lenz’s law is equivalent to Newton’s third law; I said Lenz’s law is an instance of Newton’s third law.)
Sorry, I was not very careful with my choice of words above. Let me restate my view:
(I have presented a fictitious situation in which Lenz’s law can be violated (either removing the negative sign from Faraday’s Law or changing F = Q v*B to F = – Q v*B) without violating Newton’s Third Law. To prove that I am wrong to say that these two laws are NOT related, you need to show to me that this (violating Len’s Law without violating Newton’s Third Law is impossible no matter how the force laws are modified.)

Thank you very much !

• Dear Guan Kheng Sze,

If you didn’t know the force law F = QvB, how would you use Lenz’s law to derive it?

After all, if we argue that Lenz’s Law is an instance of Newton’s third law based solely on possible inconsistency of physical laws and not direct linkage, then there is no reason why one cannot say that Lenz’s Law is an instance of say, second law of thermodynamics, or Newton’s second law of motion.

This is not what I am arguing. Your phrase “based solely on” suggests an argument that I am not making.

All the best,
SD

3. Athif says:

Hi there
My question is whether the force exerted by the coil on the magnet which is being pushed into the coil is equal to the force with which we push the magnet in? I have encountered physics problems involving conducting loops which are allowed to fall under gravity where there is a magnetic field perpendicular to the plane of its motion which achieve a terminal velocity. this is impossible if the force on them is always equal to its weigh please explain

4. Slevin says:

Real simple, Newton’s Law, for every action there is an opposite and equal reaction, The opposing field caused by generating current in a coil of wire is the reaction of trying to grab generated power in the first place.

The action is taking current off a coil of wire while a magnet is passing it, the reaction is the opposing field that fights the magnetic field that cause the generated current in the first place, real easy peasy. That’s how they are related. Use you noodle.

5. Newton’s third law of motion is simpel indeed, however, Grassman’s force law (= the Lorentz Maxwell force law in case of magnetostatic charge-current distribution) does not satisfy Newton’s 3d law. We are not even talking about Faraday’s law and Lenz law here, and the solution of this problem is rather unexpected and falls beyond the scope of the Maxwell theory.

I agree that Lenz law has something to do with Newton’s 3d law, however, the exact proof is missing, which is a much more exact treatment of Lenz law than the description of an induced counter magnetic flux that opposes the primary magnetic flux.

A varying current induces a varying magnetic field, which induces a rotational electric field (Faraday’s law), so we have an electric field force we are dealing with. As a consequence of this rotational electric field, a current is induced according to Ohm’s law. Since this current is also varying in time (that is what the word “induction” implies) then the induced current implies another (counter) magnetic field that is varying in time, which induces another (counter) rotational electric field that counters the change (in time) of the primary varying current.

Examining the Jefimenko expressions of a dynamic electric field and dynamic magnetic field, see
http://physicspages.com/2014/11/23/jefimenkos-equation-for-time-dependent-electric-field/
then the Faraday term is the last term in equation (18). It is not obvious how Newton’s third law follows from this term (multiplied with rho(r,t) to obtain the induced EMF), for instance, try to switch place vector r and r’, in order to derive the ‘counter’ force associated with Newton’s third law, then you will see how ‘asymmetric’ the Jefimenko electric field expression is, actually.

It seems that the exact expression of Lenz law is simply missing, and that is food for new scientific papers or at least extensive literature research.

6. The only electrodynamic force law that I know of, which satisfies Newton’s third law, is Weber’s force law (that describes the force between two moving charges with relative distance, relative speed and relative acceleration).