The Glade 4.0

Background:

In it's integral form, Faraday's Law states:



Lenz's Law just rephrases the implication of Gauss's Law: EMF induced by a change in magnetic flux will always act in opposition to the original change in magnetic flux. This is the principle that underlies an inductor (i.e. solenoid/coil). For instance, if you try to increase the current through an inductor, this has the effect of increasing the magnetic flux through the plane of the loops of the inductor. By Lenz's Law, the induced emf will try to reduce the magnetic flux in opposition to the increase. The induced emf will have the opposite sense of the externally applied emf across the inductor, partially cancelling it and thus reducing the current through the inductor. In other words, the inductor tries (within the limits of its inductance), to maintain the existing current flowing through it.

So:

The behavior of Lenz's Law smells an awful lot like inertia to me, but I'm not certain if there's a connection.

The photon is the mediating particle for the electromagnetic force. So if you have a magnetic field, what that entails (as I understand it) is a continuous exchange of virtual photons between electrons. My uneducated guess is that at any moment in time, the sum of the energies of all these particles must be equal (within the limits of the uncertainty principle) to the potential energy of the field. That is, E=(1/2)(B^2/μ).

Furthermore, E^2=p^2*c^2 + m^2*c^4. Photons have no rest mass, so m=0, but by the preceding equation, they still have momentum: E=pc. So if my assumption above is correct, there is a momentum associated with a magnetic (or electric) field. Intuitively, I would guess that altering the magnitude or direction of a magnetic field would be equivalent to altering the momentum of the virtual photons that mediate it.

Here's the question:

Is there a deeper connection between Lenz's Law and momentum? Might it just be a consequence of the classical notion of momentum as applied to photons, or is it merely similar, but mathematically unrelated behavior?

_________________
Sail forth! steer for the deep waters only!
Reckless, O soul, exploring, I with thee, and thou with me;
For we are bound where mariner has not yet dared to go,
And we will risk the ship, ourselves and all.

Uh...

Ask your professor?

_________________
"Hysterical children shrieking about right-wing anything need to go sit in the corner and be quiet while the adults are talking."

I am pretty certain Stathol is asking Corolinth, and with good cause.

_________________

Primarily, yeah, though I wouldn't be surprised if there are some other people around here with physics backgrounds.

I did already ask my professor, but he was a little pressed for time. Off the top of his head, he felt like it could be put into terms of conservation of energy, but couldn't be immediately sure whether it could also be put into terms of conservation of momentum.

_________________
Sail forth! steer for the deep waters only!
Reckless, O soul, exploring, I with thee, and thou with me;
For we are bound where mariner has not yet dared to go,
And we will risk the ship, ourselves and all.

I think this is what you're looking for Stathol. Short answer, yes, there is a momentum associated with the magnetic field.
http://en.wikipedia.org/wiki/Momentum#Electromagnetism

Have to reply later today.

_________________
Buckle your pants or they might fall down.

I know. that was more of a roundabout way of saying "I remember having done this at some point but damn.. it looks hard now!"

_________________
"Hysterical children shrieking about right-wing anything need to go sit in the corner and be quiet while the adults are talking."

No hurry. This is just an idle curiosity.

_________________
Sail forth! steer for the deep waters only!
Reckless, O soul, exploring, I with thee, and thou with me;
For we are bound where mariner has not yet dared to go,
And we will risk the ship, ourselves and all.

Although my dealings with physicists is hardly exhaustive, it is more extensive than most people's. It is my experience that physicists tend to regard Lenz's Law as more of a cute trick than a deep insight into the fabric of reality - that would be Faraday's Law. Your premise is accurate, that Lenz's Law could be thought of as inertia for fields and waves, and therefore imply a connection between momentum and photons. It is at this point that I have to ask what your background in physics is thus far, because what you mean by "deeper connection between Lenz's Law and momentum" could differ greatly depending on where you are. This time last year you were preparing for a Calc I final, which means you likely took Calc II in the Spring, and Calc III either in the Summer or this past semester. I am assuming you are currently wrapping up Physics II and are asking questions typical of a student performing significantly "above grade level," as it were, and not as a student pursuing formal training in quantum mechanics.

About ten years ago, I made a remark in a thread where Deeger, Taly, and Khross were arguing about various interpretations of quantum mechanics that they took some amount of umbrage to, which basically amounted to, "None of you know enough to be having this discussion." I was a younger, less civilized man back then, but the basic point is still sound, if somewhat undiplomatic. Quantum mechanics is even more bizarre than most of us think, for reasons the vast majority of us do not understand, and can not understand without more information.

It is difficult to explain adequately without a picture to show you and without my notes to refer to, unfortunately those notes are at home and not in my office. The electron has certain parts of its momentum-energy curve where it behaves as a particle, and others where it behaves as a photon. You have presented the relationships in your post. Particles have a parabolic relationship between momentum and energy, while photons have a linear relationship. The electron therefore has certain circumstances where it behaves as a particle, and others where it behaves as a photon. Another equally fascinating detail is that holes (the adequately named positive charge-carriers that represent spaces where the electrons aren't, but should be) have mass. In some cases, the hole has more mass than the electron that would go in it. These masses change based on what material you mean to transport electrons through, and I believe our favorite semiconductor, silicon, is one such material where the holes are heavier than the electrons that aren't in them. I would have to double-check. If not silicon, then I'm almost positive galium-arsenide is one.

