Energy transfer in electromagnetic induction

BruceW

Well, when the magnet is given a 'nudge', this will cause a change in the surrounding EM field which will propagate at c. (Which is due to the theory of retarded potentials in the Lorenz gauge). (And by 'propagate at c', I simply mean that after time t, the furthest point at which the EM field is affected by the nudge is at distance ct).

This is true whether or not the coil is there. And if the coil is there, it will take time x/c after the nudge, for the EM field at the coil to be affected. So it is at this time that the current can be induced in the coil, Then the propagation of this effect will again travel at c, so it will take another time of x/c for the magnet to experience the field which is due to currents which have been induced in the coil.

BruceW

I know I haven't immediately answered your questions, but I have not thought about this problem before, so I am trying to start with the principles that I am most certain should apply to this situation.

So, for the questions. Are they real waves? Um, I guess they satisfy the inhomogeneous wave equations. So yes? But then by this definition, every classical electromagnetic phenomena involves real waves. If we instead define a real wave as being oscillatory, then I guess generally the waves in this case would be partly real and partly not.

I would expect the accelerating magnet to emit EM energy even if the coil was not there. For example, if the magnet was made of a coil with current flowing through it, then when we nudge the magnet, we are accelerating charges, which generally gives off EM radiation.

Drakkith

I would guess that EM waves would be created by the acceleration, but I really don't know. I know accelerating a single charge will do this, but I don't know about a magnet. IF it emits EM waves then it must feel a small amount of "resistance" from the emission. This would be separate resistance from the resistance felt due to induction in the coil, and would only happen during the acceleration. Once it was moving steadily it would not emit EM radiation since it isn't accelerating. (That of course is IF it emits EM radiation in the first place)

entropy15

The energy transfer here should be taking place due to the induction and not through radiation (which even if present should be small and can be neglected as well).

The delay for the magnet to experience resistance due to the current in the coil seems to be
2x/c.

Indeed, If we follow the classical approach the delay seems to be 2x/c. But if we look at it from QED perspective it seems to be 0. (Wheeler Feynman absorber theory)

Can we throw more light on it looking at it from the relativity perspective?
After all magnetic field can be explained as an effect due to the "length contraction" involving moving charges using theory of relativity.

I have seen papers which explain the origin of magnetic fields using relativity. Most of them have to do with the forces of attraction/repulsion between two wires carrying current.
But I have not come across any which explain electromagnetic induction with the same approach.

I know it would be too complicated, but in case any of you have come across it, please help.

BruceW

Drakkith has got a good point. There are two different situations 1) considering a magnet which is being given a 'nudge' by some person. 2) considering a magnet which is moving toward the coil, with no 'outsider' forces.

I have mostly been talking about what happens when the magnet is given a 'nudge' i.e. acceleration. In this case, there is a delay before the effect is felt by the coil. I am pretty sure that Wheeler Feynman absorber theory also predicts that there is a time delay. But because I don't know much QFT, I can't say with complete certainty.

Now in the case when the magnet is moving freely (not being 'nudged'), then I am not so sure about whether there is a delay in the reaction force being felt, because in this case, at any particular time, there is already a magnetic field travelling between the magnet and the coil (which is due to the magnet moving at an earlier time).

Jano L.

It is just normal electromagnetic wave. When the magnet is nudged, the surface currents on it get nudged too and electromagnetic waves start to propagate out of the surface. In standard theory with retarded fields, there is no immediate effect of the coil on the magnet; this comes only later, after time 2x/c.

However, the magnet will feel small resistance immediately. This is due to the fact that one part of the surface current will act on another, an in case the magnet is accelerated, these forces do not cancel each other but tend to act against the acceleration. This is sort of a "self-inductance" effect.

Later, after time 2x/c, the wave from the coil will come and damp the magnet as well. This will be typically much greater effect (sort of "mutual inductance").

entropy15

Since the self inductance effect is very small lets consider only the mutual inductance.

Suppose that the magnet is initially at rest.
It is given a push at time t=t1. Lets assume that the kinetic energy transferred to the magnet due to this push is E_k.
This kinetic energy may decrease immediately due to radiation or self inductance but the effect is negligible.

Now at time t=t1+(x/c) we have the current induced in the coil and it has gained energy due to this induced current.

Now since the magnet does not experience any resistance till t1+(2x/c) i.e any considerable resistance we can see that the magnet does not lose the kinetic energy acquired by it during the initial push till t1+(2x/c).