The photon/particle duality of the electron may be the sort of connection between photons and momentum that you're looking for. It's something that physicists generally take for granted in their work. The photon actually does have momentum, it just has no mass, and therefore momentum serves as a substitute for mass in problems involving relativity. The Heisenberg Uncertainty Principle is also more frequently stated in quantum mechanics as the relationship between position and momentum, as opposed to position and velocity like laypersons are taught.

Back to Lenz's Law specifically, it would not be the first time an electromagnetic equation has had a mechanical analog. Voltage in an electrical network is a substitute for pressure in fluid mechanics, the word current was stolen directly from fluid mechanics, and most EEs understand fluid mechanics and plumbing as an electrical circuit where water replaces electricity. MEs are taught to model heat flow problems as a resistive electrical network, because they've had a DC circuits class long before they took thermal analysis. It is these corollaries across what appear to be radically different sciences that motivate our search for a Grand Unified Theory. Forget about the Higgs' Boson. Look at the formulas for gravitational, electrical, and magnetic forces. All three of those formulas are exactly the same template.

That statement of Lenz's Law that you've posted is missing a term compared to the form I'm most accustomed to seeing it in. Each coil of wire will produce that effect. Another important detail to remember is that Lenz's Law is most commonly encountered in an AC environment. This is important to mention because foundational science coursework focuses its attention on a DC environment - as well it should, since even people who specialize in AC use a model that is structured to resemble a DC system in order to make the math easier. 170Cos(x) is much harder to work with than 120.

Lenz's Law is simultaneously responsible for the transmission of power beyond about ten city blocks, as well as over 90% of the energy losses between your building and the local substation that provides its power. You probably did a few physics problems where you looked at a loop of wire and calculated a voltage due to changing the area of the loop (i.e. pulling it tighter) or by strengthening/weakening the magnetic field. It's the latter part that is actually happening, because the current that produces the magnetic field is oscillating.

Conducting materials under AC conditions induce a voltage in themselves that opposes any change to the amount of current they carry. This is due to Lenz's Law and it has the exact effect you would expect. A large inductor will block AC current. It carries DC just fine, but it doesn't like to change its current. It's exactly like getting a big truck moving and then trying to apply the brakes. This is very important in the design of electrical protective equipment on large scale systems. If you have a large inductive load, such as a big motor running at high current, and you suddenly interrupt the circuit, the large inductor is still carrying all that current, and it doesn't care that the circuit is now broken. It still requires a huge amount of energy to slow that current down to 0. That will build up a huge voltage (much higher than normal system voltage) at the place where you broke the circuit. The results can be very spectacular if the circuit interrupter isn't strong enough.

_________________
Buckle your pants or they might fall down.

This is one of my favorite places to ask questions of this nature. I visit daily --
https://www.physicsforums.com/

Fair warning, the folks there will often try to guide to you the right answer rather than give it outright.

_________________

Thanks for the response.


Pretty good guesses. I wrapped up Physics II yesterday. I'm pursuing a CS undergrad, so I've completed Calc II. Oddly enough, Calc III is not a requirement for my degree. Maybe one of these days I'll dredge up the motivation to learn it on my own, but today is not that day.

By "deeper connection", I suppose all I really mean is: could you derive Faraday's Law (and thus Lenz's Law) from applying classical, mechanical laws of momentum to the momentum associated with a magnetic field? Superficially, the behavior seems similar; if you push on a field, it pushes back. But I don't know anywhere near enough about quantum field theory, or the mathematics behind it, to tell whether there's an actual analogue here, or just a superficial similarity. If they are analogous, I don't expect to understand the whys and wherefores of it.


In other words, exactly this. I read QED about a decade ago, so I have at least some sense of what you're talking about regarding electron behavior and holes. Mostly, though, I just learned that a lot of very abstract math is required to "understand" quantum behavior. Scare quotes intentional. You can analyze it, of course, but It will never make "sense" because nothing that anyone has direct experience with compares. For instance, particle/wave duality is somewhat fictitious. There's nothing schizophrenic about electrons -- at all times they follow the same set of rules. It's just that the consequences of those rules are unlike anything macroscopic. When we squeeze it one way it reminds us of a particle and when we squeeze it the other way it reminds us of a wave, but it's really neither of these things.

This I happened to be aware of. In fact, it's what got me thinking along this line to begin with. I was only aware that fields have momentum because of quantum foam. A field with 0 magnitude would have an exact position and momentum, ergo that can't happen.

I was just giving the general form of Faraday's Law. For an inductor, yes, N loops = N surfaces of integration, so emf = -N*dΦ/dt as you say.


Heh. We had plenty of material on the transient response of R-L and R-C circuits, and on the resonance of L-C and L-R-C circuits, but inductive kickback was never specifically mentioned. The last question on our third exam (he always tosses one curveball at us) involved abruptly disconnecting a 12V battery from an inductor and getting an arc across the gap. In other words, he was testing if anyone could figure out that inductive kickback is a thing without being told about it. Our exams are take-home format. I worked out the problem one evening and got a call from a friend the next morning. His car had broken down and he wanted me to take a look at it. The problem turned out to be that he had blown his ignition coil. Life is funny that way.

_________________
Sail forth! steer for the deep waters only!
Reckless, O soul, exploring, I with thee, and thou with me;
For we are bound where mariner has not yet dared to go,
And we will risk the ship, ourselves and all.