So between t1+(x/c) and t1+(2x/c) we have this energy in the coil as well as in the kinetic energy of the magnet.

The kinetic energy of the magnet is the cause for the current to be induced in the coil.
The magnet just transfers its kinetic energy to the coil through mutual induction.
So should not the kinetic energy of the magnet decrease before the current is induced in the coil?

Jano L.

Not exactly. It is better to say that the electromagnetic forces due to the magnet drive the currents in the coil.

This is also not accurate enough. If we want to use the energy concept, we can, but we have to keep in mind all contributions to the energy. In this case, besides the magnet and the coil, there is also energy distributed throughout the remaining space. It is better to say that when the wave hits the coil, the field in the vicinity of the coil and inside it supply the energy for the animation of the currents.

It should not. Again, try to think of this in terms of the force - the response from the coil has not arrived yet, so there is no force acting on the magnet in the time interval x/c,2x/c.

You are probably worried about conservation of energy, but I think there is no problem. The currents in the coil get their energy from the field near the wires. Eventually, the magnet will lose some kinetic energy, but this is merely an after-effect.

entropy15

The current strength is proportional to the velocity of the magnet. The faster the magnet moves, more is the current.
The energy stored inside the magnetic field is constant. If the current gets energy from the field surrounding the coil, how can it obtain more and more energy from this field as the velocity of the magnet increases.

If the magnet is moving with a sufficiently high velocity the current induced in the coil may increase beyond what the surrounding magnetic field can provide.

entropy15

If we consider the Wheeler Feynman absorber theory the response from the coil arrives as advanced waves from the future to the instant when the magnet starts to move.
So this should get rid of the energy conservation problems.

In this case the resistance is felt immediately rather than 2x/c.

Jano L.

The energy of the field is not just magnetic; it has electric component as well. This has to be so, for pure magnetic field could not induce currents in the coil.

The energy of the field is the higher the higher is the velocity of the magnet. We can be sure that there is always enough energy present to maintain the energy conservation, because we have the Poynting theorem; this shows that the energy is conserved locally, and moves through space like fluid..

I agree it is difficult to imagine this in such complicated process, but there is an alternative way of description, via EM forces, which makes this much more clear. According to the Faraday law, the magnitude of the electric field due to magnet animating the currents is the higher the higher is the velocity of the magnet; the higher the electric field, the stronger currents get induced.

DaleSpam

When you have a moving magnetic field then you also induce an E field according to Faraday's law. This E field also contains energy. As you push the magnet faster your dB/dt is greater and therefore the induced E field is also greater. The extra energy you are worried about is in this induced E field.

I see the scenario as follows:
1) magnet and coil at rest wrt each other, no forces
2) force on magent to accelerate magnet (extra force required in order to generate E field)
3) EM wave propagates at c to coil
4) changing current induced in coil
5) changing field produced by coil
6) EM wave propagates at c to magnet
7) coil field opposes motion of magnet

I recommend against throwing unnecessary quantum concepts into any discussion which can be done purely classically. Generally it adds more confusion than understanding.

BruceW

No, the point is that when we give the magnet a 'nudge', we are putting energy into the electromagnetic field. So there is an immediate resistance, but not due to mutual inductance. (We will feel a resistance even if there is no coil). And if there is a coil, then we also get a delayed resistance due to mutual inductance.

Edit: So I guess I'm saying that if we only considered mutual inductance, then energy is not conserved when we give the magnet a nudge. (But this is because we are not considering the whole picture).

BruceW

I don't understand what you mean here.

entropy15

Hi Bruce,
Thanks for replying to my queries.

One more thing here, suppose that we replace the magnet with an electromagnet. The current in the electromagnet is controlled by a switch. Also we will remove the coil.(which was originally placed at a distance x)

Initially the electromagnet is not powered on and hence no magnetic field would be present.
Now we nudge this electromagnet so that it starts moving at a constant velocity.

Next the electromagnet is switched on so that it produces a magnetic field. Now since it is moving, it should try putting energy to the electromagnetic field.

So wont this oppose the current flowing in the electromagnet itself? trying to decrease the magnetic field.

entropy15

I was assuming that the energy in the electric field was constant, but as DaleSpam has pointed out the faster the magnet moves more the energy in the electric field, so this should not be a problem.

Drakkith

No, from the frame of the electromagnet it is stationary and it has no field built up prior. When you had the regular magnet, you also had a magnetic field built up which has its own frame of reference. When the magnet was bumped the two frames were no longer stationary with each other and the movement caused the field to oppose the acceleration and movement for a moment.